The Teetering Towers of Abstraction

Abstraction is an abstraction. You can’t touch it or taste it or photograph it. You can barely talk about it without resorting to metaphors and analogies. Yet this ghostly concept is an essential tool in both mathematics and computer science. Oddly, it seems to inspire quite different feelings and responses in those two fields. I’ve been wondering why.

In mathematics abstraction serves as a kind of stairway to heaven—as well as a test of stamina for those who want to get there. West stairs to Grand View Park, San Francisco, October 2017. You begin the climb at an early age, at ground level, with things that are not at all abstract. Jelly beans, for example. You learn the important life lesson that if you have and you eat , you will have only left. After absorbing this bitter truth, you are invited to climb the stairs of ab­straction as far as the first landing, where you replace the tasty tangible jelly beans with sugar-free symbols: $$5 - 3 = 2$$.

Some years later you reach higher ground. The sym­bols represent­ing par­tic­ular numbers give way to the $$x$$s and $$y$$s that stand for quantities yet to be determined. They are symbols for sym­bols. Later still you come to realize that this algebra business is not just about “solving for $$x$$,” for finding a specific number that corresponds to a specific letter. It’s a magical device that allows you to make blanket statements encompassing all numbers: $$x^2 - 1 = (x + 1)(x - 1)$$ is true for any value of $$x$$.

Continuing onward and upward, you learn to manipulate symbolic expressions in various other ways, such as differentiating and integrating them, or constructing functions of functions of functions. Keep climbing the stairs and eventually you’ll be introduced to areas of mathematics that openly boast of their abstractness. There’s abstract algebra, where you build your own collections of numberlike things: groups, fields, rings, vector spaces. Cartoon by Ben Orlin, mathwithbaddrawings.com, reprinted under Creative Commons license.Another route up the stairway takes you to category theory, where you’ll find a collection of ideas with the disarming label ab­stract nonsense.

Not everyone is filled with admiration for this Jenga tower of abstrac­tions teetering atop more abstrac­tions. Con­sider Andrew Wiles’s proof of Fermat’s last theorem, and its reception by the public. The theorem, first stated by Pierre de Fermat in the 1630s, makes a simple claim about powers of integers: If $$x, y, z, n$$ are all integers greater than $$0$$, then $$x^n + y^n = z^n$$ has solutions only if $$n \le 2$$. The proof of this claim, published in the 1990s, is not nearly so simple. Wiles (with contributions from Richard Taylor) went on a scavenger hunt through much of modern mathematics, collecting a truckload of tools and spare parts needed to make the proof work: elliptic curves, modular forms, Galois groups, functions on the complex plane, L-series. It is truly a tour de force.

Diagram (borrowed from Kenneth A. Ribet and Brian Hayes, “Fermat’s Last Theorem and Modern Arithmetic“) outlines the overall strategy of the Wiles proof. If you had a counterexample to FLT, you could construct an elliptic curve E with certain properties. But the properties deduced on the left and right branches of the diagram turn out to be inconsistent, implying that E does not exist, nor does the counter­example that gave rise to it.

Is all that heavy machinery really needed to prove such an innocent-looking state­ment? Many people yearn for a simpler and more direct proof, ideally based on methods that would have been available to Fermat himself. Ken Ribet will be presenting “A 2020 View of Fermat’s Last Theorem” at the Joint Mathematics Meetings later this week. In a preview of the talk, he notes that advances made since 1994 allow a more succinct statement of the proof. But those recent advances are no easier to understand than the original proof.At least nine attempts to construct an elementary proof have been posted on the arXiv in the past 20 years, and there are lots more elsewhere. I think the sentiment motivating much of this work is, “You shouldn’t be allowed to prove a theorem I care about with methods I don’t understand.” Marilyn vos Savant, the Parade columnist, takes an even more extreme position, arguing that Wiles strayed so far from the subject matter of the theorem as to make his proof invalid. (For a critique of her critique, see Boston and Granville.)

Almost all of this grumbling about illegimate methods and excess complexity comes from outside the community of research mathematicians. Insiders see the Wiles proof differently. For them, the wide-ranging nature of the proof is actually what’s most important. The main accomp­lishment, in this view, was cementing a connection between those far-flung areas of mathematics; resolving FLT was just a bonus.

Yet even mathematicians can have misgivings about the intricacy of math­ematical arguments and the ever-taller skyscrapers of abstraction. Jeremy Gray, a historian of mathematics, believes anxiety over abstraction was already rising in the 19th century, when mathematics seemed to be “moving away from reality, into worlds of arbitrary dimension, for example, and into the habit of supplanting intuitive concepts (curves that touch, neighboring points, velocity) with an opaque language of mathematical analysis that bought rigor at a high cost in intelligibility.”

Quite apart from these comments on abstraction, the thesis is well worth reading. It offers alternating sections of “mathsplaining” and “laysplaining.” See also a review in MAA Focus by Adriana Salerno. The thesis was to be published in book form last fall by Birkhäuser, but the book doesn’t seem to be available yet.For a view of abstraction in contemporary mathematics, we have a vivid image from Piper Harron, a young mathematician who wrote an extraordinarily candid PhD thesis in 2016. The introductory chapter begins, “The hardest part about math is the level of abstraction required.” She goes on to explain:

I like to imagine abstraction (abstractly ha ha ha) as pulling the strings on a marionette. The marionette, being “real life,” is easily accessible. Everyone understands the marionette whether it’s walking or dancing or fighting. We can see it and it makes sense. But watch instead the hands of the puppeteers. Can you look at the hand movements of the puppeteers and know what the marionette is doing?… Imagine it gets worse. Much, much worse. Imagine that the marionettes we see are controlled by marionettoids we don’t see which are in turn controlled by pre-puppeteers which are finally controlled by actual puppeteers.

Keep all those puppetoids in mind. I’ll be coming back to them, but first I want to shift my attention to computer science, where the towers of abstraction are just as tall and teetery, but somehow less scary.

Suppose your computer is about to add two numbers…. No, wait, there’s no need to suppose or imagine. In the orange panel below, type some numbers into the $$a$$ and $$b$$ boxes, then press the “+” button to get the sum in box $$c$$. Now, please describe what’s happening inside the machine as that computation is performed.

a

b
c

You can probably guess that somewhere behind the curtains there’s a fragment of code that looks like c = a + b. And, indeed, that statement appears verbatim in the JavaScript program that’s triggered when you click on the plus button. But if you were to go poking around among the circuit boards under the keyboard of your laptop, you wouldn’t find anything resembling that sequence of symbols. The program statement is a high-level abstraction. If you really want to know what’s going on inside the computing engine, you need to dig deeper—down to something as tangible as a jelly bean.

How about an electron? In truth, electrons are not so tangible. The proper mental image is not a hard sphere like a BB but a diffuse probability distribution. In other words, the electron itself is an abstraction.During the computation, clouds of electrons drift through the machine’s circuitry, like swarms of migrating butterflies. Their movements are regulated by the switching action of transistors, and the transistors in turn are controlled by the moving electrons. It is this dance of the electrons that does the arithmetic and produces an answer. Yet it would be madness to describe the evaluation of c = a + b by tracing the motions of all the electrons (perhaps $$10^{23}$$ of them) through all the transistors (perhaps $$10^{11}$$).

To understand how electrons are persuaded to do arithmetic for us, we need to introduce a whole sequence of abstractions.

• First, step back from the focus on individual electrons, and reformulate the problem in terms of continuous quantities: voltage, current, capacitance, inductance.
• Replace the physical transistors, in which voltages and currents change smoothly, with idealized devices that instantly switch from totally off to fully on.
• Interpret the two states of a transistor as logical values (true and false) or as numerical values ($$1$$ and $$0$$).
• Organize groups of transistors into “gates” that carry out basic functions of Boolean logic, such as and, or, and not.
• Assemble the gates into larger functional units, including adders, multipliers, comparators, and other components for doing base-$$2$$ arithmetic.
• Build higher-level modules that allow the adders and such to be operated under the control of a program. This is the conceptual level of the instruction-set architecture, defining the basic operation codes (add, shift, jump, etc.) recognized by the computer hardware.
• Graduating from hardware to software, design an operating system, a collection of services and interfaces for abstract objects such as files, input and output channels, and concurrent processes.
• Create a compiler or interpreter that knows how to translate programming language statements such as c = a + b into sequences of machine instructions and operating-system requests.

From the point of view of most programmers, the abstractions listed above represent computational infrastructure: They lie beneath the level where you do most of your thinking—the level where you describe the algorithms and data structures that solve your problem. But computational abstractions are also a tool for building superstructure, for creating new functions beyond what the operating system and the programming language provide. For example, if your programming language handles only numbers drawn from the real number line, you can write procedures for doing arithmetic with complex numbers, such as $$3 + 5i$$. (Go ahead, try it in the orange box above.) And, in analogy with the mathematical practice of defining functions of functions, we can build compiler compilers and schemes for metaprogramming—programs that act on other programs.

In both mathematics and computation, rising through the various levels of abstraction gives you a more elevated view of the landscape, with wider scope but less detail. Even if the process is essentially the same in the two fields, however, it doesn’t feel that way, at least to me. In mathematics, abstraction can be a source of anxiety; in computing, it is nothing to be afraid of. In math, you must take care not to tangle the puppet strings; in computing, abstractions are a defense against such confusion. For the mathematician, abstraction is an intellectual challenge; for the programmer, it is an aid to clear thinking.

Why the difference? How can abstraction have such a friendly face in computation and such a stern mien in math? One possible answer is that computation is just plain easier than mathematics. In speaking of “computation,” what I have in mind is the design of algorithms and data structures suitable for a machine we can build out of material components. If you are playing with Turing machines and other toys of theoretical computer science, the game is altogether different. But in my view theoretical computer science is just a funny-looking branch of mathematics. (With apologies to those of my friends who grimace to hear me say it.) Anything that fits into the computer is necessarily discrete and finite. In principle, any computer program could be reduced to a big table mapping all possible inputs to the corresponding outputs. Mathematics is invulnerable to this kind of trivialization by brute force. It has infinities hiding under the bed and lurking behind the closet door, and that’s what makes it both fun and frightening.

Another possible explanation is that computer systems are engineered artifacts; we can build them to our own specifications. If a concept is just too hairy for the human mind to master, we can break it down into simpler pieces. Math is not so complaisant—not even for those who hold that mathematical objects are invented rather than discovered. We can’t just design number theory so that the Riemann hypothesis will be true.

But I think the crucial distinction between math abstractions and computer abstractions lies elsewhere. It’s not in the abstractions themselves but in the boundaries between them.

Warning from the abstraction police on the office door of Radhika Nagpal, Harvard University. (Photographed November 2013.)

I believe I first encountered the term abstraction barrier in Abelson and Sussman’s Structure and Inter­pretation of Computer Programs, circa 1986. The underlying idea is surely older; it’s implicit in the “structured programming” literature of the 1960s and 70s. But SICP still offers the clearest and most compelling introduction.In building computer systems, we are urged to compartmentalize, to create self-contained and sealed-off modules—black boxes whose inner workings are concealed from outside observers. In this world, information hiding is considered a virtue, not an impeachable offense. If a design has a layered structure, with abstractions piled one atop the other, the layers are separated by abstraction barriers. A high-level module can reach across the barrier to make use of procedures from lower levels, but it won’t know anything about the implementation of those procedures. When you are writing programs in Lisp or Python, you shouldn’t need to think about how the operating system carries out its chores; and when you’re writing routines for the operating system, you needn’t think about the physics of electrons meandering through the crystal lattice of a semiconductor. Each level of the hierarchy can be treated (almost) independently.

Mathematics also has its abstraction barriers, although I’ve never actually heard the term used by mathematicians. A notable example comes from Giuseppe Peano’s formulation of the foundations of arithmetic, circa 1900. Peano posits the existence of a number $$0$$, and a function called successor, $$S(n)$$, which takes a number $$n$$ and returns the next number in the counting sequence. Thus the natural numbers begin $$0, S(0), S(S(0)), S(S(S(0)))$$, and so on. Peano deliberately refrains from saying anything more about what these numbers look like or how they work. They might be implemented as sets, with $$0$$ being the empty set and successor the operation of adjoining an element to a set. Or they could be unary lists: (), (|), (||), (|||), . . . The most direct approach is to use Church numerals, in which the successor function itself serves as a counting token, and the number $$n$$ is represented by $$n$$ nested applications of $$S$$.

From these minimalist axioms we can define the rest of arithmetic, starting with addition. In calculating $$a + b$$, if $$b$$ happens to be $$0$$, the problem is solved: $$a + 0 = a$$. If $$b$$ is not $$0$$, then it must be the successor of some number, which we can call $$c$$. Then $$a + S(c) = S(a + c)$$. Notice that this definition doesn’t depend in any way on how the number $$0$$ and the successor function are represented or implemented. Under the hood, we might be working with sets or lists or abacus beads; it makes no difference. An abstraction barrier separates the levels. From addition you can go on to define multiplication, and then exponentiation, and again abstraction barriers protect you from the lower-level details. There’s never any need to think about how the successor function works, just as the computer programmer doesn’t think about the flow of electrons.

The importance of not thinking was stated eloquently by Alfred North Whitehead, more than a century ago:

Alfred North Whitehead, An Introduction of Mathematics, 1911, pp. 45–46.It is a profoundly erroneous truism, repeated by all copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle—they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

If all of mathematics were like the Peano axioms, we would have a watertight structure, compartmentalized by lots of leakproof abstraction barriers. And abstraction would probably not be considered “the hardest part about math.” But, of course, Peano described only the tiniest corner of mathematics. We also have the puppet strings.

In Piper Harron’s unsettling vision, the puppeteers high above the stage pull strings that control the pre-puppeteers, who in turn operate the marionettoids, who animate the marionettes. Each of these agents can be taken as representing a level of abstraction. The problem is, we want to follow the action at both the top and the bottom of the hierarchy, and possibly at the middle levels as well. The commands coming down from the puppeteers on high embody the abstract ideas that are needed to build theorems and proofs, but the propositions to be proved lie at the level of the marionettes. There’s no separating these levels; the puppet strings tie them together.

In the case of Fermat’s Last Theorem, you might choose to view the Wiles proof as nothing more than an elevated statement about elliptic curves and modular forms, but the proof is famous for something else—for what it tells us about the elementary equation $$x^n + y^n = z^n$$. Thus the master puppeteers work at the level of algebraic geometry, but our eyes are on the dancing marionettes of simple number theory. What I’m suggesting, in other words, is that abstraction barriers in mathematics sometimes fail because events on both sides of the barrier make simultaneous claims on our interest.

In computer science, the programmer can ignore the trajectories of the electrons because those details really are of no consequence. Indeed, the electronic guts of the computing machinery could be ripped out and replaced by fluidic devices or fiber optics or hamsters in exercise wheels, and that brain transplant would have no effect on the outcome of the computation. Few areas of mathematics can be so cleanly floated away and rebuilt on a new foundation.

Can this notion of leaky abstraction barriers actually explain why higher mathematics looks so intimidating to most of the human population? It’s surely not the whole story, but maybe it has a role.

In closing I would like to point out an analogy with a few other areas of science, where problems that cross abstraction barriers seem to be particularly difficult. Physics, for example, deals with a vast range of spatial scales. At one end of the spectrum are the quarks and leptons, which rattle around comfortably inside a particle with a radius of $$10^{-15}$$ meter; at the other end are galaxy clusters spanning $$10^{24}$$ meters. In most cases, effective abstraction barriers separate these levels. When you’re studying celestial mechanics, you don’t have to think about the atomic composition of the planets. Conversely, if you are looking at the interactions of elementary particles, you are allowed to assume they will behave the same way anywhere in the universe. But there are a few areas where the barriers break down. For example, near a critical point where liquid and gas phases merge into an undifferentiated fluid, forces at all scales from molecular to macroscopic become equally important. Turbulent flow is similar, with whirls upon whirls upon whirls. It’s not a coincidence that critical phenomena and turbulence are notoriously difficult to describe.

Biology also covers a wide swath of territory, from molecules and single cells to whole organisms and ecosystems on a planetary scale. Again, abstraction barriers usually allow the biologist to focus on one realm at a time. To understand a predator-prey system you don’t need to know about the structure of cytochrome c. But the barriers don’t always hold. Evolution spans all these levels. It depends on molecular events (mutations in DNA), and determines the shape and fate of the entire tree of life. We can’t fully grasp what’s going on in the biosphere without keeping all these levels in mind at once.

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My God, It’s Full of Dots!

Please click or tap in the gray square below. It will fill with a jolly tableau of colored disks—first big blue ones, then somewhat smaller purple ones, and eventually lots of tiny red dots.

Sorry. My program and your browser are not getting along. None of the interactive elements of this page will work. Could you try a different browser? Current versions of Chrome, Firefox, and Safari seem to work.

The disks are scattered randomly, except that no disk is allowed to overlap another disk or extend beyond the boundary of the square. Once a disk has been placed, it never moves, so each later disk has to find a home somewhere in the nooks and crannies between the earlier arrivals. Can this go on forever?

The search for a vacant spot would seem to grow harder as the square gets more crowded, so you might expect the process to get stuck at some point, with no open site large enough to fit the next disk. On the other hand, because the disks get progressively smaller, later ones can squeeze into tighter quarters. In the specific filling protocol shown here, these two trends are in perfect balance. The process of adding disks, one after another, never seems to stall. Yet as the number of disks goes to infinity, they completely fill the box provided for them. There’s a place for every last dot, but there’s no blank space left over.

Or at least that’s the mathematical ideal. The computer program that fills the square above never attains this condition of perfect plenitude. It shuts down after placing just 5,000 disks, which cover about 94 percent of the square’s area. This early exit is a concession to the limits of computer precision and human patience, but we can still dream of how it would work in a world without such tiresome constraints.

This scheme for filling space with randomly placed objects is the invention of John Shier, a physicist who worked for many years in the semiconductor industry and who has also taught at Normandale Community College near Minneapolis. He explains the method and the mathematics behind it in a recent book, Fractalize That! A Visual Essay on Statistical Geometry. (For bibliographic details see the links and references at the end of this essay.) I learned of Shier’s work from my friend Barry Cipra.

Shier hints at the strangeness of these doings by imagining a set of 100 round tiles in graduated sizes, with a total area approaching one square meter. He would give the tiles to a craftsman with these instructions:

“Mark off an area of one square meter, either a circle or a square. Start with the largest tile, and attach it permanently anywhere you wish in the marked-off area. Continue to attach the tiles anywhere you wish, proceeding always from larger to smaller. There will always be a place for every tile regardless of how you choose to place them.” How many experienced tile setters would believe this?

Shier’s own creations go way beyond squares and circles filled with simple shapes such as disks. He has shown that the algorithm also works with an assortment of more elaborate designs, including nonconvex figures and even objects composed of multiple disconnected pieces. We get snow­flakes, nested rings, stars, butter­flies, fish eating lesser fish, faces, letters of the alphabet, and visual salads bringing together multiple ingredients. Shier’s interest in these patterns is aesthetic as well as mathematical, and several of his works have appeared in art exhibits; one of them won a best-of-show award at the 2017 Joint Mathematics Meeting.

Shier and his colleagues have also shown that the algorithm can be made to work in three-dimensional space. The book’s cover is adorned with a jumble of randomly placed toruses filling the volume of a transparent cube. If you look closely, you’ll notice that some of the rings are linked; they cannot be disentangled without breaking at least one ring. (The 3D illustration was created by Paul Bourke, who has more examples online, including 3D-printed models.)

After reading Shier’s account of his adventures, and admiring the pictures, I had to try it for myself. The experiments I’m presenting in this essay have no high artistic ambitions. I stick with plain-vanilla circular disks in a square frame, all rendered with the same banal blue-to-red color scheme. My motive is merely to satisfy my curiosity—or perhaps to overcome my skepticism. When I first read the details of how these graphics are created, I couldn’t quite believe it would work. Writing my own programs and seeing them in action has helped persuade me. So has a proof by Christopher Ennis, which I’ll return to below.

Filling a region of the plane with disks is not in itself such a remarkable trick. One well-known way of doing it goes by the name Apollonian circles. Start with three disks that are all tangent to one another, leaving a spiky three-pointed vacancy between them. Draw a new disk in the empty patch, tangent to all three of the original disks; this is the largest disk that can possibly fit in the space. Adding the new disk creates three smaller triangular voids, where you can draw three more triply tangent disks. There’s nothing to stop you from going on in this way indefinitely, approaching a limiting configuration where the entire area is filled.

