Welcome to the 130th monthly Carnival of Mathematics, a roving blog organized by The Aperiodical. The previous edition was hosted by GanitCharcha. This month’s collection includes material appearing on the Web between December 10 and January 9 (with a bit of spillover on both ends).
Tradition obliges me to say something interesting about the number 130. I could mention that 130 = 5 × 5 × 5 + 5. Is that interesting? Meh, eh? The Wikipedia page for 130 reports the following curious observation:
\(130\) is the only integer that is the sum of the squares of its first four divisors, including \(1\): \(1^2 + 2^2 + 5^2 + 10^2 = 130\).
What’s interesting here is not the fact that 130 has this property. The provocative part is the statement that 130 is the only such integer. How can one know this? It would be easy enough to check that no smaller number qualifies, but how do you rule out the possibility that somewhere far out on the number line there lurks another example? Clearly, what’s needed is a proof, but Wikipedia offers none. I spent some time trying to come up with a proof myself but failed to put the pieces together. Maybe you’ll do better. If not, Robert Munafo explains all, and traces the question to the Russian website dxdy.ru.
Enough of 130. I’d like to celebrate another number of the season: 2016. As midnight approached on December 31, Twitter brought this news:
In other words, the year we are now beginning is equal to \(2^{11} - 2^{5}\), and in another 32 years we’ll be celebrating the turn of a binary millennium, marking the passage of two kibiyears. (Year-zero deniers are welcome to wait an extra year.)
Fernando Juan, with an animated GIF, pointed out another nonobvious fact: \(2016 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3\).
Even more noteworthy, May-Li Khoe and Federico Ardillo gave graphic proof that 2016 is a triangular number:
Specifically, 2016 is the sum of the integers \(1 + 2 + 3 + \cdots + 63\), making it the \(63\)rd element of the sequence \(1, 3, 6, 10, 15, \ldots\). You could go bowling with \(2016\) pins!
According to a well-known anecdote, Carl Friedrich Gauss was a schoolboy when he discovered that the nth triangular number is equal to \(n(n + 1) / 2\), which is also the value of the binomial coefficient \(\binom{n + 1}{2}\). Hence 2016 is the number of ways of choosing two items at a time from a collection of 64 objects. A tweet by John D. Cook offers this interpretation: “2016 = The number of ways to place two pawns on a chessboard.”
Patrick Honner expressed the same idea another way: 2016 is the number of edges in a complete graph with 64 vertices.
If you attended a New Years Eve party with 64 people present, and at midnight every guest clinked glasses with every other guest, there were 2016 clinks in all. (And probably a lot of broken glassware.)
In a New Years story of another kind, Tim Harford warns of the cost of overconfidence when it comes to making resolutions, as when you buy a year’s gym membership and stop going after six weeks. “Some companies base their business models on our tendency to overestimate our willpower.”
Triangular numbers turn up in another holiday item as well. A video by Tipping Point Math offers a quantitative analysis of the gift-giving extravaganza in “The Twelve Days of Christmas.” On the \(n\)th day of Christmas you receive \(1 + 2 + 3 + \cdots + n\) gifts, which is clearly the triangular number \(T_n\). But what’s the total number of gifts when you add up the largesse over \(n\) days? These sums of triangular numbers are the tetrahedral numbers: \(1, 4, 10, 20, 35 \ldots\); the twelfth tetrahedral number is \(364\). The video shows where to find all these numbers in Pascal’s triangle.
Vince Knight, in Un Peu de Math, approaches the Christmas gift exchange as a problem in game theory. Two friends agree not to exchange gifts, but then they are both tempted to renege on the agreement and buy a present afterall. This situation is a variant of the game-theory puzzle known as prisoner’s dilemma—which sounds like a rather grim view of holiday tradition. But in fact the conclusion is cheery: “People enjoy giving gifts a lot more than receiving them.”
The December holiday season also brings Don Knuth’s annual Christmas lecture, which has been posted on YouTube. In previous years, this talk was the Christmas Tree lecture, because it touched on some aspect of trees as a data structure or a concept in graph theory. Now Knuth has branched out from the world of trees. His subject is comma-free codes—sets of words that can be jammed together without commas or spaces to mark the boundaries between words, and yet still be read without ambiguity. A set that lacks the comma-free property is {tea, ate, eat}, because a concatenation such as teateatea has within it not just tea,tea,tea but also eat,eat and ate,ate. But the words {sat, set, tea} do qualify as comma-free, because no sequence of the words can be partitioned in more than one way to yield words in the set.
The idea of comma-free codes arose in the late 1950s, when it was thought that the genetic code might have a comma-free structure. Experiments soon showed otherwise, and interest in comma-free codes waned. But Knuth has rediscovered a paper by Willard Eastman, published in 1965, that constructs an algorithm for generating a comma-free code (if possible) from a given set of symbols. Knuth’s lecture demonstrates the algorithm and gives a computer implementation.
Stephen Wolfram with a replica of Babbage’s difference
engine and a portrait of Ada Lovelace. Image courtesy
of Stephen Wolfram.
December 10, 2015, was the 200th birthday of Ada, Countess of Lovelace, who collaborated with Charles Babbage on the proposed computing machine called the analytic engine. The anniversary was commemorated in several ways, including an exhibition at the Science Museum in London, but here I want to call attention to a 12,000-word biographical essay by Stephen Wolfram, the creator of Mathematica.
The stories of Lovelace and Babbage have been told many times, but Wolfram’s account is worth reading even if you already know how the plot comes out. The principles of the machines and the algorithm that Lovelace devised as a demo (a computation of Bernoulli numbers) are described in detail, but the heart of the story is the human drama. Today we see these two figures as pioneers and heroes, but their lives were tinged with frustration and disappointment. Lovelace, who died at age 36, never got a proper opportunity to show off her ideas and abilities; in her one published work, all of her original contibutions were relegated to footnotes. Babbage in his later years felt aggrieved with the world for failing to support his vision. Wolfram makes an interesting biographer of this pair, never hesitating to see his subjects reflected in his own experience, and vice versa.
Back in the summer of 2012 Shinichi Mochizuki of Kyoto University released four long papers that claim to resolve an important problem in number theory called the abc conjecture. (I’m not going to try to explain the conjecture here; I did so in an earlier post.) More than three years later, no one in the mathematical community has been able to understand Mochizuki’s work well enough to verify that it is indeed a