There are randomized versions of the Apollonian model. For example, you might place zero-diameter seed disks at random unoccupied positions and then allow them to grow until they touch one (or more) of their neighbors. This process, too, is space-filling in the limit. And it can never fail: Because the disks are custom-fitted to the space available, you can never get stuck with a disk that can’t find a home.

Shier’s algorithm is different. You are given disks one at a time in a predetermined order, starting with the largest, then the second-largest, and so on. To place a disk in the square, you choose a point at random and test to see if the disk will fit at that location without bumping into its neighbors or poking beyond the boundaries of the square. If the tests fail, you pick another random point and try again. It’s not obvious that this haphazard search will always succeed—and indeed it works only if the successive disks get smaller according to a specific mathematical rule. But if you follow that rule, you can keep adding disks forever. Furthermore, as the number of disks goes to infinity, the fraction of the area covered approaches $$1$$. It’s convenient to have a name for series of disks that meet these two criteria; I have taken to calling them fulfilling series.

In exploring these ideas computationally, it makes sense to start with the simplest case: disks that are all the same size. This version of the process clearly cannot be fulfilling. No matter how the disks are arranged, their aggregate area will eventually exceed that of any finite container. Click in the gray square below to start filling it with equal-size disks. The square box has area $$A_{\square} = 4$$. The slider in the control panel determines the area of the individual disks $$A_k$$, in a range from $$0.0001$$ to $$1.0$$.

Program 1: Fixed-Size Disks

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If you play with this program for a while, you’ll find that the dots bloom quickly at first, but the process invariably slows down and eventually ends in a state labeled “Jammed,” indicating that the program has been unable The program gives up after trying 10 million random locations. to find a place where one more disk will fit. If you move the slider to the right, specifying larger disks, this impasse is reached very quickly, sometimes after placing just one or two disks. If you select very small disks, the program may churn away for five or ten minutes and fill the square with more than 20,000 disks before running out of options. Nevertheless, for any disk size greater than zero, a jammed outcome is inescapable.

The densest possible packing of equal-size disks places the centers on a triangular lattice with spacing equal to the disk diameter. The resulting density (for an infinite number of disks on an infinite plane) is $$\pi \sqrt{3}\, /\, 6 \approx 0.9069$$, which means more than 90 percent of the area is covered. A random filling in a finite square is much looser. My first few trials all halted with a filling fraction fairly close to one-half, and so I wondered if that nice round number might be the expectation value of the probabilistic process. Further experiments suggested otherwise. Over a broad range of disk sizes, from $$0.0001$$ up to about $$0.01$$, the area covered varied from one run to the next, but the average was definitely above one-half—perhaps $$0.54$$. After some rummaging through the voluminous literature on circle packing, I think I may have a clue to the exact expectation value: $$\pi / (3 + 2 \sqrt{2}) \approx 0.539012$$. Where does that weird number come from? The answer has nothing to do with Shier’s algorithm, but I think it’s worth a digression.

Consider an adversarial process: Alice is filling a unit square with $$n$$ equal-size disks and wants to cover as much of the area as possible. Bob, who wants to minimize the area covered, gets to choose $$n$$. If Bob chooses $$n = 1$$, Alice can produce a single disk that just fits inside the square and covers about $$79$$ percent of the space. Can Bob do better? Yes, if Bob specifies $$n = 2$$, Alice’s best option is to squeeze the two disks into diagonally opposite corners of the square as shown in the diagram at right. These disks are bounded by right isosceles triangles, which makes it easy to calculate their radii as $$r = 1 / (2 + \sqrt{2}) \approx 0.2929$$. Their combined area works out to that peculiar number $$\pi / (3 + 2 \sqrt{2}) \approx 0.54$$.

If two disks are better than one (from Bob’s point of view), could three be better still? Or four, or some larger number? Apparently not. In 2010, Erik Demaine, Sándor Fekete and Robert Lang conjectured that the two-disk configuration shown above represents the worst case for any number of equal-size disks. In 2017 Fekete, Sebastian Morr, and Christian Scheffer proved this result.

Is it just a coincidence that the worst-case density for packing disks into a square also appears to be the expected density when equal-size disks are placed randomly until no more will fit? Wish I knew.

Let us return to the questions raised in Shier’s Fractalize That! If we want to fit infinitely many disks into a finite square, our only hope is to work with disks that get smaller and smaller as the process goes on. The disk areas must come from some sequence of ever-diminishing numbers. Among such sequences, the one that first comes to mind is $$\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$$ These fractions have been known since antiquity as the harmonic numbers. (They are the wavelengths of the overtones of a plucked string.)

To see what happens when successive disks are sized according to the harmonic sequence, click in the square below.

Program 2: Disk Sizes from Harmonic Series

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Again, the process halts when no open space is large enough to accommodate the next disk in the sequence. If you move the slider all the way to the right, you’ll see a sequence of disks with areas drawn from the start of the full harmonic sequence, $$\frac{1}{1} , \frac{1}{2}, \frac{1}{3}, \dots$$; at this setting, you’ll seldom get beyond eight or nine disks. Moving the slider to the left omits the largest disks at the beginning of the sequence, leaving the infinite tail of smaller disks. For example, setting the slider to $$1/20$$ skips all the disks from $$\frac{1}{1}$$ through $$\frac{1}{19}$$ and begins filling the square with disks of area $$\frac{1}{20}, \frac{1}{21}, \frac{1}{22}, \dots$$ Such truncated series go on longer, but eventually they also end in a jammed configuration.

The slider goes no further than 1/50, but even if you omitted the first 500 disks, or the first 5 million, the result would be the same. This is a consequence of the most famous property of the harmonic numbers: Although the individual terms $$1/k$$ dwindle away to zero as $$k$$ goes to infinity, the sum of all the terms,

$\sum_{k = 1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots,$

does not converge to a finite value. As long as you keep adding terms, the sum will keep growing, though ever more slowly. This curious fact was proved in the 14th century by the French bishop and scholar Nicole Oresme. The proof is simple but ingenious. Oresme pointed out that the harmonic sequence

$\frac{1}{1} + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \cdots$

is greater than

$\frac{1}{1} + \frac{1}{2} + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \cdots$

The latter series is equivalent to $$1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} \cdots$$, and so it is clearly divergent. Since the grouped terms of the harmonic series are even greater, they too must exceed any finite bound.

The divergence of the harmonic series implies that disks whose areas are generated by the series will eventually overflow any enclosing container. Dropping a finite prefix of the sequence, such as the first 50 disks, does not change this fact.

Let me note in passing that just as the filling fraction for fixed-size disks seems to converge to a specific constant, 0.5390, disks in harmonic series also seem to have a favored filling fraction, roughly 0.71. Can this be explained by some simple geometric argument? Again, I wish I knew.

Evidently we need to make the disks shrink faster than the harmonic numbers do. Here’s an idea: Square each element of the harmonic series, yielding this:

$\sum_{k = 1}^{\infty}\frac{1}{k^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots.$

Click below (or press the Start button) to see how this one turns out, again in a square of area 4.

Program 3: Disk Sizes from Zeta(2)

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At last we have a process that won’t get stuck in a situation where there’s no place to put another disk. It could run forever, but of course it doesn’t. It quits when the area of the next disk shrinks down to about a tenth of the size of a single pixel on a computer display. The stopped state is labeled “Exhausted” rather than “Jammed.”This is an algorithm that could truly run forever. And yet the result is still not quite what we were hoping for—it’s not fulfilling. The disks are scattered sparsely in the square, leaving vast open spaces unoccupied. The configuration reminds me of deep-sky images made by large telescopes.

Why does this outcome look so different from the others? Unlike the harmonic numbers, the infinite series $$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots$$ converges to a finite sum. In the 18th century the task of establishing this fact (and determining the exact sum) was known as the Basel Problem, after the hometown of the Bernoulli family, who put much effort into the problem but never solved it. The answer came in 1735 from Leonhard Euler (another native of Basel, though he was working in St. Petersburg), who showed that the sum is equal to $$\pi^2 / 6$$. This works out to about $$1.645$$; since the area of the square we want to fill is $$4$$, even an infinite series of disks would cover only about $$41$$ percent of the territory.

Given that the numbers $$\frac{1}{1^1}, \frac{1}{2^1}, \frac{1}{3^1}, \dots$$ diminish too slowly, whereas $$\frac{1}{1^2}, \frac{1}{2^2}, \frac{1}{3^2}, \dots$$ shrink too fast, it makes sense to try an exponent somewhere between $$1$$ and $$2$$ in the hope of finding a Goldilocks solution. The computation performed below in Program 4 is meant to facilitate the search for such a happy medium. Here the disk sizes are elements of the sequence $$\frac{1}{1^s}, \frac{1}{2^s}, \frac{1}{3^s}, \dots$$, where the value of the exponent $$s$$ is determined by the setting of the slider, with a range of $$1 \lt s \le 2$$. We already know what happens at the extremes of this range. What is the behavior in the middle?

Program 4: Disk Sizes from Zeta(s)

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If you try the default setting of $$s = 1.5$$, you’ll find you are still in the regime where the disks dwindle away so quickly that the box never fills up; if you’re willing to wait long enough, the program will end in an exhausted state rather than a jammed one. Reducing the exponent to $$s = 1.25$$ puts you on the other side of the balance point, where the disks remain too large and at some point one of them will not fit into any available space. By continuing to shuttle the slider back and forth, you could carry out a binary search, closing in, step by step, on the “just right” value of $$s$$. This strategy can succeed, but it’s not quick. As you get closer to the critical value, the program will run longer and longer before halting. (After all, running forever is the behavior we’re seeking.) To save you some tedium, I offer a spoiler: the optimum setting is between 1.29 and 1.30.

At this point we have wandered into deeper mathematical waters. A rule of the form $$A_k = 1/k^s$$ is called a power law, since each $$k$$ is raised to the same power. And series of the form $$\sum 1/k^s$$ are known as zeta functions, denoted $$\zeta(s)$$. Zeta functions have quite a storied place in mathematics. The harmonic numbers correspond to $$\zeta(1) = \sum 1/k^1$$, which does not converge. If $$\zeta(1)$$ grows without limit whereas $$\zeta(2)$$ converges, you might well wonder where the boundary lies between these two behaviors. As it happens, $$\zeta(1 + \epsilon)$$ converges for any $$\epsilon \gt 0$$.As mentioned above, Euler found that $$\zeta(2) = \pi^2 / 6$$, and he went on to show that $$\zeta(s)$$ also converges for all integer values of $$s$$ greater than $$1$$. Later, Pafnuty Chebyshev extended the domain of the function beyond the integers to all real numbers $$s$$ greater than $$1$$. And then Bernhard Riemann went further still: He devised a smoke-and-mirrors trick for defining the zeta function over the entire plane of complex numbers, with the single exception of $$s = 1$$.

Today, Riemann’s version of the zeta function is the engine (or enigma!) driving a major mathematical industry. Shier’s use of this apparatus in making fractal art is far removed from that heavy-duty research enterprise—but no less fascinating. Think of it as the zeta function on vacation.

If a collection of disks are to fill a square exactly, their aggregate area must equal the area of the square. This is a necessary condition though not a sufficient one. In all the examples I’ve presented so far, the containing square has an area of 4, so what’s needed is to find a value of $$s$$ that satisfies the equation:

$\zeta(s) = \sum_{k = 1}^{\infty}\frac{1}{k^s} = 4$

Except for isolated values of $$s$$, Those isolated values are the negative integers and the even positive integers.there is no known method for solving this equation exactly. But numerical approximation works well enough for a computer program that draws pictures. The binary search described above is a crude and cumbersome version of this numerical method. SageMath presumably does something more sophisticated. When I ask it to find a root of the equation $$zeta(s) = 4$$, it returns the result $$1.2939615055572438$$.

Having this result in hand solves one part of the square-filling problem. It tells us how to construct an infinite set of disks whose total area is just enough to cover a square of area $$4$$, with adequate precision for graphical purposes. We assign each disk $$k$$ (starting at $$k = 1$$) an area of $$1/k^{1.2939615}.$$ This sequence begins 1.000, 0.408, 0.241, 0.166, 0.125, 0.098,…

In the graph above, the maroon curve with $$s = 1.29396$$ converges to a sum very close to 4. Admittedly, the rate of convergence is not quick. More than 3 million terms are needed to get within 1 percent of the target.

Our off-label use of the zeta function defines an infinite sequence of disks whose aggregate area is equal to $$4$$. The disks in this unique collection will exactly fill our square box (assuming they can be properly arranged). It’s satisfying to have a way of reliably achieving this result, after our various earlier failures. On the other hand, there’s something irksome about that number $$4$$ appearing in the equation. It’s so arbitrary! I don’t dispute that $$4$$ is a perfectly fine and foursquare number, but there are many other sizes of squares we might want to fill with dots. Why give all our attention to the $$2 \times 2$$ variety?

This is all my fault. When I set out to write some square-filling programs, I knew I couldn’t use the unit square—which seems like the obvious default choice—because of the awkward fact that $$\zeta(s) = 1$$ has no finite solution. The unit square is also troublesome in the case of the harmonic numbers; the first disk, with area $$A_1 = 1$$, is too large to fit. So I picked the next squared integer for the box size in those first programs. Having made my choice, I stuck with it, but now I feel hemmed in by that decision made with too little forethought.

We have all the tools we need to fill squares of other sizes (as long as the size is greater than $$1$$). Given a square of area $$A_{\square}$$, we just solve for $$s$$ in $$\zeta(s) = A_{\square}$$. A square of area 8 can be covered by disks sized according to the rule $$A_k = 1/k^s$$ with $$s = \zeta(8) \approx 1.1349$$. For $$A_{\square} = 100$$, the corresponding value of $$s$$ is $$\zeta(100) \approx 1.0101$$. For any $$A_{\square} \gt 1$$ there is an $$s$$ that yields a fulfilling set of disks, and vice versa for any $$s \gt 1$$.

This relation between the exponent $$s$$ and the box area $$A_{\square}$$ suggests a neat way to evade the whole bother of choosing a specific container size. We can just scale the disks to fit the box, or else scale the box to accommodate the disks. Shier adopts the former method. Each disk in the infinite set is assigned an area of

$A_k = \frac{A_{\square}}{\zeta(s)} \frac{1}{k^s},$

where the first factor is a scaling constant that adjusts the disk sizes to fit the container. In my first experiments with these programs I followed the same approach. Later, however, when I began writing this essay, it seemed easier to think about the scaling—and explain it—if I transformed the size of the box rather than the sizes of the disks. In this scheme, the area of disk $$k$$ is simply $$1 / k^s$$, and the area of the container is $$A_{\square} = \zeta(s)$$. (The two scaling procedures are mathematically equivalent; it’s only the ratio of disk size to container size that matters.)

Program 5 offers an opportunity to play with such scaled zeta functions. I’m not actually changing the physical size of the box—the number of pixels it occupies on-screen remains the same. I’m scaling the units of measure.No matter where you move the $$s$$ slider, the area of the square container will adjust to match the total area of the infinite set of disks. As the ratio of disk size to container size shifts, so does the overall texture or appearance of the pattern. At a setting of $$s = 1.35$$, the largest disk fills almost a third of the square; at $$s = 1.08$$, the first disk occupies only about $$8$$ percent of box area. At those lower settings it takes a very long time to reach a high filling percentage, but if you have true faith in mathematical certainties, your patience will be rewarded.

Program 5: Disk Sizes from Scaled Zeta(s)

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At the other end of the scale, if you push the value of $$s$$ up beyond about $$1.40$$, you’ll discover something else: The program more often than not halts after placing just a few disks. At $$s = 1.50$$ or higher, it seldom gets beyond the first disk. This failure is similar to what we saw with the harmonic numbers, but more interesting. In the case of the harmonic numbers, the total area of the disks is unbounded, making an overflow inevitable. With this new scaled version of the zeta function, the total area of the disks is always equal to that of the enclosing square. In principle, all the disks could all be made to fit, if you could find the right arrangement. I’ll return below to the question of why that doesn’t happen.

In Fractalize That! Shier introduces another device for taming space-filling sets. He not only scales the object sizes so that their total area matches the space available; he also adopts a variant zeta function that has two adjustable parameters rather than just one:

A note on notation: Shier writes the Hurwitz zeta function as $$\zeta(c, N)$$, whereas most of the mathematical literature seems to favor $$\zeta(s, a)$$. I’m going with the majority.$\zeta(s, a) = \sum_{k=0}^{\infty} \frac{1}{(a + k)^s}$

This is the Hurwitz zeta function, named for the German mathematician Adolf Hurwitz (1859–1919). Before looking into the details of the function, let’s play with the program and see what happens. Try a few settings of the $$s$$ and $$a$$ controls:

Program 6: Disk Sizes from Scaled Hurwitz Zeta Function

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Different combinations of $$s$$ and $$a$$ produce populations of disks with different size distributions. The separate contributions of the two parameters are not always easy to disentangle, but in general decreasing $$s$$ or increasing $$a$$ leads to a pattern dominated by smaller disks. Here are snapshots of four outcomes:

Within the parameter range shown in these four panels, the filling process always continues to exhaustion, but at higher values of $$s$$ it can jam, just as it does with the scaled Riemann zeta function.

Adolf Hurwitz in the 1880s. Photo from Wikipedia.Until I began this project, I knew nothing of Adolf Hurwitz or his work, although he is hardly an obscure figure in the history of mathematics. He earned his Ph.D. under Felix Klein and also studied with Karl Weierstrass, Ernst Eduard Kummer, and Leopold Kronecker—key figures in the founding of modern analysis and number theory. Among his own pupils (and close friends) were David Hilbert and Hermann Minkowski. Albert Einstein was another of his students, although apparently he seldom went to class.

Hurwitz wrote just one paper on the zeta function. It was published in 1882, when he was still quite young and just beginning his first academic appointment, at the University of Göttingen. (The paper is available from the Göttinger Digitalisierungszentrum; see pp. 86–101.)

Hurwitz modified the Riemann zeta function in two ways. First, the constant $$a$$ is added to each term, turning $$1/k^s$$ into $$1/(a + k)^s$$. Second, the summation begins with $$k = 0$$ rather than $$k = 1$$. By letting $$a$$ take on any value in the range $$0 \lt a \le 1$$ we gain access to a continuum of zeta functions. The elements of the series are no longer just reciprocals of integers but reciprocals of real numbers. Suppose $$a = \frac{1}{3}$$. Then $$\zeta(s, a)$$ becomes:

$\frac{1}{\left(\frac{1}{3} + 0\right)^s} + \frac{1}{\left(\frac{1}{3} + 1\right)^s} + \frac{1}{\left(\frac{1}{3} + 2\right)^s} + \cdots\ = \left(\frac{3}{1}\right)^s + \left(\frac{3}{4}\right)^s + \left(\frac{3}{7}\right)^s + \cdots$

The Riemann zeta function and the Hurwitz zeta function differ substantially only for small values of $$k$$ or large values of $$a$$. When $$k$$ is large, adding a small $$a$$ to it makes little difference in the value of the function. Thus as $$k$$ grows toward infinity, the two functions are asymptotically equal, as suggested in the graph at right. When the Hurwitz function is put to work packing disks into a square, a rule with $$a > 1$$ causes the first several disks to be smaller than they would be with the Riemann rule. A value of $$a$$ between $$0$$ and $$1$$ enlarges the early disks. In either case, the later disks in the sequence are hardly affected at all.

If $$a$$ is a positive integer, the interpretation of $$\zeta(s, a)$$ is even simpler. The case $$a = 1$$ corresponds to the Riemann zeta sum. When $$a$$ is a larger integer, the effect is to omit the first $$a - 1$$ entries, leaving only the tail of the series. For example,

$\zeta(s, 5) = \frac{1}{5^s} + \frac{1}{6^s} + \frac{1}{7^s} + \cdots.$

In his fractal artworks, Shier chooses various values of $$a$$ as a way of controlling the size distribution of the placed objects, and thereby fine-tuning the appearance of the patterns. Having this adjustment knob available is very convenient, but in the interests of simplicity, I am going to revert to the Riemann function in the rest of this essay.

Before going on, however, I also have to confess that I don’t really understand the place of the Hurwitz zeta function in modern mathematical research, or what Hurwitz himself had in mind when he formulated it. Zeta functions have been an indispensable tool in the long struggle to understand how the prime numbers are sprinkled among the integers. The connection between these two realms was made by Euler, with his remarkable equation linking a sum of powers of integers with a product of powers of primes:

Euler’s other famous equation, $$e^{i\pi} + 1 = 0$$, has a bigger fan club, but this is the one that revs my motor.$\sum_{k = 1}^{\infty} \frac{1}{k^s} = \prod_{p \text{ prime}} \frac{1}{1 - \frac{1}{p^s}}.$

Riemann went further, showing that everything we might want to know about the distribution of primes is encoded in the undulations of the zeta function over the complex plane. Indeed, if we could simply pin down all the complex values of $$s$$ for which $$\zeta(s) = 0$$, we would have a master key to the primes. Hurwitz, in his 1882 paper, was clearly hoping to make some progress toward this goal, but I have not been able to figure out how his work fits into the larger story. The Hurwitz zeta function gets almost no attention in standard histories and reference works (in contrast to the Riemann version, which is everywhere). Wikipedia notes: “At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa”—which sounds interesting, but I don’t know if it’s useful or important. A recent article by Nicola Oswald and Jörn Steuding puts Hurwitz’s work in historical context, but it does not answer these questions—at least not in a way I’m able to understand.

But again I digress. Back to dots in boxes.

If a set of circular disks and a square container have the same total area, can you always arrange the disks so that they completely fill the square without overflowing? Certainly not! Suppose the set consists of a single disk with area equal to that of the square; the disk’s diameter is greater than the side length of the square, so it will bulge through the sides while leaving the corners unfilled. A set of two disks won’t work either, no matter how you apportion the area between them. Indeed, when you are putting round pegs in a square hole, no finite set of disks can ever fill all the crevices.

Only an infinite set—a set with no smallest disk—can possibly fill the square completely. But even with an endless supply of ever-smaller disks, it seems like quite a delicate task to find just the right arrangement, so that every gap is filled and every disk has a place to call home. It’s all the more remarkable, then, that simply plunking down the disks at random locations seems to produce exactly the desired result. This behavior is what intrigued and troubled me when I first saw Shier’s pictures and read about his method for generating them. If a random arrangement works, it’s only a small step to the proposition that any arrangement works. Could that possibly be true?

Computational experiments offer strong hints on this point, but they can never be conclusive. What we need is a proof. Ennis’s proof was published in Math Horizons, a publication of the Mathe­matical Association of America, which keeps it behind a paywall. If you have no library access and won’t pay the 50 ransom, I can recommend a video of Ennis explaining his proof in a talk at St. Olaf College.And so I turn to the work of Christopher Ennis, a mathematician at Normandale Community College, who met Shier when they were both teaching there. As a warm-up exercise, Ennis proves a one-dimensional version of the area-filling conjecture, where the geometry is simpler and some of the constraints are easier to satisfy. In one dimension a disk is merely a line segment; its area is its length, and its radius is half that length. As in the two-dimensional model, disks are placed in descending order of size at random positions, with the usual proviso that no disk can overlap another disk or extend beyond the end points of the containing interval. In Program 7 you can play with this scheme. Program 7: One-Dimensional Disks Sorry, the program will not run in this browser. I have given the line segment some vertical thickness to make it visible. The resulting pattern of stripes may look like a supermarket barcode or an atomic spectrum, but please imagine it as one-dimensional. If you adjust the slider in this program, you’ll notice a difference from the two-dimensional system. In 2D, the algorithm is fulfilling only if the exponent $$s$$ is less than a critical value, somewhere in the neighborhood of 1.4. In one dimension, the process continues without impediment for all values of $$s$$ throughout the range $$1 \lt s \lt 2$$. Try as you might, you won’t find a setting that produces a jammed state. (In practice, the program halts after placing no more than 10,000 disks, but the reason is exhaustion rather than jamming.) Ennis titles his Math Horizons article “(Always) room for one more.” He proves this assertion by keeping track of the set of points where the center of a new disk can legally be placed, and showing the set is never empty. Suppose $$n - 1$$ disks have already been randomly scattered in the container. The next disk to be placed, disk $$n$$, will have an area (or length) of $$A_n = 1 / n^s$$. Since the geometry is one-dimensional, the corresponding disk radius is simply $$r_n = A_n / 2$$. The center of this new disk cannot lie any closer than $$r_n$$ to the perimeter of another disk. It must also be at a distance of at least $$r_n$$ from the boundary of the containing segment. We can visualize these constraints by adding bumpers, or buffers, of thickness $$r_n$$ to the outside of each existing disk and to the inner edges of the containing segment. A few stages of the process are illustrated below. Placed disks are blue, the excluded buffer areas are orange, and open areas—the set of all points where the center of the next disk could be placed—are black. In the top line, before any disks have been placed, the entire containing segment is open except for the two buffers at the ends. Each of these buffers has a length equal to $$r_1$$, the radius of the first disk to be placed; the center of that disk cannot lie in the orange regions because the disk would then overhang the end of the containing segment. After the first disk has been placed (second line), the extent of the open area is reduced by the area of the disk itself and its appended buffers. On the other hand, all of the buffers have also shrunk; each buffer is now equal to the radius of disk $$2$$, which is smaller than disk $$1$$. The pattern continues as subsequent disks are added. Note that although the blue disks cannot overlap, the orange buffers can. For another view of how this process evolves, click on the Next button in Program 8. Each click inserts one more disk into the array and adjusts the buffer and open areas accordingly. Program 8: One-Dimensional with Buffers Sorry, the program will not run in this browser. Because the blue disks are never allowed to overlap, the total blue area must increase monotonically as disks are added. It follows that the orange and black areas, taken together, must steadily decrease. But there’s nothing steady about the process when you keep an eye on the separate area measures for the orange and black regions. Changes in the amount of buffer overlap cause erratic, seesawing tradeoffs between the two subtotals. If you keep clicking the Next button (especially with $$s$$ set to a high value), you may see the black area falling below $$1$$ percent. Can we be sure it will never vanish entirely, leaving no opening at all for the next disk? Ennis answers this question through worst-case analysis. He considers only configurations in which no buffers overlap, thereby squeezing the black area to its smallest possible extent. If the black area is always positive under these conditions, it cannot be smaller when buffer overlaps are allowed. The basic idea of the proofI have altered some of the notation and certain details of presentation to conform with choices I made elsewhere in this exposition. is to start with $$n - 1$$ buffered disks already in place, arranged so that none of the orange buffer areas intersect. The total area of the blue disks is $$\sum_{k=1}^{k=n-1} 1/k^s$$. Each buffer zone has a width equal to $$r_{n}$$, the radius of the next disk to be added to the tableau; since each disk has two buffers, the total area of the orange buffers is $$2(n-1)r_{n}$$. The black area is whatever’s left over. In other words, $A_{\square} = \zeta(s), \quad A_{\color{blue}{\mathrm{blue}}} = \sum_{k=1}^{k = n - 1} \frac{1}{k^s}, \quad A_{\color{orange}{\mathrm{orange}}} = 2(n-1)r_{n}.$ Then we need to prove that $A_{\square} - (A_{\color{blue}{\mathrm{blue}}} + A_{\color{orange}{\mathrm{orange}}}) \gt 0.$ A direct proof of this statement would require an exact, closed-form expression for $$\zeta(s)$$, which we already know is problematic. Ennis evades this difficulty by turning to calculus. He needs to evaluate the remaining tail of the zeta series, $$\sum_{k = n}^\infty 1/k^s$$, but this discrete sum is intractable. On the other hand, by shifting from a sum to an integral, the problem becomes an exercise in undergraduate calculus. Exchanging the discrete variable $$k$$ for a continuous variable $$x$$, we want to find the area under the curve $$1/x^s$$ in the interval from $$n$$ to infinity; this will provide a lower bound on the corresponding discrete sum. Evaluating the integral yields: $\int_{x = n}^{\infty} \frac{1}{x^{s}} d x = \frac{1}{(s-1) n^{s-1}}.$ Some further manipulation reveals that the area of the black regions is never smaller than $\frac{2 - s}{(s - 1)n^{s - 1}}.$ If $$s$$ lies strictly between $$1$$ and $$2$$, this expression must be greater than zero, since both the numerator and the denominator will be positive. Thus for all $$n$$ there is at least one black point where the center of a new disk can be placed. Ennis’s proof is a stronger one than I expected. When I first learned there was a proof, I guessed that it would take a probabilistic approach, showing that although a jammed configuration may exist, it has probability zero of turning up in a random placement of the disks. Instead, Ennis shows that no such arrangement exists at all. Even if you replaced the randomized algorithm with an adversarial one that tries its best to block every disk, the process would still run to fulfillment. The proof for a two-dimensional system follows the same basic line of argument, but it gets more complicated for geometric reasons. In one dimension, as the successive disk areas get smaller, the disk radii diminish in simple proportion: $$r_k = A_k / 2$$. In two dimensions, disk radius falls off only as the square root of the disk area: $$r_k = \sqrt{A_k / \pi}$$. As a result, the buffer zone surrounding a disk excludes neighbors at a greater distance in two dimensions than it would in one dimension. There is still a range of $$s$$ values where the process is provably unstoppable, but it does not extend across the full interval from $$s \gt 1$$ to $$s \lt 2$$. Program 9, running in the panel below, is one I find very helpful in gaining intuition into the behavior of Shier’s algorithm. As in the one-dimensional model of Program 8, each press of the Next button adds a single disk to the containing square, and shows the forbidden buffer zones surrounding the disks. Program 9: Two-Dimensional Disks with Buffers Sorry, the program will not run in this browser. Move the $$s$$ slider to a position somewhere near 1.40. In the control panel for this program, the registers showing fill percentages for blue, orange, and black areas are not entirely trustworthy. In the one-dimensional case, it’s easy to calculate the areas to high precision. That’s much harder in two dimensions, where the overlapping regions of buffers can have complex shapes. I have resorted to counting pixels, a procedure that has limited resolution and is subject to errors caused by fuzzy boundaries.At this setting, there’s a fair chance (maybe 10 or 20 percent) that the very first disk and its orange buffer zone will entirely cover the open black region, creating a jammed state. On other runs you might get as far as two or three or 10 disks before you get stuck. If you make it beyond that point, however, you are likely to continue unimpeded for as long as you have the patience to keep pressing Next. Shier describes this phenomenon as “infant mortality”: If the placement process survives the high-risk early period, it is all but immortal. There’s a certain whack-a-mole dynamic to the behavior of this system. Maybe the first disk covers all but one small corner of the black zone. It looks like the next disk will completely obliterate that open area. And so it does—but at the same time the shrinking of the orange buffer rings opens up another wedge of black elsewhere. The third disk blots out that spot, but again the narrowing of the buffers allows a black patch to peek out from still another corner. Later on, when there are dozens of disks, there are also dozens of tiny black spots where there’s room for another disk. You can often guess which of the openings will be filled next, because the random search process is likely to land in the largest of them. Again, however, as these biggest targets are buried, many smaller ones are born. Ennis’s two-dimensional proof addresses the case of circular disks inside a circular boundary, rather than a square one. (The higher symmetry and the absence of corners streamlines certain calculations.) The proof strategy, again, is to show that after $$n - 1$$ disks have been placed, there is still room for the $$n$$th disk, for any value of $$n \ge 1$$. The argument follows the same logic as in one dimension, relying on an integral to provide a lower bound for the sum of a zeta series. But because of the $$\pi r^2$$ area relation, the calculation now includes quadratic as well as linear terms. As a result, the proof covers only a part of the range of $$s$$ values. The black area is provably nonempty if $$s$$ is greater than $$1$$ but less than roughly $$1.1$$; outside that interval, the proof has nothing to say. As mentioned above, Ennis’s proof applies only to circular disks in a circular enclosure. Nevertheless, in what follows I am going to assume the same ideas carry over to disks in a square frame, although the location of the boundary will doubtless be somewhat different. I have recently learned that Ennis has written a further paper on the subject, expected to be published in the American Mathematical Monthly. Perhaps he addresses this question there. With Program 9, we can explore the entire spectrum of behavior for packing disks into a square. The possibilities are summarized in the candybar graph below. • The leftmost band, in darker green, is the interval for which Ennis’s proof might hold. The question mark at the upper boundary line signifies that we don’t really know where it lies. • In the lighter green region no proof is known, but in Shier’s extensive experiments the system never jams there. • The transition zone sees the probability of jamming rise from $$0$$ to $$1$$ as $$s$$ goes from about $$1.3$$ to about $$1.5$$. • Beyond $$s \approx 1.5$$, experiments suggest that the system always halts in a jammed configuration. • At $$s \approx 1.6$$ we enter a regime where the buffer zone surrounding the first disk invariably blocks the entire black region, leaving nowhere to place a second disk. Thus we have a simple proof that the system always jams. • Still another barrier arises at $$s \approx 2.7$$. Beyond this point, not even one disk will fit. The diameter of a disk with area $$1$$ is greater than the side length of the enclosing square. Can we pin down the exact locations of the various threshold points in the diagram above? This problem is tractable in those situations where the placement of the very first disk determines the outcome. At high values of $$s$$ (and thus low values of $$\zeta(s)$$, the first disk can obliterate the black zone and thereby preclude placement of a second disk. What is the lowest value of $$s$$ for which this can happen? As in the image at right, the disk must lie at the center of the square box, and the orange buffer zone surrounding it must extend just far enough out to cover the corners of the inner black square, which defines the locus of all points that could accommodate the center of the second disk. Finding the value of $$s$$ that satisfies this condition is a messy but straightforward bit of geometry and algebra. With the help of SageMath I get the answer $$s = 1.282915$$. This value—let’s call it $$\overline{s}$$—is an upper bound on the “never jammed” region. Above this limit there is always a nonzero probability that the filling process will end after placing a single disk. The value of $$\overline{s}$$ lies quite close to the experimentally observed boundary between the never-jammed range and the transition zone, where jamming first appears. Is it possible that $$\overline{s}$$ actually marks the edge of the transition zone—that below this value of $$s$$ the program can never fail? To prove that conjecture, you would have to show that when the first disk is successfully placed, the process never stalls on a subsequent disk. That’s certainly not true in higher ranges of $$s$$. Yet the empirical evidence near the threshold is suggestive. In my experiments I have yet to see a jammed outcome at $$s \lt \overline{s}$$, not even in a million trials just below the threshold, at $$s = 0.999 \overline{s}$$. In contrast, at $$s = 1.001 \overline{s}$$, a million trials produced 53 jammed results—all of them occuring immediately after the first disk was placed. The same kind of analysis leads to a lower bound on the region where every run ends after the first disk (medium pink in the diagram above). In this case the critical situation puts the first disk as close as possible to a corner of the square frame, rather than in the middle. If the disk and its orange penumbra are large enough to block the second disk in this extreme configuration, then they will also block it in any other position. Putting a number on this bound again requires some fiddly equation wrangling; the answer I get is $$\underline{s} = 1.593782$$. No process with higher $$s$$ can possibly live forever, since it will die with the second disk. In analogy with the lower-bound conjecture, one might propose that the probability of being jammed remains below $$1$$ until $$s$$ reaches $$\underline{s}$$. If both conjectures were true, the transition region would extend from $$\overline{s}$$ to $$\underline{s}$$. The final landmark, way out at $$s \approx 2.7$$, marks the point where the first disk threatens to burst the bounds of the enclosing square. In this case the game is over before it begins. In program 9, if you push the slider far to the right, you’ll find that the black square in the middle of the orange field shrinks away and eventually winks out of existence. This extinction event comes when the diameter of the disk equals the side length of the square. Given a disk of area $$1$$, and thus radius $$1/\sqrt{\pi}$$, we want to find the value of $$s$$ that satisfies the equation $\frac{2}{\sqrt{\pi}} = \sqrt{\zeta(s)}.$ Experiments with Program 9 show that the value is just a tad more than 2.7. That’s an interesting numerical neighborhood, no? A famous number lives nearby. Do you suppose? Another intriguing set of questions concerns the phenomenon that Shier calls infant mortality. If you scroll back up to Program 5 and set the slider to $$s = 1.45$$, you’ll find that roughly half the trials jam. The vast majority of these failures come early in the process, after no more than a dozen disks have been placed. At $$s = 1.50$$ death at an early age is even more common; three-fourths of all the trials end with the very first disk. On the other hand, if a sequence of disks does manage to dodge all the hazards of early childhood, it may well live on for a very long time—perhaps forever. Should we be surprised by this behavior? I am. As Shier points out, the patterns formed by our graduated disks are fractals, and one of their characteristic properties is self-similarity, or scale invariance. If you had a fully populated square—one filled with infinitely many disks—you could zoom in on any region to any magnification, and the arrangement of disks would look the same as it does in the full-size square. By “look the same” I don’t mean the disks would be in the same positions, but they would have the same size distribution and the same average number of neighbors at the same distances. This is a statistical concept of identity. And since the pattern looks the same and has the same statistics, you would think that the challenge of finding a place for a new disk would also be the same at any scale. Slipping in a tiny disk late in the filling operation would be no different from plopping down a large disk early on. The probability of jamming ought to be constant from start to finish. But there’s a rejoinder to this argument: Scale invariance is broken by the presence of the enclosing square. The largest disks are strongly constrained by the boundaries, whereas most of the smaller disks are nowhere near the edges and are little influenced by them. The experimental data offer some support for this view. The graph below summarizes the outcomes of $$20{,}000$$ trials at $$s = 1.50$$. The red bars show the absolute numbers of trials ending after placing $$n$$ disks, for each $$n$$ from $$0$$ through $$35$$. The blue lollipops indicate the proportion of trials reaching disk $$n$$ that halted after placing disk $$n$$. This ratio can be interpreted (if you’re a frequentist!) as the probability of stopping at $$n$$. It certainly looks like there’s something odd happening on the left side of this graph. More than three fourths of the trials end after a single disk, but none at all jam at the second or third disks, and very few (a total of $$23$$) at disks $$4$$ and $$5$$. Then, suddenly, $$1{,}400$$ more fall by the wayside at disk $$6$$, and serious attrition continues through disk $$11$$. Geometry can explain some of this weirdness. It has to do with the squareness of the container; other shapes would produce different results. At $$s = 1.50$$ we are between $$\overline{s}$$ and $$\underline{s}$$, in a regime where the first disk is large enough to block off the entire black zone but not so large that it must do so. This is enough to explain the tall red bar at $$n = 1$$: When you place the first disk randomly, roughly $$75$$ percent of the time it will block the entire black region, ending the parade of disks. If the first disk doesn’t foreclose all further action, it must be tucked into one of the four corners of the square, leaving enough room for a second disk in the diagonally opposite corner. The sequence of images below (made with Program 9) tells the rest of the story. The placement of the second disk blocks off the open area in that corner, but the narrowing of the orange buffers also creates two tiny openings in the cross-diagonal corners. The third and fourth disks occupy these positions, and simultaneously allow the black background to peek through in two other spots. Finally the fifth and sixth disks close off the last black pixels, and the system jams. This stereotyped sequence of disk placements accounts for the near absence of mortality at ages $$n = 2$$ through $$n = 5$$, and the sudden upsurge at age $$6$$. The elevated levels at $$n = 7$$ through $$11$$ are part of the same pattern; depending on the exact positioning of the disks, it may take a few more to expunge the last remnants of black background. At still higher values of $$n$$—for the small subset of trials that get there—the system seems to shift to a different mode of behavior. Although numerical noise makes it hard to draw firm conclusions, it doesn’t appear that any of the $$n$$ values beyond $$n = 12$$ are more likely jamming points than others. Indeed, the data are consistent with the idea that the probability of jamming remains constant as each additional disk is added to the array, just as scale invariance would suggest. A much larger data set would be needed to test this conjecture, and collecting such data is painfully slow. Furthermore, when it comes to rare events, I don’t have much trust in the empirical data. During one series of experiments, I noticed a program run that stalled after $$290$$ disks—unusually late. The 290-disk configuration, produced at $$s = 1.47$$, is shown at left below. This pattern, compared with those produced at lower values of $$s$$, has a strongly Apollonian texture. Many of the disks are nestled tightly among their neighbors, and they form the recursive triangular motifs characteristic of Apollonian circles. I wondered if it was truly jammed. My program gives up on finding a place for a disk after $$10^7$$ random attempts. Perhaps if I had simply persisted, it would have gone on. So I reset the limit on random attempts to $$10^9$$, and sat back to wait. After some minutes the program discovered a place where disk $$291$$ would fit, and then another for disk $$292$$, and kept going as far as 300 disks. The program had an afterlife! Could I revive it again? Upping the limit to $$10^{10}$$ allowed another $$14$$ disks to squueze in. The final configuration is shown at right above (with the original $$290$$ disks faded, in order to make the $$24$$ posthumous additions more conspicuous). Is it really finished now, or is there still room for one more? I have no reliable way to answer that question. Checking $$10$$ billion random locations sounds like a lot, but it is still a very sparse sampling of the space inside the square box. Using 64-bit floating-point numbers to define the coordinate system allows for more than $$10^{30}$$ distinguishable points. And to settle the question mathematically, we would need unbounded precision. We know from Ennis’s proof that at values of $$s$$ not too far above $$1.0$$, the filling process can always go on forever. And we know that beyond $$s \approx 1.6$$, every attempt to fill the square is doomed. There must be some kind of transition between these two conditions, but the details are murky. The experimental evidence gathered so far suggests a smooth transition along a sigmoid curve, with the probability of jamming gradually increasing from $$0$$ to $$1$$. As far as I can tell, however, nothing we know for certain rules out a single hard threshold, below which all disk sequences are immortal and above which all of them die. Thus the phase diagram would be reduced to this simple form: The softer transition observed in computational experiments would be an artifact of our inability to perform infinite random searches or place infinite sequences of disks. Here’s a different approach to understanding the random dots-in-a-box phenomenon. It calls for a mental reversal of figure and ground. Instead of placing disks on a square surface, we drill holes in a square metal plate. And the focus of attention is not the array of disks or holes but rather the spaces between them. Shier has a name for the perforated plate: the gasket. Program 10 allows you to observe a gasket as it evolves from a solid black square to a delicate lace doily with less than 1 percent of its original substance. Program 10: The Gasket Sorry, the program will not run in this browser. The gasket is quite a remarkable object. When the number of holes becomes infinite, the gasket must disappear entirely; its area falls to zero. Up until that very moment, however, it retains its structural integrity. This statement about the con­nectedness of the gasket requires a more careful consideration of what it means for two disks or holes to overlap. Are they allowed to touch, or in other words to be tangent, sharing a single point? The answer makes no difference to the calculation of areas, but it does matter for connectivity. Allowing tangency (as in Apollonian circles) would shatter the gasket, leaving tiny shards. To preserve connectivity, a computer program must test for overlaps with “$$\gt$$” rather than “$$\ge$$”.Although it may be reduced to a fine filigree, the perforated square never tears apart into multiple pieces; it remains a single, connected component, a network with multiple paths linking any two points you might choose. As the gasket is etched away, can we measure the average thickness of the surviving wisps and tendrils? I can think of several methods that involve elaborate sampling schemes. Shier has a much simpler and more ingenious proposal: To find the average thickness of the gasket, divide its area by its perimeter. It was not immediately obvious to me why this number would serve as an appropriate measure of the width, but at least the units come out right: We are dividing a length squared by a length and so we get a length. And the operation does make basic sense: The area of the gasket represents the amount of substance in it, and the perimeter is the distance over which it is stretched. (The widths calculated in Program 10 differ slightly from those reported by Shier. The reason, I think, is that I include the outer boundary of the square in the perimeter, and he does not.) Calculating the area and perimeter of a complicated shape such as a many-holed gasket looks like a formidable task, but it’s easy if we just keep track of these quantities as we go along. Initially (before any holes are drilled), the gasket area $$A_0^g$$ is the area of the full square, $$A_\square$$. The initial gasket perimeter $$P_0^g$$ is four times the side length of the square, which is $$\sqrt{A_\square}$$. Thereafter, as each hole is drilled, we subtract the new hole’s area from $$A^g$$ and add its perimeter to $$P^g$$. The quotient of these quantities is our measure of the average gasket width after drilling hole $$k$$: $$\widehat{W}_k^g$$. Since the gasket area is shrinking while the perimeter is growing, $$\widehat{W}_k^g$$ must dwindle away as $$k$$ increases. The importance of $$\widehat{W}_k^g$$ is that it provides a clue to how large a vacant space we’re likely to find for the next disk or hole. If we take the idea of “average” seriously, there must always be at least one spot in the gasket with a width equal to or greater than $$\widehat{W}_k^g$$. From this observation Shier makes the leap to a whole new space-filling algorithm. Instead of choosing disk diameters according to a power law and then measuring the resulting average gasket width, he determines the radius of the next disk from the observed $$\widehat{W}_k^g$$: $r_{k+1} = \gamma \widehat{W}_k^g = \gamma \frac{A_k^g}{P_k^g}.$ Here $$\gamma$$ is a fixed constant of proportionality that determines how tightly the new disks or holes fit into the available openings. The area-perimeter algorithm has a recursive structure, in which each disk’s radius depends on the state produced by the previous disks. This raises the question of how to get started: What is the size of the first disk? Shier has found that it doesn’t matter very much. Initial disks in a fairly wide range of sizes yield jam-proof and aesthetically pleasing results. Graphics produced by the original power-law algorithm and by the new recursive one look very similar. One way to understand why is to rearrange the equation of the recursion: Perhaps this equation would be a little easier to interpret if the average width were defined in terms of hole diameters rather than hole perimeters. Then the denominator of the right hand side would be the sum of the first $$k$$ diameters scaled by the $$(k+1)$$st diameter.$\frac{1}{2 \gamma} = \frac{A_k^g}{2 r_{k+1} P_k^g}.$ On the right side of this equation we are dividing the average gasket width by the diameter of the next disk to be placed. The result is a dimensionless number—dividing a length by a length cancels the units. More important, the quotient is a constant, unchanging for all $$k$$. If we calculate this same dimensionless gasket width when using the power-law algorithm, it also turns out to be nearly constant in the limit of karge $$k$$, showing that the two methods yield sequences with similar statistics. Setting aside Shier’s recursive algorithm, all of the patterns we’ve been looking at are generated by a power law (or zeta function), with the crucial requirement that the series must converge to a finite sum. The world of mathematics offers many other convergent series in addition to power laws. Could some of them also create fulfilling patterns? The question is one that Ennis discusses briefly in his talk at St. Olaf and that Shier also mentions. Among the obvious candidates are geometric series such as $$\frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$$ A geometric series is a close cousin of a power law, defined in a similar way but exchanging the roles of $$s$$ and $$k$$. That is, a geometric series is the sum: $\sum_{k=0}^{\infty} \frac{1}{s^k} = \frac{1}{s^0} + \frac{1}{s^1} + \frac{1}{s^2} + \frac{1}{s^3} + \cdots$ For any $$s > 1$$, the infinite geiometric series has a finite sum, namely $$\frac{s}{s - 1}$$. Thus our task is to construct an infinite set of disks with individual areas $$1/s^k$$ that we can pack into a square of area $$\frac{s}{s - 1}$$. Can we find a range of $$s$$ for which the series is fulfilling? As it happens, this is where Shier began his adventures; his first attempts were not with power laws but with geometric series. They didn’t turn out well. You are welcome to try your own hand in Program 11. Program 11: Disk Sizes from Geometric Series Sorry, the program will not run in this browser. There’s a curious pattern to the failures you’ll see in this program. No matter what value you assign to $$s$$ (within the available range $$1 \lt s \le 2$$), the system jams when the number of disks reaches the neighborhood of $$A_\square = \frac{s}{s-1}$$. For example, at $$s = 1.01$$, $$\frac{s}{s - 1}$$ is 101 and the program typically gets stuck somewhere between $$k = 95$$ and $$k = 100$$. At $$s = 1.001$$, $$\frac{s}{s - 1}$$ is $$1{,}001$$ and there’s seldom progress beyond about $$k = 1,000$$. For a clue to what’s going wrong here, consider the graph at right, plotting the values of $$1 / k^s$$ (red) and $$1 / s^k$$ (blue) for $$s = 1.01$$. These two series converge on nearly the same sum (roughly $$100$$), but they take very different trajectories in getting there. On this log-log plot, the power-law series $$1 / s^k$$ is a straight line. The geometric series $$1 / s^k$$ falls off much more slowly at first, but there’s a knee in the curve at about $$k = 100$$ (dashed mauve line), where it steepens dramatically. If only we could get beyond this turning point, it looks like the rest of the filling process would be smooth sledding, but in fact we never get there. Whereas the first $$100$$ disks of the power-law series fill up only about $$5$$ percent of the available area, they occuy 63 percent in the geometric case. This is where the filling process stalls. Even in one dimension, the geometric series quickly succumbs. (This is in sharp contrast to the one-dimensional power-law model, where any $$s$$ between $$1$$ and $$2$$ yields a provably infinite progression of disks.) Program 12: Disk Sizes from Geometric Series in One Dimension Sorry, the program will not run in this browser. And just in case you think I’m pulling a fast one here, let me demonstrate that those same one-dimensional disks will indeed fit in the available space, if packed efficiently. In Program 13 they are placed in order of size from left to right. Program 13: Deterministic One-Dimensional Geometric Series Sorry, the program will not run in this browser. I have made casual attempts to find fulfillment with a few other convergent series, such as the reciprocals of the Fibonacci numbers (which converge to about $$3.36$$) and the reciprocals of the factorials (whose sum is $$e \approx 2.718$$). Both series jam after the first disk. There are plenty of other convergent series one might try, but I doubt this is a fruitful line of inquiry. All the variations discussed above leave one important factor unchanged: The objects being fitted together are all circular. Exploring the wider universe of shapes has been a major theme of Shier’s work. He asks: What properties of a shape make it suitable for forming a statistical fractal pattern? And what shapes (if any) refuse to cooperate with this treatment? (The images in this section were created by John Shier and are reproduced here with his per­mission.) Shier’s first experiments were with circular disks and axis-parallel squares; the filling algorithm worked splendidly in both cases. He also succeeded with axis-parallel rectangles of various aspect ratios, even when he mixed vertical and horizontal orientations in the same tableau. In collaboration with Paul Bourke he tried randomizing the orientation of squares as well as their positions. Again the outcome was positive, as the illustration above left shows. Equilateral triangles were less cooperative, and at first Shier believed the algorithm would consistently fail with this shape. The triangles tended to form orderly arrays with the sharp point of one triangle pressed close against the broad side of another, leaving little “wiggle room.” Further efforts showed that the algorithm was not truly getting stuck but merely slowing down. With an appropriate choice of parameters in the Hurwitz zeta function, and with enough patience, the triangles did come together in boundlessly extendable space-filling patterns. The casual exploration of diverse shapes eventually became a deliberate quest to stake out the limits of the space-filling process. Surely there must be some geometric forms that the algorithm would balk at, failing to pack an infinite number of objects into a finite area. Perhaps nonconvex shapes such as stars and snowflakes and flowers would expose a limitation—but no, the algorithm worked just fine with these figures, fitting smaller stars into the crevices between the points of larger stars. The next obvious test was “hollow” objects, such as annular rings, where an internal void is not part of the object and is therefore available to be filled with smaller copies. The image at right is my favorite example of this phenomenon. The bowls of the larger nines have smaller nines within them. It’s nines all the way down. When we let the process continue indefinitely, we have a whimsical visual proof of the proposition that $$.999\dots = 1$$. These successes with nonconvex forms and objects with holes led to an Aha moment, as Shier describes it. The search for a shape that would break the algorithm gave way to a suspicion that no such shape would be found, and then the suspicion gradually evolved into a conviction that any “reasonably compact” object is suitable for the Fractalize That! treatment. The phrase “reasonably compact” would presumably exclude shapes that are in fact dispersed sets of points, such as Cantor dust. But Shier has shown that shapes formed of disconnected pieces, such as the words in the pair of images below, present no special difficulty. Fractalize That! is not all geometry and number theory. Shier is eager to explain the mathematics behind these curious patterns, but he also presents the algorithm as a tool for self-expression. MATH and ART both have their place. Finally, I offer some notes on what’s needed to turn these algorithms into computer programs. Shier’s book includes a chapter for do-it-yourselfers that explains his strategy and provides some crucial snippets of code (written in C). My own source code (in JavaScript) is available on GitHub. And if you’d like to play with the programs without all the surrounding verbiage, try the GitHub Pages version. The inner loop of a typical program looks something like this: let attempt = 1; while (attempt <= maxAttempts) { disk.x = randomCoord(); disk.y = randomCoord(); if (isNotOverlapping(disk)) { return disk; } attempt++; } return false; We generate a pair of random $$x$$ and $$y$$ coordinates, which mark the center point of the new disk, and check for overlaps with other disks already in place. If no overlaps are discovered, the disk stays put and the program moves on. Otherwise the disk is discarded and we jump back to the top of the loop to try a new $$xy$$ pair. The main computational challenge lies in testing for overlaps. For any two specific disks, the test is easy enough: They overlap if the sum of their radii is greater than the distance between their centers. The problem is that the test might have to be repeated many millions of times. My program makes $$10$$ million attempts to place a disk before giving up. If it has to test for overlap with $$100{,}000$$ other disks on each attempt, that’s a trillion tests. A trillion is too many for an interactive program where someone is staring at the screen waiting for things to happen. To speed things up a little I divide the square into a $$32 \times 32$$ grid of smaller squares. The largest disks—those whose diameter is greater than the width of a grid cell—are set aside in a special list, and all new candidate disks are checked for overlap with them. Below this size threshold, each disk is allocated to the grid cell in which its center lies. A new candidate is checked against the disks in its own cell and in that cell’s eight neighbors. The net result is an improvement by two orders of magnitude—lowering the worst-case total from $$10^{12}$$ overlap tests to about $$10{10}$$. All of this works smoothly with circular disks. Devising overlap tests for the variety of shapes that Shier has been working with is much harder. From a theoretical point of view, the whole rigmarole of overlap testing is hideously wasteful and unnecessary. If the box is already 90 percent full, then we know that 90 percent of the random probes will fail. A smarter strategy would be to generate random points only in the “black zone” where new disks can legally be placed. If you could do that, you would never need to generate more than one point per disk, and there’d be no need to check for overlaps. But keeping track of the points that comprise the black zone—scattered throughout multiple, oddly shaped, transient regions—would be a serious exercise in computational geometry. For the actual drawing of the disks, Shier relies on the technology known as SVG, or scalable vector graphics. As the name suggests, these drawings retain full resolution at any size, and they are definitely the right choice if you want to create works of art. They are less suitable for the interactive programs embedded in this document, mainly because they consume too much memory. The images you see here rely on the HTML canvas element, which is simply a fixed-size pixel array. Another point of possible interest is the evaluation of the zeta function. If we want to scale the disk sizes to match the box size (or vice versa), we need to compute a good approximation of the Riemann function $$\zeta(s)$$ or the Hurwitz function $$\zeta(s, a)$$. I didn’t know how to do that, and most of the methods I read about seemed overwhelming. Before I could get to zeta, I’d have to hack my way through thickets of polygamma functions and Stieltjes constants. For the Riemann zeta function I found a somewhat simpler algorithm published by Peter Borwein in 1995. It’s based on a polynomial approximation that yields ample precision and runs in less than a millisecond. For the Hurwitz zeta function I stayed with a straightforward translation of Shier’s code, which takes more of a brute-force approach. (There are alternatives for Hurwitz too, but I couldn’t understand them well enough to make them work.) The JavaScript file in the GitHub repository has more discussion of implementation details. Bibliography: Works by John Shier and colleagues Shier, John. 2018. Fractalize That! A Visual Essay on Statistical Geometry. Singapore: World Scientific. Publisher’s website. Shier, John. Website: http://www.john-art.com/ Shier, John. 2011. The dimensionless gasket width $$b(c,n)$$ in statistical geometry. http://www.john-art.com/gasket_width.pdf Shier, John. 2012. Random fractal filling of a line segment. http://www.john-art.com/gasket_width.pdf Dunham, Douglas, and John Shier. 2014. The art of random fractals. In Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture pp. 79–86. PDF. Shier, John. 2015. A new recursion for space-filling geometric fractals. http://www.john-art.com/gasket_width.pdf Dunham, Douglas, and John Shier. 2015. An algorithm for creating aesthetic random fractal patterns. Talk delivered at the Joint Mathematics Meetings January 2015, San Antonio, Texas. Dunham, Douglas, and John Shier. 2018. A property of area and perimeter. In ICGG 2018: Proceedings of the 18th International Conference on Geometry and Graphics, Milano, August 2018, pp. 228–237. Dunham, Douglas, and John Shier. 2017. New kinds of fractal patterns. In Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture, pp. 111–116. Preprint. Shier, John, and Paul Bourke. 2013. An algorithm for random fractal filling of space. Computer Graphics Forum 32(8):89–97. PDF. Preprint. Proof Ennis, Christopher. 2016. (Always) room for one more. Math Horizons 23(3):8–12. PDF (paywalled). Apollonian circles Dodds, Peter Sheridan, and Joshua S. Weitz. 2002. Packing-limited growth. Physical Review E 65: 056108. Lagarias, Jeffrey C., Colin L. Mallows, and Allan R. Wilks. 2001. Beyond the Descartes circle theorem. https://arxiv.org/abs/math/0101066. (Also published in American Mathematical Monthly, 2002, 109:338–361.) Mackenzie, Dana. 2010. A tisket, a tasket, an Apollonian gasket. American Scientist 98:10–14. https://www.americanscientist.org/article/a-tisket-a-tasket-an-apollonian-gasket. Manna, S. S. 1992. Space filling tiling by random packing of discs. Physica A 187:373–377. Zeta functions Bailey, David H., and Jonathan M. Borwein. 2015. Crandall’s computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet L-series. Applied Mathematics and Computation, 268, 462–477. Borwein, Peter. 1995. An efficient algorithm for the Riemann zeta function. http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf Coffey, Mark W. 2009. An efficient algorithm for the Hurwitz zeta and related functions. Journal of Computational and Applied Mathematics 225:338–346. Hurwitz, Adolf. 1882. Einige Eigenschaften der Dirichletschen Funktionen $$F(s) = \sum \left(\frac{D}{n} \frac{1}{n^s}\right)$$, die bei der Bestimmung der Klassenzahlen binärer quadratischer Formen auftreten. Zeitschrift für Mathematik und Physik 27:86–101. https://gdz.sub.uni-goettingen.de/id/PPN599415665_0027. Oswald, Nicola, and Jörn Steuding. 2015. Aspects of zeta-function theory in the mathematical works of Adolf Hurwitz. https://arxiv.org/abs/1506.00856. Xu, Andy. 2018. Approximating the Hurwitz zeta function. PDF. Posted in computing, featured, mathematics | 9 Comments 737: The MAX Mess Controlled Flight into Terrain is the aviation industry’s term for what happens when a properly functioning airplane plows into the ground because the pilots are distracted or disoriented. What a nightmare. Even worse, in my estimation, is Automated Flight into Terrain, when an aircraft’s control system forces it into a fatal nose dive despite the frantic efforts of the crew to save it. That is the conjectured cause of two recent crashes of new Boeing 737 MAX 8 airplanes. I’ve been trying to reason my way through to an understanding of how those accidents could have happened. Disclaimer: The investigations of the MAX 8 disasters are in an early stage, so much of what follows is based on secondary sources—in other words, on leaks and rumors and the speculations of people who may or may not know what they’re talking about. As for my own speculations: I’m not an aeronautical engineer, or an airframe mechanic, or a control theorist. I’m not even a pilot. Please keep that in mind if you choose to read on. The accidents Early on the morning of October 29, 2018, Lion Air Flight 610 departed Jakarta, Indonesia, with 189 people on board. The airplane was a four-month-old 737 MAX 8—the latest model in a line of Boeing aircraft that goes back to the 1960s. Takeoff and climb were normal to about 1,600 feet, where the pilots retracted the flaps (wing extensions that increase lift at low speed). At that point the aircraft unexpectedly descended to 900 feet. In radio conversations with air traffic controllers, the pilots reported a “flight control problem” and asked about their altitude and speed as displayed on the controllers’ radar screens. Cockpit instruments were giving inconsistent readings. The pilots then redeployed the flaps and climbed to 5,000 feet, but when the flaps were stowed again, the nose dipped and the plane began to lose altitude. Over the next six or seven minutes the pilots engaged in a tug of war with their own aircraft, as they struggled to keep the nose level but the flight control system repeatedly pushed it down. In the end the machine won. The airplane plunged into the sea at high speed, killing everyone aboard. The second crash happened March 8, when Ethiopian Airlines Flight 302 went down six minutes after taking off from Addis Ababa, killing 157. The aircraft was another MAX 8, just two months old. The pilots reported control problems, and data from a satellite tracking service showed sharp fluctuations in altitude. The similarities to the Lion Air crash set off alarm bells: If the same malfunction or design flaw caused both accidents, it might also cause more. Within days, the worldwide fleet of 737 MAX aircraft was grounded. Data recovered since then from the Flight 302 wreckage has reinforced the suspicion that the two accidents are closely related. The grim fate of Lion Air 610 can be traced in brightly colored squiggles extracted from the flight data recorder. (The chart was published in November in a preliminary report from the Indonesian National Committee on Transportation Safety.) The outline of the story is given in the altitude traces at the bottom of the chart. The initial climb is interrupted by a sharp dip; then a further climb is followed by a long, erratic roller coaster ride. At the end comes the dive, as the aircraft plunges 5,000 feet in a little more than 10 seconds. (Why are there two altitude curves, separated by a few hundred feet? I’ll come back to that question at the end of this long screed.) All those ups and downs were caused by movements of the horizontal stabilizer, the small winglike control surface at the rear of the fuselage. The stabilizer controls the airplane’s pitch attitude—nose-up vs. nose-down. On the 737 it does so in two ways. A mechanism for pitch trim tilts the entire stabilizer, whereas push­ing or pulling on the pilot’s control yoke moves the elevator, a hinged tab at the rear of the stabilizer. In either case, moving the trailing edge of the surface upward tends to force the nose of the airplane up, and vice versa. Here we’re mainly concerned with trim changes rather than elevator movements. Commands to the pitch-trim system and their effect on the airplane are shown in three traces from the flight data, which I reproduce here for convenience: The line labeled “trim manual” (light blue) reflects the pilots’ inputs, “trim automatic” (orange) shows commands from the airplane’s electronic systems, and “pitch trim position” (dark blue) represents the tilt of the stabilizer, with higher position on the scale denoting a nose-up command. This is where the tug of war between man and machine is clearly evident. In the latter half of the flight, the automatic trim system repeatedly commands nose down, at intervals of roughly 10 seconds. In the breaks between those automated commands, the pilots dial in nose-up trim, using buttons on the control yoke. In response to these conflicting commands, the position of the horizontal stabilizer oscillates with a period of 15 or 20 seconds. The see-sawing motion continues for at least 20 cycles, but toward the end the unrelenting automatic nose-down adjustments prevail over the briefer nose-up commands from the pilots. The stabilizer finally reaches its limiting nose-down deflection and stays there as the airplane plummets into the sea. Angle of attack What’s to blame for the perverse behavior of the automatic pitch trim system? The accusatory finger is pointing at something called MCAS, a new feature of the 737 MAX series. MCAS stands for Maneuvering Characteristics Augmentation System—an im­pressively polysyllabic name that tells you nothing about what the thing is or what it does. As I understand it, MCAS is not a piece of hardware; there’s no box labeled MCAS in the airplane’s electronic equipment bays. MCAS consists entirely of software. It’s a program running on a computer. MCAS has just one function. It is designed to help prevent an aerodynamic stall, a situation in which an airplane has its nose pointed up so high with respect to the surrounding airflow that the wings can’t keep it aloft. A stall is a little like what happens to a bicyclist climbing a hill that keeps getting steeper and steeper: Eventually the rider runs out of oomph, wobbles a bit, and then rolls back to the bottom. Pilots are taught to recover from stalls, but it’s not a skill they routinely practice with a planeful of passengers. In commercial aviation the emphasis is on avoiding stalls—forestalling them, so to speak. Airliners have mechanisms to detect an imminent stall and warn the pilot with lights and horns and a “stick shaker” that vibrates the control yoke. On Flight 610, the captain’s stick was shaking almost from start to finish. Some aircraft go beyond mere warnings when a stall threatens. If the aircraft’s nose continues to pitch upward, an automated system intervenes to push it back down—if necessary overriding the manual control inputs of the pilot. MCAS is designed to do exactly this. It is armed and ready whenever two criteria are met: The flaps are up (generally true except during takeoff and landing) and the airplane is under manual control (not autopilot). Under these conditions the system is triggered whenever an aerodynamic quantity called angle of attack, or AoA, rises into a dangerous range. Angle of attack is a concept subtle enough to merit a diagram:Adapted from Lisa R. Le Vie, Review of Research on Angle-of-Attack Indi­cator Effectiveness. The various angles at issue are rotations of the aircraft body around the pitch axis, a line parallel to the wings, perpendicular to the fuselage, and passing through the airplane’s center of gravity. If you’re sitting in an exit row, the pitch axis might run right under your seat. Rotation about the pitch axis tilts the nose up or down. Pitch attitude is defined as the angle of the fuselage with respect to a horizontal plane. The flight-path angle is measured between the horizontal plane and the aircraft’s velocity vector, thus showing how steeply it is climbing or descending. Angle of attack is the difference between pitch attitude and flight-path angle. It is the angle at which the aircraft is moving through the surrounding air (assuming the air itself is motionless, i.e., no wind). AoA affects both lift (the upward force opposing the downward tug of gravity) and drag (the dissipative force opposing forward motion and the thrust of the engines). As AoA increases from zero, lift is enhanced because of air impinging on the underside of the wings and fuselage. For the same reason, however, drag also increases. As the angle of attack grows even steeper, the flow of air over the wings becomes turbulent; beyond that point lift diminishes but drag continues increasing. That’s where the stall sets in. The critical angle for a stall depends on speed, weight, and other factors, but usually it’s no more than 15 degrees. Neither the Lion Air nor the Ethiopian flight was ever in danger of stalling, so if MCAS was activated, it must have been by mistake. The working hypothesis mentioned in many press accounts is that the system received and acted upon erroneous input from a failed AoA sensor. A sensor to measure angle of attack is conceptually simple. It’s essentially a weather­vane poking out into the airstream. In the photo below, the angle-of-attack sensor is the small black vane just forward of the “737 MAX” legend. Hinged at the front, the vane rotates to align itself with the local airflow and generates an electrical signal that rep­resents the vane’s angle with respect to the axis of the fuselage. The 737 MAX has two angle-of-attack vanes, one on each side of the nose. (The protruding devices above the AoA vane are pitot tubes, used to measure air speed. Another device below the word MAX is probably a temperature sensor.) Angle of attack was not among the variables displayed to the pilots of the Lion Air 737, but the flight data recorder did capture signals derived from the two AoA sensors: There’s something dreadfully wrong here. The left sensor is indicating an angle of attack about 20 degrees steeper than the right sensor. That’s a huge discrepancy. There’s no plausible way those disparate readings could reflect the true state of the airplane’s motion through the air, with the left side of the nose pointing sky-high and the right side near level. One of the measurements must be wrong, and the higher reading is the suspect one. If the true angle of attack ever reached 20 degrees, the airplane would already be in a deep stall. Unfortunately, on Flight 610 MCAS was taking data only from the left-side AoA sensor. It interpreted the nonsensical measurement as a valid indicator of aircraft attitude, and worked relentlessly to correct it, up to the very moment the airplane hit the sea. Cockpit automation The tragedies in Jakarta and Addis Ababa are being framed as a cautionary tale of automation run amok, with computers usurping the authority of pilots. The Washington Post editorialized: A second fatal airplane accident involving a Boeing 737 MAX 8 may have been a case of man vs. machine…. The debacle shows that regulators should apply extra review to systems that take control away from humans when safety is at stake. Tom Dieusaert, a Belgian journalist who writes often on aviation and computation, offered this opinion: What can’t be denied is that the Boeing of Flight JT610 had serious computer problems. And in the hi-tech, fly-by-wire world of aircraft manufacturers, where pilots are reduced to button pushers and passive observers, these accidents are prone to happen more in the future. The button-pushing pilots are particularly irate. Gregory Travis, who is both a pilot and software developer, summed up his feelings in this acerbic comment: “Raise the nose, HAL.” “I’m sorry, Dave, I can’t do that.” Even Donald Trump tweeted on the issue: Airplanes are becoming far too complex to fly. Pilots are no longer needed, but rather computer scientists from MIT. I see it all the time in many products. Always seeking to go one unnecessary step further, when often old and simpler is far better. Split second decisions are…. ….needed, and the complexity creates danger. All of this for great cost yet very little gain. I don’t know about you, but I don’t want Albert Einstein to be my pilot. I want great flying professionals that are allowed to easily and quickly take control of a plane! There’s considerable irony in the complaint that the 737 is too automated; in many respects the aircraft is in fact quaintly old-fashioned. The basic design goes back more than 50 years, and even in the latest MAX models quite a lot of 1960s technology survives. The primary flight controls are hydraulic, with a spider web of high-pressure tubing running directly from the control yokes in the cockpit to the ailerons, elevator, and rudder. If the hydraulic systems should fail, there’s a purely mechanical backup, with cables and pulleys to operate the various control surfaces. For stabilizer trim the primary actuator is an electric motor, but again there’s a mechanical fallback, with crank wheels near the pilots’ knees pulling on cables that run all the way back to the tail. Other aircraft are much more dependent on computers and electronics. The 737′s principal competitor, the Airbus A320, is a thoroughgoing fly-by-wire vehicle. The pilot flies the computer, and the computer flies the airplane. Specifically, the pilot decides where to go—up, down, left, right—but the computer decides how to get there, choosing which control surfaces to deflect and by how much. Boeing’s own more recent designs, the 777 and 787, also rely on digital controls. Indeed, the latest models from both companies go a step beyond fly-by-wire to fly-by-network. Most of the communication from sensors to computers and onward to control surfaces consists of digital packets flowing through a variant of Ethernet. The airplane is a computer peripheral. Thus if you want to gripe about the dangers and indignities of automation on the flight deck, the 737 is not the most obvious place to start. And a Luddite campaign to smash all the avionics and put pilots back in the seat of their pants would be a dangerously off-target response to the current predicament. There’s no question the 737 MAX has a critical problem. It’s a matter of life and death for those who would fly in it and possibly also for the Boeing Company. But the problem didn’t start with MCAS. It started with earlier decisions that made MCAS necessary. Furthermore, the problem may not end with the remedy that Boeing has proposed—a software update that will hobble MCAS and leave more to the discretion of pilots. Maxing out the 737 The 737 flew its first passengers in 1968. It was (and still is) the smallest member of the Boeing family of jet airliners, and it is also the most popular by far. More than 10,000 have been sold, and Boeing has orders for another 4,600. Of course there have been changes over the years, especially to engines and instruments. A 1980s update came to be known as 737 Classic, and a 1997 model was called 737 NG, for “next generation.” (Now, with the MAX, the NG has become the previous generation.) Through all these revisions, however, the basic structure of the airframe has hardly changed. Ten years ago, it looked like the 737 had finally come to the end of its life. Boeing announced it would develop an all-new design as a replacement, with a hull built of lightweight composite materials rather than aluminum. Competitive pressures forced a change of course. Airbus had a head start on the A320neo, an update that would bring more efficient engines to their entry in the same market segment. The revised Airbus would be ready around 2015, whereas Boeing’s clean-slate project would take a decade. Customers were threatening to defect. In particular, American Airlines—long a Boeing loyalist—was negotiating a large order of A320neos. In 2011 Boeing scrapped the plan for an all-new design and elected to do the same thing Airbus was doing: bolt new engines onto an old airframe. This would eliminate most of the up-front design work, as well as the need to build tooling and manufacturing facilities. Testing and certification by the FAA would also go quicker, so that the first deliveries might be made in five or six years, not too far behind Airbus. A 737-800 (a pre-MAX model) burns about 800 gallons of jet fuel per hour aloft. That comes to2,000 at $2.50 per gallon. If the airplane flies 10 hours a day, the annual fuel bill is$7.3 million. Fourteen percent of that is just over $1 million.The new engines mated to the 737 promised a 14 percent gain in fuel efficiency, which might save an airline a million dollars a year in operating costs. The better fuel economy would also increase the airplane’s range. And to sweeten the deal Boeing proposed to keep enough of the airframe unchanged that the new model would operate under the same “type certificate” as the old one. A pilot qualified to fly the 737 NG could step into the MAX without extensive retraining. Sources: (left) Bryan via Wikimedia, CC BY 2.0; (right) Steve Lynes via Wikimedia, CC BY 2.0. The original 1960s 737 had two cigar-shaped engines, long and skinny, tucked up under the wings (left photo above). Since then, jet engines have grown fat and stubby. They derive much of their thrust not from the jet exhaust coming out of the tailpipe but from “bypass” air moved by a large-diameter fan. Such engines would scrape on the ground if they were mounted under the wings of the 737; instead they are perched on pylons that extend forward from the leading edge of the wing. The engines on the MAX models (right photo) are the fattest yet, with a fan 69 inches in diameter. Compared with the NG series, the MAX engines are pushed a few inches farther forward and hang a few inches lower. A New York Times article by David Gelles, Natalie Kitroeff, Jack Nicas, and Rebecca R. Ruiz describes the plane’s development as hurried and hectic. Months behind Airbus, Boeing had to play catch-up. The pace of the work on the 737 Max was frenetic, according to current and former employees who spoke with The New York Times…. Engineers were pushed to submit technical drawings and designs at roughly double the normal pace, former employees said. The Times article also notes: “Although the project had been hectic, current and former employees said they had finished it feeling confident in the safety of the plane.” Pitch instability Sometime during the development of the MAX series, Boeing got an unpleasant surprise. The new engines were causing unwanted pitch-up movements under certain flight con­ditions. When I first read about this problem, soon after the Lion Air crash, I found the following explanation is an article by Sean Broderick and Guy Norris in Aviation Week and Space Technology (Nov. 26–Dec. 9, 2018, pp. 56–57): Like all turbofan-powered airliners in which the thrust lines of the engines pass below the center of gravity (CG), any change in thrust on the 737 will result in a change of flight path angle caused by the vertical component of thrust. In other words, the low-slung engines not only push the airplane forward but also tend to twirl it around the pitch axis. It’s like a motorcycle doing wheelies. Because the MAX engines are mounted farther below and in front of the center of gravity, they act through a longer lever arm and cause more severe pitch-up motions. I found more detail on this effect in an earlier Aviation Week article, a 2017 pilot report by Fred George, describing his first flight at the controls of the new MAX 8. The aircraft has sufficient natural speed stability through much of its flight envelope. But with as much as 58,000 lb. of thrust available from engines mounted well below the center of gravity, there is pronounced thrust-versus-pitch coupling at low speeds, especially with aft center of gravity (CG) and at light gross weights. Boeing equips the aircraft with a speed-stability augmen­tation function that helps to compensate for the coupling by automatically trimming the horizontal stabilizer according to indicated speed, thrust lever position and CG. Pilots still must be aware of the effect of thrust changes on pitching moment and make purposeful control-wheel and pitch-trim inputs to counter it. The reference to an “augmentation function” that works by “automatically trimming the horizontal stabilizer” sounded awfully familiar, but it turns out this is not MCAS. The system that compensates for thrust-pitch coupling is known as speed-trim. Like MCAS, it works “behind the pilot’s back,” making adjustments to control surfaces that were not directly commanded. There’s yet another system of this kind called mach-trim that silently corrects a different pitch anomally when the aircraft reaches transonic speeds, at about mach 0.6. Neither of these systems is new to the MAX series of aircraft; they have been part of the control algorithm at least since the NG came out in 1997. MCAS runs on the same computer as speed-trim and mach-trim and is part of the same software system, but it is a distinct function. And according to what I’ve been reading in the past few weeks, it addresses a different problem—one that seems more sinister. Most aircraft have the pleasant property of static stability. When an airplane is properly trimmed for level flight, you can let go of the controls—at least briefly—and it will continue on a stable path. Moreover, if you pull back on the control yoke to point the nose up, then let go again, the pitch angle should return to neutral. The layout of the airplane’s various airfoil surfaces accounts for this behavior. When the nose goes up, the tail goes down, pushing the underside of the horizontal stabilizer into the airstream. The pressure of the air against this tail surface provides a restoring force that brings the tail back up and the nose back down. (That’s why it’s called a stabilizer!) This negative feedback loop is built in to the structure of the airplane, so that any departure from equilibrium creates a force that opposes the disturbance. However, the tail surface, with its helpful stablizing influence, is not the only structure that affects the balance of aerodynamic forces. Jet engines are not designed to contribute lift to the airplane, but at high angles of attack they can do so, as the airstream impinges on the lower surface of each engine’s outer covering, or nacelle. When the engines are well forward of the center of gravity, the lift creates a pitch-up turning moment. If this moment exceeds the counterbalancing force from the tail, the aircraft is unstable. A nose-up attitude generates forces that raise the nose still higher, and positive feedback takes over. Is the 737 MAX vulnerable to such runaway pitch excursions? The possibility had not occurred to me until I read a commentary on MCAS on the Boeing 737 Technical Site, a web publication produced by Chris Brady, a former 737 pilot and flight instructor. He writes: MCAS is a longitudinal stability enhancement. It is not for stall prevention or to make the MAX handle like the NG; it was introduced to counteract the non-linear lift of the LEAP-1B engine nacelles and give a steady increase in stick force as AoA increases. The LEAP engines are both larger and relocated slightly up and forward from the previous NG CFM56-7 engines to accommodate their larger fan diameter. This new location and size of the nacelle cause the vortex flow off the nacelle body to produce lift at high AoA; as the nacelle is ahead of the CofG this lift causes a slight pitch-up effect (ie a reducing stick force) which could lead the pilot to further increase the back pressure on the yoke and send the aircraft closer towards the stall. This non-linear/reducing stick force is not allowable underFAR = Federal Air Regulations. Part 25 deals with airworthiness standards for transport category airplanes. FAR §25.173 “Static longitudinal stability”. MCAS was therefore introduced to give an automatic nose down stabilizer input during steep turns with elevated load factors (high AoA) and during flaps up flight at airspeeds approaching stall. Brady cites no sources for this statement, and as far as I know Boeing has neither confirmed nor denied. But Aviation Week, which earlier mentioned the thrust-pitch linkage, has more recently (issue of March 20) gotten behind the nacelle-lift instability hypothesis: The MAX’s larger CFM Leap 1 engines create more lift at high AOA and give the aircraft a greater pitch-up moment than the CFM56-7-equipped NG. The MCAS was added as a certification requirement to minimize the handling difference between the MAX and NG. Assuming the Brady account is correct, an interesting question is when Boeing noticed the instability. Were the designers aware of this hazard from the outset? Did it emerge during early computer simulations, or in wind tunnel testing of scale models? A story by Dominic Gates in the Seattle Times hints that Boeing may not have recognized the severity of the problem until flight tests of the first completed aircraft began in 2015. According to Gates, the safety analysis that Boeing submitted to the FAA specified that MCAS would be allowed to move the horizontal stabilizer by no more than 0.6 degree. In the airplane ultimately released to the market, MCAS can go as far as 2.5 degrees, and it can act repeatedly until reaching the mechanical limit of motion at about 5 degrees. Gates writes: That limit was later increased after flight tests showed that a more powerful movement of the tail was required to avert a high-speed stall, when the plane is in danger of losing lift and spiraling down. The behavior of a plane in a high angle-of-attack stall is difficult to model in advance purely by analysis and so, as test pilots work through stall-recovery routines during flight tests on a new airplane, it’s not uncommon to tweak the control software to refine the jet’s performance. The high-AoA instability of the MAX appears to be a property of the aerodynamic form of the entire aircraft, and so a direct way to suppress it would be to alter that form. For example, enlarging the tail surface might restore static stability. But such airframe modifications would have delayed the delivery of the airplane, especially if the need for them was discovered only after the first prototypes were already flying. Structural changes might also jeopardize inclusion of the new model under the old type certificate. Modifying software instead of aluminum must have looked like an attractive alternative. Someday, perhaps, we’ll learn how the decision was made. By the way, according to Gates, the safety document filed with the FAA specifying a 0.6 degree limit has yet to be amended to reflect the true range of MCAS commands. Flying while unstable Instability is not necessarily the kiss of death in an airplane. There have been at least a few successful unstable designs, starting with the 1903 Wright Flyer. The Wright brothers deliberately put the horizontal stabilizer in front of the wing rather than behind it because their earlier experiments with kites and gliders had shown that what we call stability can also be described as sluggishness. The Flyer’s forward control surfaces (known as canards) tended to amplify any slight nose-up or nose-down motions. Maintaining a steady pitch attitude demanded high alertness from the pilot, but it also allowed the airplane to respond more quickly when the pilot wanted to pitch up or down. (The pros and cons of the design are reviewed in a 1984 paper by Fred E. C. Culick and Henry R. Jex.) Orville at the controls, Wilbur running alongside, at Kitty Hawk on December 17, 1903. In this view we are seeing the airplane from the stern. The canards—dual adjustable horizontal surfaces at the front—seem to be calling for nose-up pitch. (Photo from WikiMedia. Another dramatically unstable aircraft was the Grumman X-29, a research platform designed in the 1980s. The X-29 had its wings on backwards; to make matters worse, the primary surfaces for pitch control were canards mounted in front of the wings, as in the Wright Flyer. The aim of this quirky project was to explore designs with exceptional agility, sacrificing static stability for tighter maneuvering. No unaided human pilot could have mastered such a twitchy vehicle. It required a digital fly-by-wire system that sampled the state of the airplane and adjusted the control surfaces up to 80 times per second. The controller was successful—perhaps too much so. It allowed the airplane to be flown safely, but in taming the instability it also left the plane with rather tame handling characteristics. I have a glancing personal connection with the X-29 project. In the 1980s I briefly worked as an editor with members of the group at Honeywell who designed and built the X-29 control system. I helped prepare publications on the control laws and on their implementation in hardware and software. That experience taught me just enough to recognize something odd about MCAS: It is way too slow to be suppressing aerodynamic instability in a jet aircraft. Whereas the X-29 controller had a response time of 25 milliseconds, MCAS takes 10 seconds to move the 737 stabilizer through a 2.5-degree adjustment. At that pace, it cannot possibly keep up with forces that tend to flip the nose upward in a positive feedback loop. There’s a simple explanation. MCAS is not meant to control an unstable aircraft. It is meant to restrain the aircraft from entering the regime where it becomes unstable. This is the same strategy used by other mechanisms of stall prevention—intervening before the angle of attack reaches the critical point. However, if Brady is correct about the instability of the 737 MAX, the task is more urgent for MCAS. Instability implies a steep and slippery slope. MCAS is a guard rail that bounces you back onto the road when you’re about to drive over the cliff. Which brings up the question of Boeing’s announced plan to fix the MCAS problem. Reportedly, the revised system will not keep reactivating itself so persistently, and it will automatically disengage if it detects a large difference between the two AoA sensors. These changes should prevent a recurrence of the recent crashes. But do they provide adequate protection against the kind of mishap that MCAS was designed to prevent in the first place? With MCAS shut down, either manually or automatically, there’s nothing to stop an unwary or misguided pilot from wandering into the corner of the flight envelope where the MAX becomes unstable. Without further information from Boeing, there’s no telling how severe the instability might be—if indeed it exists at all. The Brady article at the Boeing 737 Technical Site implies the problem is partly pilot-induced. Normally, to make the nose go higher and higher you have to pull harder and harder on the control yoke. In the unstable region, however, the resistance to pulling suddenly fades, and so the pilot may unwittingly pull the yoke to a more extreme position. Is this human interaction a necessary part of the instability, or is it just an exacer­bating factor? In other words, without the pilot in the loop, would there still be positive feedback causing runaway nose-up pitch? I have yet to find answers. Another question: If the root of the problem is a deceptive change in the force resisting a nose-up movements of the control yoke, why not address that issue directly? The elevator feel computer and the elevator feel and centering unit pro­vide “fake” forces to the pilot’s control yoke. Figure borrowed from B737 NG Flight controls, a presentation by theoryce. The presentation is for the 737 NG, not the MAX series; it’s possible the architecture has changed.In the 737 (and most other large aircraft) the forces that the pilot “feels” through the control yoke are not simple reflections of the aerodynamic forces acting on the elevator and other control surfaces. The feedback forces are largely synthetic, generated by an “elevator feel computer” and an “elevator feel and centering unit,” devices that monitor the state of the airplane and gen­erate appro­priate hydraulic pressures push­ing the yoke one way or another. Those systems could have been given the addi­tional task of maintaining or increasing back force on the yoke when the angle of attack approaches the instability. Artificially en­hanced resis­tance is already part of the stall warning system. Why not extend it to MCAS? (There may be a good answer; I just don’t know it.) Where’s the off switch? Even after the spurious activation of MCAS on Lion Air 610, the crash and the casualties would have been avoided if the pilots had simply turned the damn thing off. Why didn’t they? Apparently because they had never heard of MCAS, and didn’t know it was installed on the airplane they were flying, and had not received any instruction on how to disable it. There’s no switch or knob in the cockpit labeled “MCAS ON/OFF.” The Flight Crew Operation Manual does not mention it (except in a list of abbreviations), and neither did the transitional training program the pilots had completed before switching from the 737 NG to the MAX. The training consisted of either one or two hours (reports differ) with an iPad app. Boeing’s explanation of these omissions was captured in a Wall Street Journal story: One high-ranking Boeing official said the company had decided against disclos­ing more details to cockpit crews due to concerns about inundating average pilots with too much information—and significantly more technical data—than they needed or could digest. To call this statement disingenuous would be disingenuous. What it is is preposterous. In the first place, Boeing did not withhold “more details”; they failed to mention the very existence of MCAS. And the too-much-information argument is silly. I don’t have access to the Flight Crew Operation Manual for the MAX, but the NG edition runs to more than 1,300 pages, plus another 800 for the Quick Reference Handbook. A few paragraphs on MCAS would not have sunk any pilot who wasn’t already drowning in TMI. Moreover, the manual carefully documents the speed-trim and mach-trim features, which seem to fall in the same category as MCAS: They act autonomously, and offer the pilot no direct interface for monitoring or adjusting them. In the aftermath of the Lion Air accident, Boeing stated that the procedure for disabling MCAS was spelled out in the manual, even though MCAS itself wasn’t mentioned. That procedure is given in a checklist for “runaway stabilizer trim.” It is not complicated: Hang onto the control yoke, switch off the autopilot and autothrottles if they’re on; then, if the problem persists, flip two switches labeled “STAB TRIM” to the “CUTOUT” position. Only the last step will actually matter in the case of an MCAS malfunction. This checklist is considered a “memory item”; pilots must be able to execute the steps without looking it up in the handbook. The Lion Air crew should certainly have been familiar with it. But could they recognize that it was the right checklist to apply in an airplane whose behavior was unlike anything they had seen in their training or previous 737 flying experience? According to the handbook, the condition that triggers use of the runaway checklist is “Uncommanded stabilizer trim movement occurs continuously.” The MCAS commands were not continuous but repetitive, so some leap of inference would have been needed to make this diagnosis. By the time of the Ethiopian crash, 737 pilots everywhere knew all about MCAS and the procedure for disabling it. A preliminary report issued last week by Ethiopian Airlines indicates that after a few minutes of wrestling with the control yoke, the pilots on Flight 302 did invoke the checklist procedure, and moved the STAB TRIM switches to CUTOUT. The stabilizer then stopped responding to MCAS nose-down commands, but the pilots were unable to regain control of the airplane. It’s not entirely clear why they failed or what was going on in the cockpit in those last minutes. One factor may be that the cutout switch disables not only automatic pitch trim movements but also manual ones requested through the buttons on the control yoke. The switch cuts all power to the electric motor that moves the stabilizer. In this situation the only way to adjust the trim is to turn the hand crank wheels near the pilots’ knees. During the crisis on Flight 302 that mechanism may have been too slow to correct the trim in time, or the pilots may have been so fixated on pulling the control yoke back with maximum force that they did not try the manual wheels. It’s also possible that they flipped the switches back to the NORMAL setting, restoring power to the stabilizer motor. The report’s narrative doesn’t mention this possibility, but the graph from the flight data recorder suggests it (see below). The single point of failure There’s room for debate on whether the MCAS system is a good idea when it is operating correctly, but when it activates mistakenly and sends an airplane diving into the sea, no one would defend it. By all appearances, the rogue behavior in both the Lion Air and the Ethiopian accidents was triggered by a malfunction in a single sensor. That’s not supposed to happen in aviation. It’s unfathomable that any aircraft manufacturer would knowingly build a vehicle in which the failure of a single part would lead to a fatal accident. Protection against single failures comes from redundancy, and the 737 is so committed to this principle that it almost amounts to two airplanes wrapped up in a single skin. Aircraft that rely more heavily on automation generally have three of everything—sensors, computers, and actuators.The cockpit has stations for two pilots, who look at separate sets of instruments and operate separate sets of controls. The left and right instrument panels receive signals from separate sets of sensors, and those signals are processed by separate computers. Each side of the cockpit has its own inertial guidance system, its own navigation computer, its own autopilot. There are two electric power supplies and two hydraulic systems—plus mechanical backups in case of a dual hydraulic failure. The two control yokes normally move in unison—they are linked under the floor—but if one yoke should get stuck, the connection can be broken, allowing the other pilot to continue flying the airplane. There’s one asterisk in this roster of redundancy: A device called the flight control computer, or FCC, apparently gets special treatment. There are two FCCs, but according to the Boeing 737 Technical Site only one of them operates during any given flight. All the other duplicated components run in parallel, receiving independent inputs, doing independent computations, emitting independent control actions. But for each flight just one FCC does all the work, and the other is put on standby. The scheme for choosing the active computer seems strangely arbitrary. Each day when the airplane is powered up, the left side FCC gets control for the first flight, then the right side unit takes over for the second flight of the day, and the two sides alternate until the power is shut off. After a restart, the alternation begins again with the left FCC. Aspects of this scheme puzzle me. I don’t understand why redundant FCC units are treated differently from other components. If one FCC dies, does the other automatically take over? Can the pilots switch between them in flight? If so, would that be an effective way to combat MCAS misbehavior? I’ve tried to find answers in the manuals, but I don’t trust my interpretation of what I read. I’ve also had a hard time learning anything about the FCC itself. I don’t know who makes it, or what it looks like, or how it is programmed. On a website called Closet Wonderfuls an item identified as a 737 flight control computer is on offer for$43.82, with free shipping.A website called Airframer lists many suppliers of parts and materials for the 737, but there’s no entry for a flight control computer. It has a Honeywell label. I’m tempted, but I’m pretty sure this is not the unit installed in the latest MAX models. I’ve learned that the FCC was once the FCE, for flight control elec­tronics, suggesting it was an analog device, doing its integrations and differ­entiations with capacitors and resis­tors. By now I’m sure the FCC has caught up with the digital age, but it might still be special-purpose, custom-built hardware. Or it might be an off-the-shelf Intel CPU in a fancy box, maybe even running Linux or Windows. I just don’t know.

In the context of the MAX crashes, the flight control computer is important for two reasons. First, it’s where MCAS lives; this is the computer on which the MCAS software runs. Second, the curious procedure for choosing a different FCC on alternating flights also winds up choosing which AoA sensor is providing input to MCAS. The left and right sensors are connected to the corresponding FCCs.

If the two FCCs are used in alternation, that raises an interesting question about the history of the aircraft that crashed in Indonesia. The preliminary crash report describes trouble with various instruments and controls on five flights over four days (including the fatal flight). All of the problems were on the left side of the aircraft or involved a dis­agreement between the left and right sides.
The flight in the gray row is not mentioned in the preliminary report, but the airplane had to get from Manado to Denpasar for the following day’s flight.

date route trouble reports maintenance
Oct 26 Tianjin → Manado left side: no airspeed
or altitude indications
test left Stall Management and
Yaw Damper computer; passed
? Manado → Denpasar ? ?
Oct 27 Denpasar → Manado left side: no airspeed
or altitude indications

speed trim and mach trim
warning lights
test left Stall Management and
Yaw Damper computer; failed

reset left Air Data and Inertial
Reference Unit

retest left Stall Management and
Yaw Damper computer; passed

clean electrical connections
Oct 27 Manado → Denpasar left side: no airspeed
or altitude indications

speed trim and mach trim
warning lights

autothrottle disconnect
test left Stall Management and
Yaw Damper computer; failed

reset left Air Data and Inertial
Reference Unit

replace left AoA sensor
Oct 28 Denpasar → Jakarta left/right disagree warning
on airspeed and altitude

stick shaker

[MCAS activation]
flush left pitot tube
and static port

clean electrical connectors
on elevator “feel” computer
Oct 29 Jakarta → Pangkal Pinang stick shaker

[MCAS activation]

Which of the five flights had the left-side FCC as active computer? The final two flights (red), where MCAS activated, were both first-of-the-day flights and so presumably under control of the left FCC. For the rest it’s hard to tell, especially since maintenance operations may have entailed full shutdowns of the aircraft, which would have reset the alternation sequence.

The revised MCAS software will reportedly consult signals from both AoA sensors. What will it do with the additional information? Only one clue has been published so far: If the readings differ by more than 5.5 degrees, MCAS will shut down. What if the readings differ by 4 or 5 degrees? A recent paper by Daniel Ossmann of the German Aerospace Center dis­cusses algorithmic detection of fail­ures in AoA sensors.Which sensor will MCAS choose to believe? Conservative (or pessimistic) engineering practice would seem to favor the higher reading, in order to provide better protection against instability and a stall. But that choice also raises the risk of dangerous “corrections” mandated by a faulty sensor.

The present MCAS system, with its alternating choice of left and right, has a 50 percent chance of disaster when a single random failure causes an AoA sensor to spew out falsely high data. With the same one-sided random failure, the updated MCAS will have a 100 percent chance of ignoring a pilot’s excursion into stall territory. Is that an improvement?

The broken sensor

Although a faulty sensor should not bring down an airplane, I would still like to know what went wrong with the AoA vane.

It’s no surprise that AoA sensors can fail. They are mechanical devices operating in a harsh environment: winds exceeding 500 miles per hour and temperatures below –40. A common failure mode is a stuck vane, often caused by ice (despite a built-in de-icing heater). But a seized vane would produce a constant output, regardless of the real angle of attack, which is not the symptom seen in Flight 610. The flight data recorder shows small fluctuations in the signals from both the left and the right instruments. Furthermore, the jiggles in the two curves are closely aligned, suggesting they are both tracking the same movements of the aircraft. In other words, the left-hand sensor appears to be functioning; it’s just giving measurements offset by a constant deviation of roughly 20 degrees.

Is there some other failure mode that might produce the observed offset? Sure: Just bend the vane by 20 degrees. Maybe a catering truck or an airport jetway blundered into it. Another creative thought is that the sensor might have been installed wrong, with the entire unit rotated by 20 degrees. Several writers on a website called the Professional Pilots Rumour Network explored this possibility, but they ultimately concluded it was impossible. The manufacturer, doubtless aware of the risk, placed the mounting screws and locator pins asymmetrically, so the unit will only go into the hull opening one way.

You might get the same effect through an assembly error during the manufacture of the sensor. The vane could be incorrectly attached to the shaft, or else the internal transducer that converts angular position into an electrical signal might be mounted wrong. Did the designers also ensure that such mistakes are impossible? I don’t know; I haven’t been able to find any drawings or photographs of the sensor’s innards.

Looking for other ideas about what might have gone wrong, I made a quick, scattershot survey of FAA airworthiness directives that call for servicing or replacing AoA sensors. I found dozens of them, including several that discuss the same sensor installed on the 737 MAX (the Rosemount 0861). But none of the reports I read describes a malfunction that could cause a consistent 20-degree error.

For a while I thought that the fault might lie not in the sensor itself but farther along the data path. It could be something as simple as a bad cable or connector. Signals from the AoA sensor go to the Air Data and Inertial Reference Unit (ADIRU), where the sine and cosine components are combined and digitized to yield a number representing the measured angle of attack. The ADIRU also receives inputs from other sensors, including the pitot tubes for measuring airspeed and the static ports for air pressure. And it houses the gyroscopes and accelerometers of an inertial guidance system, which can keep track of aircraft motion without reference to external cues. (There’s a separate ADIRU for each side of the airplane.) Maybe there was a problem with the digitizer—a stuck bit rather than a stuck vane.

Further information has undermined this idea. For one thing, the AoA sensor removed by the Lion Air maintenance crew on October 27 is now in the hands of investigators. According to news reports, it was “deemed to be defective,” though I’ve heard no hint of what the defect might be. Also, it turns out that one element of the control system, the Stall Management and Yaw Damper (SMYD) computer, receives the raw sine and cosine voltages directly from the sensor, not a digitized angle calculated by the ADIRU. It is the SMYD that controls the stick-shaker function. On both the Lion Air and the Ethiopian flights the stick shaker was active almost continuously, so those undigitized sine and cosine voltages must have been indicating a high angle of attack. In other words the error already existed before the signals reached the ADIRU.

I’m still stumped by the fixed angular offset in the Lion Air data, but the question now seems a little less important. The release of the preliminary report on Ethiopian Flight 302 shows that the left-side AoA sensor on that aircraft also failed badly, but in a way that looks totally different. Here are the relevant traces from the flight data recorder:

The readings from the AoA sensors are the uppermost lines, red for the left sensor and blue for the right. At the left edge of the graph they differ somewhat when the airplane has just begun to move, but they fall into close coincidence once the roll down the runway has built up some speed. At takeoff, however, they suddenly diverge dramtically, as the left vane begins reading an utterly implausible 75 degrees nose up. Later it comes down a few degrees but otherwise shows no sign of the ripples that would suggest a response to airflow. At the very end of the flight there are some more unexplained excursions.

By the way, in this graph the light blue trace of automatic trim commands offers another clue to what might have happened in the last moments of Flight 302. Around the middle of the graph, the STAB TRIM switches were pulled, with the result that an automatic nose-down command had no effect on the stabilizer position. But at the far right, another automatic nose-down command does register in the trim-position trace, suggesting that the cutout switches may have been turned on again.

Still more stumpers

There’s so much I still don’t understand.

Puzzle 1. If the Lion Air and Ethiopian accidents were both caused by faulty AoA sensors, then there were three parts with similar defects in brand new aircraft (including the replacement sensor installed by Lion Air on October 27). A recent news item says the replacement was not a new part but one that had been refurbished by a Florida shop called XTRA Aerospace. This fact offers us somewhere else to point the accusatory finger, but presumably the two sensors installed by Boeing were not retreads, so XTRA can’t be blamed for all of them.

There are roughly 400 MAX aircraft in service, with 800 AoA sensors. Is a failure rate of 3 out of 800 unusual or unacceptable? Does that judgment depend on whether or not it’s the same defect in all three cases?

Puzzle 2. Let’s look again at the traces for pitch trim and angle of attack in the Lion Air 610 data. The conflicting manual and automatic commands in the second half of the flight have gotten lots of attention, but I’m also baffled by what was going on in the first few minutes.

During the roll down the runway, the pitch trim system was set near its maximum pitch-up position (dark blue line). Immediately after takeoff, the automatic trim system began calling for further pitch-up movement, and the stabilizer probably reached its mechanical limit. At that point the pilots manually trimmed it in the pitch-down direction, and the automatic system replied with a rapid sequence of up adjustments. In other words, there was already a tug-of-war underway, but the pilots and the automated controls were pulling in directions opposite to those they would choose later on. All this happened while the flaps were still deployed, which means that MCAS could not have been active. Some other element of the control system must have been issuing those automatic pitch-up orders. Deepening the mystery, the left side AoA sensor was already feeding its spurious high readings to the left-side flight control computer. If the FCC was acting on that data, it should not have been commanding nose-up trim.

Puzzle 3. The AoA readings are not the only peculiar data in the chart from the Lion Air preliminary report. Here are the altitude and speed traces:

The left-side altitude readings (red) are low by at least a few hundred feet. The error looks like it might be multiplicative rather than additive, perhaps 10 percent. The left and right computed airspeeds also disagree, although the chart is too squished to allow a quantitative comparison. It was these discrepancies that initially upset the pilots of Flight 610; they could see them on their instruments. (They had no angle of attack indicators in the cockpit, so that conflict was invisible to them.)

Altitude, airspeed, and angle of attack are all measured by different sensors. Could they all have gone haywire at the same time? Or is there some common point of failure that might explain all the weird behavior? In particular, is it possible a single wonky AoA sensor caused all of this havoc? My guess is yes. The sensors for altitude and airspeed and even temperature are influenced by angle of attack. The measured speed and pressure are therefore adjusted to compensate for this confounding variable, using the output of the AoA sensor. That output was wrong, and so the adjustments allowed one bad data stream to infect all of the air data measurements.

Man or machine

Six months ago, I was writing about another disaster caused by an out-of-control control system. In that case the trouble spot was a natural gas distribution network in Massa­chusetts, where a misconfigured pressure-regulating station caused fires and explosions in more than 100 buildings, with one fatality and 20 serious injuries. I lamented: “The special pathos of technological tragedies is that the engines of our destruction are machines that we ourselves design and build.”

In a world where defective automatic controls are blowing up houses and dropping aircraft out of the sky, it’s hard to argue for more automation, for adding further layers of complexity to control systems, for endowing machines with greater autonomy. Public sentiment leans the other way. Like President Trump, most of us trust pilots more than we trust computer scientists. We don’t want MCAS on the flight deck. We want Chesley Sullenberger III, the hero of USAir Flight 1549, who guided his crippled A320 to a dead-stick landing in the Hudson River and saved all 155 souls on board. No amount of cockpit automation could have pulled off that feat.

Nevertheless, a cold, analytical view of the statistics suggests a different reaction. The human touch doesn’t always save the day. On the contrary, pilot error is responsible for more fatal crashes than any other cause. One survey lists pilot error as the initiating event in 40 percent of fatal accidents, with equipment failure accounting for 23 percent. No one is (yet) advocating a pilotless cockpit, but at this point in the history of aviation technology that’s a nearer prospect than a computer-free cockpit.

The MCAS system of the 737 MAX represents a particularly awkward compromise between fully manual and fully automatic control. The software is given a large measure of responsibility for flight safety and is even allowed to override the decisions of the pilot. And yet when the system malfunctions, it’s entirely up to the pilot to figure out what went wrong and how to fix it—and the fix had better be quick, before MCAS can drive the plane into the ground.

Two lost aircraft and 346 deaths are strong evidence that this design was not a good idea. But what to do about it? Boeing’s plan is a retreat from automatic control, returning more responsibility and authority to the pilots:

• Flight control system will now compare inputs from both AOA sensors. If the sensors disagree by 5.5 degrees or more with the flaps retracted, MCAS will not activate. An indicator on the flight deck display will alert the pilots.
• If MCAS is activated in non-normal conditions, it will only provide one input for each elevated AOA event. There are no known or envisioned failure conditions where MCAS will provide multiple inputs.
• MCAS can never command more stabilizer input than can be counter­acted by the flight crew pulling back on the column. The pilots will continue to always have the ability to override MCAS and manually control the airplane.

A statement from Dennis Muilenburg, Boeing’s CEO, says the software update “will ensure accidents like that of Lion Air Flight 610 and Ethiopian Airlines Flight 302 never happen again.” I hope that’s true, but what about the accidents that MCAS was designed to prevent? I also hope we will not be reading about a 737 MAX that stalled and crashed because the pilots, believing MCAS was misbehaving, kept hauling back on the control yokes.

If Boeing were to take the opposite approach—not curtailing MCAS but enhancing it with still more algorithms that fiddle with the flight controls—the plan would be greeted with hoots of outrage and derision. Indeed, it seems like a terrible idea. MCAS was installed to prevent pilots from wandering into hazardous territory. A new supervisory system would keep an eye on MCAS, stepping in if it began acting suspiciously. Wouldn’t we then need another custodian to guard the custodians, ad infinitum? Moreoever, with each extra layer of complexity we get new side effects and unintended consequences and opportunities for something to break. The system becomes harder to test, and impossible to prove correct.

Those are serious objections, but the problem being addressed is also serious.

Suppose the 737 MAX didn’t have MCAS but did have a cockpit indicator of angle of attack. On the Lion Air flight, the captain would have felt the stick-shaker warning him of an incipient stall and would have seen an alarmingly high angle of attack on his instrument panel. His training would have impelled him to do the same thing MCAS did: Push the nose down to get the wings working again. Would he have continued pushing it down until the plane crashed? Surely not. He would have looked out the window, he would have cross-checked the instruments on the other side of the cockpit, and after some scary moments he would have realized it was a false alarm. (In darkness or low visibility, where the pilot can lose track of the horizon, the outcome might be worse.)

I see two lessons in this hypothetical exercise. First, erroneous sensor data is dangerous, whether the airplane is being flown by a computer or by Chesley Sullenberger. A prudently designed instrument and control system would take steps to detect (and ideally correct) such errors. At the moment, redundancy is the only defense against these failures—and in the unpatched version of MCAS even that protection is compromised. It’s not enough. One key to the superiority of human pilots is that they exercise judgment and sometimes skepticism about what the instruments tell them. That kind of reasoning is not beyond the reach of automated systems. There’s plenty of information to be exploited. For example, inconsistencies between AoA sensors, pitot tubes, static pressure ports, and air temperature probes not only signal that something’s wrong but can offer clues about which sensor has failed. The inertial reference unit provides an independent check on aircraft attitude; even GPS signals might be brought to bear. Admittedly, making sense of all this data and drawing a valid conclusion from it—a problem known as sensor fusion—is a major challenge.

Second, a closed-loop controller has yet another source of information: an implicit model of the system being controlled. If you change the angle of the horizontal stabilizer, the state of the airplane is expected to change in known ways—in angle of attack, pitch angle, airspeed, altitude, and in the rate of change in all these parameters. If the result of the control action is not consistent with the model, something’s not right. To persist in issuing the same commands when they don’t produce the expected results is not reasonable behavior. Autopilots include rules to deal with such situations; the lower-level control laws that run in manual-mode flight could incorporate such sanity checks as well.

I don’t claim to have the answer to the MCAS problem. And I don’t want to fly in an airplane I designed. (Neither do you.) But there’s a general principle here that I believe should be taken to heart: If an autonomous system makes life-or-death decisions based on sensor data, it ought to verify the validity of the data.

Update 2019-04-11

Boeing continues to insist that MCAS is “not a stall-protection function and not a stall-prevention function. It is a handling-qualities function. There’s a misconception it is something other than that.” This statement comes from Mike Sinnett, who is vice president of product development and future airplane development at Boeing; it appears in an Aviation Week article by Guy Norris published online April 9.

I don’t know exactly what “handling qualities” means in this context. To me the phrase connotes something that might affect comfort or aesthetics or pleasure more than safety. An airplane with different handling qualities would feel different to the pilot but could still be flown without risk of serious mishap. Is Sinnett implying something along those lines? If so—if MCAS is not critical to the safety of flight—I’m surprised that Boeing wouldn’t simply disable it temporarily, as a way of getting the fleet back in the air while they work out a permanent solution.

The Norris article also quote Sinnett as saying: “The thing you are trying to avoid is a situation where you are pulling back and all of a sudden it gets easier, and you wind up overshooting and making the nose higher than you want it to be.” That situation, with the nose higher than you want it to be, sounds to me like an airplane that might be approaching a stall.

A story by Jack Nicas, David Gelles, and James Glanz in today’s New York Times offers a quite different account, suggesting that “handling qualities” may have motivated the first version of MCAS, but stall risks were part of the rationale for later beefing it up.

The system was initially designed to engage only in rare circumstances, namely high-speed maneuvers, in order to make the plane handle more smoothly and predictably for pilots used to flying older 737s, according to two former Boeing employees who spoke on the condition of anonymity because of the open investigations.

For those situations, MCAS was limited to moving the stabilizer—the part of the plane that changes the vertical direction of the jet—about 0.6 degrees in about 10 seconds.

It was around that design stage that the F.A.A. reviewed the initial MCAS design. The planes hadn’t yet gone through their first test flights.

After the test flights began in early 2016, Boeing pilots found that just before a stall at various speeds, the Max handled less predictably than they wanted. So they suggested using MCAS for those scenarios, too, according to one former employee with direct knowledge of the conversations

Finally, another Aviation Week story by Guy Norris, published yesterday, gives a convincing account of what happened to the angle of attack sensor on Ethiopian Airlines Flight 302. According to Norris’s sources, the AoA vane was sheared off moments after takeoff, probably by a bird strike. This hypothesis is consistent with the traces extracted from the flight data recorder, including the strange-looking wiggles at the very end of the flight. I wonder if there’s hope of finding the lost vane, which shouldn’t be far from the end of the runway.

Posted in computing, technology | 13 Comments

Divisive factorials!

The other day I was derailed by this tweet from Fermat’s Library:

The moment I saw it, I had to stop in my tracks, grab a scratch pad, and check out the formula. The result made sense in a rough-and-ready sort of way. Since the multiplicative version of $$n!$$ goes to infinity as $$n$$ increases, the “divisive” version should go to zero. And $$\frac{n^2}{n!}$$ does exactly that; the polynomial function $$n^2$$ grows slower than the exponential function $$n!$$ for large enough $$n$$:

$\frac{1}{1}, \frac{4}{2}, \frac{9}{6}, \frac{16}{24}, \frac{25}{120}, \frac{36}{720}, \frac{49}{5040}, \frac{64}{40320}, \frac{81}{362880}, \frac{100}{3628800}.$

But why does the quotient take the particular form $$\frac{n^2}{n!}$$? Where does the $$n^2$$ come from?

To answer that question, I had to revisit the long-ago trauma of learning to divide fractions, but I pushed through the pain. Proceeding from left to right through the formula in the tweet, we first get $$\frac{n}{n-1}$$. Then, dividing that quantity by $$n-2$$ yields

$\cfrac{\frac{n}{n-1}}{n-2} = \frac{n}{(n-1)(n-2)}.$

Continuing in the same way, we ultimately arrive at:

$n \mathbin{/} (n-1) \mathbin{/} (n-2) \mathbin{/} (n-3) \mathbin{/} \cdots \mathbin{/} 1 = \frac{n}{(n-1) (n-2) (n-3) \cdots 1} = \frac{n}{(n-1)!}$

To recover the tweet’s stated result of $$\frac{n^2}{n!}$$, just multiply numerator and denominator by $$n$$. (To my taste, however, $$\frac{n}{(n-1)!}$$ is the more perspicuous expression.)

I am a card-carrying factorial fanboy. You can keep your fancy Fibonaccis; this is my favorite function. Every time I try out a new programming language, my first exercise is to write a few routines for calculating factorials. Over the years I have pondered several variations on the theme, such as replacing $$\times$$ with $$+$$ in the definition (which produces triangular numbers). But I don’t think I’ve ever before considered substituting $$\mathbin{/}$$ for $$\times$$. It’s messy. Because multiplication is commutative and associative, you can define $$n!$$ simply as the product of all the integers from $$1$$ through $$n$$, without worrying about the order of the operations. With division, order can’t be ignored. In general, $$x \mathbin{/} y \ne y \mathbin{/}x$$, and $$(x \mathbin{/} y) \mathbin{/} z \ne x \mathbin{/} (y \mathbin{/} z)$$.

The Fermat’s Library tweet puts the factors in descending order: $$n, n-1, n-2, \ldots, 1$$. The most obvious alternative is the ascending sequence $$1, 2, 3, \ldots, n$$. What happens if we define the divisive factorial as $$1 \mathbin{/} 2 \mathbin{/} 3 \mathbin{/} \cdots \mathbin{/} n$$? Another visit to the schoolroom algorithm for dividing fractions yields this simple answer:

$1 \mathbin{/} 2 \mathbin{/} 3 \mathbin{/} \cdots \mathbin{/} n = \frac{1}{2 \times 3 \times 4 \times \cdots \times n} = \frac{1}{n!}.$

In other words, when we repeatedly divide while counting up from $$1$$ to $$n$$, the final quotient is the reciprocal of $$n!$$. (I wish I could put an exclamation point at the end of that sentence!) If you’re looking for a canonical answer to the question, “What do you get if you divide instead of multiplying in $$n!$$?” I would argue that $$\frac{1}{n!}$$ is a better candidate than $$\frac{n}{(n - 1)!}$$. Why not embrace the symmetry between $$n!$$ and its inverse?

Of course there are many other ways to arrange the n integers in the set $$\{1 \ldots n\}$$. How many ways? As it happens, $$n!$$ of them! Thus it would seem there are $$n!$$ distinct ways to define the divisive $$n!$$ function. However, looking at the answers for the two permutations discussed above suggests there’s a simpler pattern at work. Whatever element of the sequence happens to come first winds up in the numerator of a big fraction, and the denominator is the product of all the other elements. As a result, there are really only $$n$$ different outcomes—assuming we stick to performing the division operations from left to right. For any integer $$k$$ between $$1$$ and $$n$$, putting $$k$$ at the head of the queue creates a divisive $$n!$$ equal to $$k$$ divided by all the other factors. We can write this out as:

$\cfrac{k}{\frac{n!}{k}}, \text{ which can be rearranged as } \frac{k^2}{n!}.$

And thus we also solve the minor mystery of how $$\frac{n}{(n-1)!}$$ became $$\frac{n^2}{n!}$$ in the tweet.

It’s worth noting that all of these functions converge to zero as $$n$$ goes to infinity. Asymptotically speaking, $$\frac{1^2}{n!}, \frac{2^2}{n!}, \ldots, \frac{n^2}{n!}$$ are all alike.

Ta dah! Mission accomplished. Problem solved. Done and dusted. Now we know everything there is to know about divisive factorials, right?

Well, maybe there’s one more question. What does the computer say? If you take your favorite factorial algorithm, and do as the tweet suggests, replacing any appearance of the $$\times$$ (or *) operator with /, what happens? Which of the $$n$$ variants of divisive $$n!$$ does the program produce?

Here’s my favorite algorithm for computing factorials, in the form of a Julia program:

function mul!(n)
if n == 1
return 1
else
return n * mul!(n - 1)
end
end


This is the algorithm that has introduced generations of nerds to the concept of recursion. In narrative form it says: If $$n$$ is $$1$$, then $$mul!(n)$$ is $$1$$. Otherwise, evaluate the function $$mul!(n-1)$$, then multiply the result by $$n$$. You might ask what happens if $$n$$ is zero or negative. You might ask, but please don’t. For present purposes, $$n \in \mathbb{N}$$.Starting with any positive $$n$$, the sequence of recursive calls must eventually bottom out with $$n = 1$$.

The function can be written more tersely using Julia’s one-liner style of definition:.

mul!(n)  =  n == 1 ? 1 : n * mul!(n - 1)

The right side of the assignment statement is a conditional expression, or ternary operator, which has the form a ? b : c. Here a is a boolean test clause, which must return a value of either true or false. If a is true, clause b is evaluated, and the result becomes the value of the entire expression. Otherwise clause c is evaluated.

Just to be sure I’ve got this right, here are the first 10 factorials, as calculated by this program:

[mul!(n) for n in 1:10]
10-element Array{Int64,1}:
1
2
6
24
120
720
5040
40320
362880
3628800

Now let’s edit that definition and convert the single occurence of * to a /, leaving everything else (except the name of the function) unchanged.

div!(n)  =  n == 1 ? 1 : n / div!(n - 1)

And here’s what comes back when we run the program for values of $$n$$ from $$1$$ through $$20$$:

[div!(n) for n in 1:20]
20-element Array{Real,1}:
1
2.0
1.5
2.6666666666666665
1.875
3.2
2.1875
3.657142857142857
2.4609375
4.063492063492063
2.70703125
4.432900432900433
2.9326171875
4.773892773892774
3.14208984375
5.092152292152292
3.338470458984375
5.391690662278897
3.523941040039063
5.675463855030418 

Huh? That sure doesn’t look like it’s converging to zero—not as $$\frac{1}{n!}$$ or as $$\frac{n}{n - 1}$$. As a matter of fact, it doesn’t look like it’s going to converge at all. The graph below suggests the sequence is made up of two alternating components, both of which appear to be slowly growing toward infinity as well as diverging from one another.

In trying to make sense of what we’re seeing here, it helps to change the output type of the div! function. Instead of applying the division operator /, which returns the quotient as a floating-point number, we can substitute the // operator, which returns an exact rational quotient, reduced to lowest terms.

div!(n)  =  n == 1 ? 1 : n // div!(n - 1)

Here’s the sequence of values for n in 1:20:

20-element Array{Real,1}:
1
2//1
3//2
8//3
15//8
16//5
35//16
128//35
315//128
256//63
693//256
1024//231
3003//1024
2048//429
6435//2048
32768//6435
109395//32768
65536//12155
230945//65536
262144//46189 

The list is full of curious patterns. It’s a double helix, with even numbers and odd numbers zigzagging in complementary strands. The even numbers are not just even; they are all powers of $$2$$. Also, they appear in pairs—first in the numerator, then in the denominator—and their sequence is nondecreasing. But there are gaps; not all powers of $$2$$ are present. The odd strand looks even more complicated, with various small prime factors flitting in and out of the numbers. (The primes have to be small—smaller than $$n$$, anyway.)

This outcome took me by surprise. I had really expected to see a much tamer sequence, like those I worked out with pencil and paper. All those jagged, jitterbuggy ups and downs made no sense. Nor did the overall trend of unbounded growth in the ratio. How could you keep dividing and dividing, and wind up with bigger and bigger numbers?

At this point you may want to pause before reading on, and try to work out your own theory of where these zigzag numbers are coming from. If you need a hint, you can get a strong one—almost a spoiler—by looking up the sequence of numerators or the sequence of denominators in the Online Encyclopedia of Integer Sequences.

Here’s another hint. A small edit to the div! program completely transforms the output. Just flip the final clause, changing n // div!(n - 1) into div!(n - 1) // n.

div!(n)  =  n == 1 ? 1 : div!(n - 1) // n

Now the results look like this:

10-element Array{Real,1}:
1
1//2
1//6
1//24
1//120
1//720
1//5040
1//40320
1//362880
1//3628800

This is the inverse factorial function we’ve already seen, the series of quotients generated when you march left to right through an ascending sequence of divisors $$1 \mathbin{/} 2 \mathbin{/} 3 \mathbin{/} \cdots \mathbin{/} n$$.

It’s no surprise that flipping the final clause in the procedure alters the outcome. After all, we know that division is not commutative or associative. What’s not so easy to see is why the sequence of quotients generated by the original program takes that weird zigzag form. What mechanism is giving rise to those paired powers of 2 and the alternation of odd and even?

I have found that it’s easier to explain what’s going on in the zigzag sequence when I describe an iterative version of the procedure, rather than the recursive one. (This is an embarrassing admission for someone who has argued that recursive definitions are easier to reason about, but there you have it.) Here’s the program:

function div!_iter(n)
q = 1
for i in 1:n
q = i // q
end
return q
end

I submit that this looping procedure is operationally identical to the recursive function, in the sense that if div!(n) and div!_iter(n) both return a result for some positive integer n, it will always be the same result. Here’s my evidence:

[div!(n) for n in 1:20]    [div!_iter(n) for n in 1:20]
1                         1//1
2//1                       2//1
3//2                       3//2
8//3                       8//3
15//8                      15//8
16//5                      16//5
35//16                     35//16
128//35                    128//35
315//128                   315//128
256//63                    256//63
693//256                   693//256
1024//231                  1024//231
3003//1024                 3003//1024
2048//429                  2048//429
6435//2048                 6435//2048
32768//6435                32768//6435
109395//32768              109395//32768
65536//12155               65536//12155
230945//65536              230945//65536
262144//46189              262144//46189

To understand the process that gives rise to these numbers, consider the successive values of the variables $$i$$ and $$q$$ each time the loop is executed. Initially, $$i$$ and $$q$$ are both set to $$1$$; hence, after the first passage through the loop, the statement q = i // q gives $$q$$ the value $$\frac{1}{1}$$. Next time around, $$i = 2$$ and $$q = \frac{1}{1}$$, so $$q$$’s new value is $$\frac{2}{1}$$. On the third iteration, $$i = 3$$ and $$q = \frac{2}{1}$$, yielding $$\frac{i}{q} \rightarrow \frac{3}{2}$$. If this is still confusing, try thinking of $$\frac{i}{q}$$ as $$i \times \frac{1}{q}$$. The crucial observation is that on every passage through the loop, $$q$$ is inverted, becoming $$\frac{1}{q}$$.

If you unwind these operations, and look at the multiplications and divisions that go into each element of the series, a pattern emerges:

$\frac{1}{1}, \quad \frac{2}{1}, \quad \frac{1 \cdot 3}{2}, \quad \frac{2 \cdot 4}{1 \cdot 3}, \quad \frac{1 \cdot 3 \cdot 5}{2 \cdot 4} \quad \frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5}$

The general form is:

$\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot n}{2 \cdot 4 \cdot \cdots \cdot (n-1)} \quad (\text{odd } n) \qquad \frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)} \quad (\text{even } n).$

The functions $$1 \cdot 3 \cdot 5 \cdot \cdots \cdot n$$ for odd $$n$$ and $$2 \cdot 4 \cdot 6 \cdot \cdots \cdot n$$ for even $$n$$ have a name! They are known as double factorials, with the notation $$n!!$$. Terrible terminology, no? Better to have named them “semi-factorials.” And if I didn’t know better, I would read $$n!!$$ as “the factorial of the factorial.” The double factorial of n is defined as the product of n and all smaller positive integers of the same parity. Thus our peculiar sequence of zigzag quotients is simply $$\frac{n!!}{(n-1)!!}$$.

A 2012 article by Henry W. Gould and Jocelyn Quaintance (behind a paywall, regrettably) surveys the applications of double factorials. They turn up more often than you might guess. In the middle of the 17th century John Wallis came up with this identity:

$\frac{\pi}{2} = \frac{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdots}{1 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdots} = \lim_{n \rightarrow \infty} \frac{((2n)!!)^2}{(2n + 1)!!(2n - 1)!!}$

An even weirder series, involving the cube of a quotient of double factorials, sums to $$\frac{2}{\pi}$$. That one was discovered by (who else?) Srinivasa Ramanujan.

Gould and Quaintance also discuss the double factorial counterpart of binomial coefficients. The standard binomial coefficient is defined as:

$\binom{n}{k} = \frac{n!}{k! (n-k)!}.$

The double version is:

$\left(\!\binom{n}{k}\!\right) = \frac{n!!}{k!! (n-k)!!}.$

Note that our zigzag numbers fit this description and therefore qualify as double factorial binomial coefficients. Specifically, they are the numbers:

$\left(\!\binom{n}{1}\!\right) = \left(\!\binom{n}{n - 1}\!\right) = \frac{n!!}{1!! (n-1)!!}.$

The regular binomial $$\binom{n}{1}$$ is not very interesting; it is simply equal to $$n$$. But the doubled version $$\left(\!\binom{n}{1}\!\right)$$, as we’ve seen, dances a livelier jig. And, unlike the single binomial, it is not always an integer. (The only integer values are $$1$$ and $$2$$.)

Seeing the zigzag numbers as ratios of double factorials explains quite a few of their properties, starting with the alternation of evens and odds. We can also see why all the even numbers in the sequence are powers of 2. Consider the case of $$n = 6$$. The numerator of this fraction is $$2 \cdot 4 \cdot 6 = 48$$, which acquires a factor of $$3$$ from the $$6$$. But the denominator is $$1 \cdot 3 \cdot 5 = 15$$. The $$3$$s above and below cancel, leaving $$\frac{16}{5}$$. Such cancelations will happen in every case. Whenever an odd factor $$m$$ enters the even sequence, it must do so in the form $$2 \cdot m$$, but at that point $$m$$ itself must already be present in the odd sequence.

Is the sequence of zigzag numbers a reasonable answer to the question, “What happens when you divide instead of multiply in $$n!$$?” Or is the computer program that generates them just a buggy algorithm? My personal judgment is that $$\frac{1}{n!}$$ is a more intuitive answer, but $$\frac{n!!}{(n - 1)!!}$$ is more interesting.

Furthermore, the mere existence of the zigzag sequence broadens our horizons. As noted above, if you insist that the division algorithm must always chug along the list of $$n$$ factors in order, at each stop dividing the number on the left by the number on the right, then there are only $$n$$ possible outcomes, and they all look much alike. But the zigzag solution suggests wilder possibilities. We can formulate the task as follows. Take the set of factors $$\{1 \dots n\}$$, select a subset, and invert all the elements of that subset; now multiply all the factors, both the inverted and the upright ones. If the inverted subset is empty, the result is the ordinary factorial $$n!$$. If all of the factors are inverted, we get the inverse $$\frac{1}{n!}$$. And if every second factor is inverted, starting with $$n - 1$$, the result is an element of the zigzag sequence.

These are only a few among the many possible choices; in total there are $$2^n$$ subsets of $$n$$ items. For example, you might invert every number that is prime or a power of a prime $$(2, 3, 4, 5, 7, 8, 9, 11, \dots)$$. For small $$n$$, the result jumps around but remains consistently less than $$1$$:

If I were to continue this plot to larger $$n$$, however, it would take off for the stratosphere. Prime powers get sparse farther out on the number line.

Here’s a question. We’ve seen factorial variants that go to zero as $$n$$ goes to infinity, such as $$1/n!$$. We’ve seen other variants grow without bound as $$n$$ increases, including $$n!$$ itself, and the zigzag numbers. Are there any versions of the factorial process that converge to a finite bound other than zero?

function greedy_balance(n)
q = 1
while n > 0
q = q > 1 ? q /= n : q *= n
n -= 1
end
return q
end

We loop through the integers from $$n$$ down to $$1$$, calculating the running product/quotient $$q$$ as we go. At each step, if the current value of $$q$$ is greater than $$1$$, we divide by the next factor; otherwise, we multiply. This scheme implements a kind of feedback control or target-seeking behavior. If $$q$$ gets too large, we reduce it; too small and we increase it. I conjectured that as $$n$$ goes to infinity, $$q$$ would settle into an ever-narrower range of values near $$1$$.

Running the experiment gave me another surprise:

That sawtooth wave is not quite what I expected. One minor peculiarity is that the curve is not symmetric around $$1$$; the excursions above have higher amplitude than those below. But this distortion is more visual than mathematical. Because $$q$$ is a ratio, the distance from $$1$$ to $$10$$ is the same as the distance from $$1$$ to $$\frac{1}{10}$$, but it doesn’t look that way on a linear scale. The remedy is to plot the log of the ratio:

Now the graph is symmetric, or at least approximately so, centered on $$0$$, which is the logarithm of $$1$$. But a larger mystery remains. The sawtooth waveform is very regular, with a period of $$4$$, and it shows no obvious signs of shrinking toward the expected limiting value of $$\log q = 0$$. Numerical evidence suggests that as $$n$$ goes to infinity the peaks of this curve converge on a value just above $$q = \frac{5}{3}$$, and the troughs approach a value just below $$q = \frac{3}{5}$$. (The corresponding base-$$10$$ logarithms are roughly $$\pm0.222$$. I have not worked out why this should be so. Perhaps someone will explain it to me.

The failure of this greedy algorithm doesn’t mean we can’t find a divisive factorial that converges to $$q = 1$$. If we work with the logarithms of the factors, this procedure becomes an instance of a well-known compu­tational problem called the number partitioning problem. You are given a set of real numbers and asked to divide it into two sets whose sums are equal, or as close to equal as possible. It’s a certifiably hard problem, but it has also been called (PDF) “the easiest hard problem.”For any given $$n$$, we might find that inverting some other subset of the factors gives a better approximation to $$n! = 1$$. For small $$n$$, we can solve the problem by brute force: Just look at all $$2^n$$ subsets and pick the best one.

I have computed the optimal partitionings up to $$n = 30$$, where there are a billion possibilities to choose from.

The graph is clearly flatlining. You could use the same method to force convergence to any other value between $$0$$ and $$n!$$.

And thus we have yet another answer to the question in the tweet that launched this adventure. What happens when you divide instead of multiply in n!? Anything you want.

Posted in computing, mathematics | 6 Comments

A Room with a View

On my visit to Baltimore for the Joint Mathematics Meetings a couple of weeks ago, I managed to score a hotel room with a spectacular scenic view. My seventh-floor perch overlooked the Greene Street substation of the Baltimore Gas and Electric Company, just around the corner from the Camden Yards baseball stadium.

Some years ago, writing about such technological landscapes, I argued that you can understand what you’re looking at if you’re willing to invest a little effort:

At first glance, a substation is a bewildering array of hulking steel machines whose function is far from obvious. Ponderous tanklike or boxlike objects are lined up in rows. Some of them have cooling fins or fans; many have fluted porcelain insulators poking out in all directions…. If you look closer, you will find there is a logic to this mélange of equipment. You can make sense of it. The substation has inputs and outputs, and with a little study you can trace the pathways between them.

If I were writing that passage now, I would hedge or soften my claim that an electrical substation will yield its secrets to casual observation. Each morning in Baltimore I spent a few minutes peering into the Greene Street enclosure. I was able to identify all the major pieces of equipment in the open-air part of the station, and I know their basic functions. But making sense of the circuitry, finding the logic in the arrangement of devices, tracing the pathways from inputs to outputs—I have to confess, with a generous measure of chagrin, that I failed to solve the puzzle. I think I have the answers now, but finding them took more than eyeballing the hardware.

Basics first. A substation is not a generating plant. BGE does not “make” electricity here. The substation receives electric power in bulk from distant plants and repackages it for retail delivery. At Greene Street the incoming supply is at 115,000 volts (or 115 kV). The output voltage is about a tenth of that: 13.8 kV. How do I know the voltages? Not through some ingenious calculation based on the size of the insulators or the spacing between conductors. In an enlargement of one of my photos I found an identify­ing plate with the blurry and partially obscured but still legible notation “115/13.8 KV.”

The biggest hunks of machinery in the yard are the transformers (photo below), which do the voltage conversion. Each transformer is housed in a steel tank filled with oil, which serves as both insulator and coolant. Immersed in the oil bath are coils of wire wrapped around a massive iron core. Stacks of radiator panels, with fans mounted underneath, help cool the oil when the system is under heavy load. A bed of crushed stone under the transformer is meant to soak up any oil leaks and reduce fire hazards.

Electricity enters and leaves the transformer through the ribbed gray posts, called bushings, mounted atop the casing. A bushing is an insulator with a conducting path through the middle. It works like the rubber grommet that protects the power cord of an appliance where it passes through the steel chassis. The high-voltage inputs attach to the three tallest bushings, with red caps; the low-voltage bushings, with dark gray caps, are shorter and more closely spaced. Notice that each high-voltage input travels over a single slender wire, whereas each low-voltage output has three stout conductors. That’s because reducing the voltage to one-tenth increases the current tenfold.

What about the three slender gray posts just to the left of the high-voltage bushings? They are lightning arresters, shunting sudden voltage surges into the earth to protect the transformer from damage.

Perhaps the most distinctive feature of this particular substation is what’s not to be seen. There are no tall towers carrying high-voltage transmission lines to the station. Clearing a right of way for overhead lines would be difficult and destructive in an urban center, so the high-voltage “feeders” run under­ground. In the photo at right, near the bottom left corner, a bundle of three metal-sheathed cables emerges from the earth. Each cable, about as thick as a human forearm, has a copper or aluminum conductor running down the middle, surrounded by insulation. I suspect these cables are insulated with layers of paper impregnated with oil under pressure; some of the other feeders entering the station may be of a newer design, with solid plastic insulation. Each cable plugs into the bottom of a ceramic bushing, which carries the current to a copper wire at the top. (You can tell the wire is copper because of the green patina.)

Connecting the feeder input to the transformer is a set of three hollow aluminum conductors called bus bars, held high overhead on steel stanchions and ceramic insulators. At both ends of the bus bars are mechanical switches that open like hinged doors to break the circuit. I don’t know whether these switches can be opened when the system is under power or whether they are just used to isolate components for maintenance after a feeder has been shut down. Beyond the bus bars, and hidden behind a concrete barrier, we can glimpse the bushings atop a different kind of switch, which I’ll return to below.

At this point you might be asking, why does everything come in sets of three—the bus bars, the feeder cables, the terminals on the transformer? It’s because electric power is distributed as three-phase alternating current. Each conductor carries a voltage oscillating at 60 Hertz, with the three waves offset by one-third of a cycle. If you recorded the voltage between each of the three pairs of conductors (AB, AC, BC), you’d see a waveform like the one above at left.

At the other end of the conducting pathway, connected to three more bus bars on the low-voltage side of the transformer, is an odd-looking stack of three large drums. These

are current-limiting reactors (no connection with nuclear reactors). They are coils of thick conductors wound on a stout concrete armature. Under normal operating conditions they have little effect on the transmission of power, but in the milliseconds following a short circuit, the sudden rush of current generates a strong magnetic field in the coils, absorbing the energy of the fault current and preventing damage to other equipment.

So those are the main elements of the substation I was able to spot from my hotel window. They all made sense individually, and yet I realized over the course of a few days that I didn’t really understand how it all works together. My doubts are easiest to explain with the help of a bird’s eye view of the substation layout, cribbed from Google Maps:

My window vista was from off to the right, beyond the eastern edge of the compound. In the Google Maps view, the underground 115 kV feeders enter at the bottom or southern edge, and power flows northward through the transformers and the reactor coils, finally entering the building that occupies the northeast corner of the lot. Neither Google nor I can see inside this windowless building, but I know what’s in there, in a general way. That’s where the low-voltage (13.8 kV) distribution lines go underground and fan out to their various destinations in the neighborhood.

Let’s look more closely at the outdoor equipment. There are four high-voltage feeders, four transformers, and four sets of reactor coils. Apart from minor differences in geometry (and one newer-looking, less rusty transformer), these four parallel pathways all look alike. It’s a symmetric four-lane highway. Thus my first hypothesis was that four independent 115 kV feeders supply power to the station, presumably bringing it from larger substations and higher-voltage transmission lines outside the city.

However, something about the layout continued to bother me. If we label the four lanes of the highway from left to right, then on the high-voltage side, toward the bottom of the map view, it looks like there’s something connecting lanes 1 and 2 and, and there’s a similar link between lanes 3 and 4. From my hotel window the view of this device is blocked by a concrete barricade, and unfortunately the Google Maps image does not show it clearly either. (If you zoom in for a closer view, the goofy Google compression algorithm will turn the scene into a dreamscape where all the components have been draped in Saran Wrap.) Nevertheless, I’m quite sure of what I’m looking at. The device connecting the pairs of feeders is a high-voltage three-phase switch, or circuit breaker, something like the ones seen in the image at right (photographed at another substation, in Missouri.) The function of this device is essentially the same as that of a circuit breaker in your home electrical panel. You can turn it off manually to shut down a circuit, but it may also “trip” automatically in response to an overload or a short circuit. The concrete barriers flanking the two high-voltage breakers at Greene Street hint at one of the problems with such switches. Interrupting a current of hundreds of amperes at more than 100,000 volts is like stopping a runaway truck: It requires absorbing a lot of energy. The switch does not always survive the experience.

When I first looked into the Greene Street substation, I was puzzled by the absence of breakers at the input end of each main circuit. I expected to see them there to protect the transformers and other components from overloads or lightning strikes. I think there are breakers on the low-voltage side, tucked in just behind the transformers and thus not clearly visible from my window. But there’s nothing on the high side. I could only guess that such protection is provided by breakers near the output of the next substation upstream, the one that sends the 115 kV feeders into Greene Street.

That leaves the question of why pairs of circuits within the substation are cross-linked by breakers. I drew a simplified diagram of how things are wired up:

Two adjacent 115 kV circuits run from bottom to top; the breaker between them connects corresponding conductors—left to left, middle to middle, right to right. But what’s the point of doing so?

I had some ideas. If one transformer were out of commission, the pathway through the breaker could allow power to be rerouted through the remaining transformer (assuming it could handle the extra load). Indeed, maybe the entire design simply reflects a high level of redundancy. There are four incoming feeders and four transformers, but perhaps only two are expected to operate at any given time. The breaker provides a means of switching between them, so that you could lose a circuit (or maybe two) and still keep all the lights on. After all, this is a substation supplying power to many large facilities—the convention center (where the math meetings were held), a major hospital, large hotels, the ball park, theaters, museums, high-rise office buildings. Reliability is important here.

After further thought, however, this scheme seemed highly implausible. There are other substation layouts that would allow any of the four feeders to power any of the four transformers, allowing much greater flexibility in handling failures and making more efficient use of all the equipment. Linking the incoming feeders in pairs made no sense.

I would love to be able to say that I solved this puzzle on my own, just by dint of analysis and deduction, but it’s not true. When I got home and began looking at the photographs, I was still baffled. The answer eventually came via Google, though it wasn’t easy to find. Before revealing where I went wrong, I’ll give a couple of hints, which might be enough for you to guess the answer.

Hint 1. I was led astray by a biased sample. I am much more familiar with substations out in the suburbs or the countryside, partly because they’re easier to see into. Most of them are surrounded by a chain-link fence rather than a brick wall. But country infrastructure differs from the urban stuff.

Hint 2. I was also fooled by geometry when I should have been thinking about topology. To understand what you’re seeing in the Greene Street compound, you have to get beyond individual components and think about how it’s all connected to the rest of the network.

The web offers marvelous resources for the student of infrastructure, but finding them can be a challenge. You might suppose that the BGE website would have a list of the company’s facilities, and maybe a basic tutorial on where Baltimore’s electricity comes from. There’s nothing of the sort (although the utility’s parent company does offer thumbnail descriptions of some of their generating plants). Baltimore City websites were a little more helpful—not that they explained any details of substation operation, but they did report various legal and regulatory filings concerned with proposals for new or updated facilities. From those reports I learned the names of several BGE installations, which I could take back to Google to use as search terms.

One avenue I pursued was figuring out where the high-voltage feeders entering Greene Street come from. I discovered a substation called Pumphrey about five miles south of the city, near BWI airport, which seemed to be a major nexus of transmission lines. In particular, four 115 kV feeders travel north from Pumphrey to a substation in the Westport neighborhood, which is about a mile south of downtown. The Pumphrey-Westport feeders are overhead lines, and I had seen them already. Their right of way parallels the light rail route I had taken into town from the airport. Beyond the Westport substation, which is next to a light rail stop of the same name, the towers disappear. An obvious hypothesis is that the four feeders dive underground at Westport and come up at Greene Street. This guess was partly correct: Power does reach Greene Street from Westport, but not exclusively.

At Westport BGE has recently built a small, gas-fired generating plant, to help meet peak demands. The substation is also near the Baltimore RESCO waste-to-energy power plant (photo above), which has become a local landmark. (It’s the only garbage burner I know that turns up on postcards sold in tourist shops.) Power from both of these sources could also make its way to the Greene Street substation, via Westport.

I finally began to make sense of the city’s wiring diagram when I stumbled upon some documents published by the PJM Interconnection, the administrator and coordinator of the power “pool” in the mid-Atlantic region. PJM stands for Pennsylvania–New Jersey–Maryland, but it covers a broader territory, including Delaware, Ohio, West Virginia, most of Virginia, and parts of Kentucky, Indiana, Michigan, and Illinois. Connecting to such a pool has important advantages for a utility. If an equipment failure means you can’t meet your customers’ demands for electricity, you can import power from elsewhere in the pool to make up the shortage; conversely, if you have excess generation, you can sell the power to another utility. The PJM supervises the market for such exchanges.

The idea behind power pooling is that neighbors can prop each other up in times of trouble; however, they can also knock each other down. As a condition of membership in the pool, utilities have to maintain various standards for engineering and reliability. PJM committees review plans for changes or additions to a utility’s network. It was a set of Powerpoint slides prepared for one such committee that first alerted me to my fundamental misconception. One of the slides included the map below, tracing the routes of 115 kV feeders (green lines) in and around downtown Baltimore.

I had been assuming—even though I should have known better—that the distribution network is essentially treelike, with lines radiating from each node to other nodes but never coming back together. For low-voltage distribution lines in sparsely settled areas, this assumption is generally correct. If you live in the suburbs or in a small town, there is one power line that runs from the local substation to your neighborhood; if a tree falls on it, you’re in the dark until the problem is fixed. There is no alternative route of supply. But that is not the topology of higher-voltage circuits. The Baltimore network consists of rings, where power can reach most nodes by following either of two pathways.

In the map we can see the four 115 kV feeders linking Pumphrey to Westport. From Westport, two lines run due north to Greene Street, then make a right turn to another station named Concord Street. As far as I can tell, there is no Concord Street in Baltimore. There’s a Concord Road, but it’s miles away in the northwest corner of the city. The substation is actually at 750 East Pratt Street, occupying the lower floors of an 18-story office tower.They continue east to Monument Street, then north again to Erdman, where the ring receives additional power from other lines coming down from the north. The ring then continues west to Center Street and returns to Westport, closing the loop. The arrangement has some clear advantages for reliability. You can break any one link in a ring without cutting power to any of the substations; the power simply flows around the ring in the other direction.

This double-ring architecture calls for a total reinterpretation of how the Greene Street substation works. I had imagined the four 115 kV inputs as four lanes of one-way traffic, all pouring into the substation and dead-ending in the four transformers. In reality we have just two roadways, both of which enter the substation and then leave again, continuing on to further destinations. And they are not one-way; they can both carry traffic in either direction. The transformers are like exit ramps that siphon off a portion of the traffic while the main stream passes by.

At Greene Street, two of the underground lines entering the compound come from Westport, but the other two proceed to Concord Street, the next station around the ring. What about the breakers that sit between the incoming and outgoing branches of each circuit? They open up the ring to isolate any section that experiences a serious failure. For example, a short circuit in one of the cables running between Greene Street and Concord Street would cause breakers at both of those stations to open up, but both stations would continue to receive power coming around the other branch of the loop.

This revised interpretation was confirmed by another document made available by PJM, this one written by BGE engineers as an account of their engineering practices for transmission lines and substations. It includes a schematic diagram of a typical downtown Baltimore substation. The diagram makes no attempt to reproduce the geometric layout of the components; it rearranges them to make the topology clearer.

The two 115 kV feeders that run through the substation are shown as horizontal lines; the solid black squares in the middle are the breakers that join the pairs of feeders and thereby close the two rings that run through all the downtown substations. The transformers are the W-shaped symbols at the ends of the branch lines.

A mystery remains. The symbol represents a disconnect switch, a rather simple mechanical device that generally cannot be operated when the power line is under load. The symbol is identified in the BGE document as a circuit switcher, a more elaborate device capable of interrupting a heavy current. In the Greene Street photos, however, the switches at the two ends of the high-voltage bus bars appear almost identical. I’m not seeing any circuit switchers there. But, as should be obvious by now, I’m capable of misinterpreting what I see.

Posted in technology | 1 Comment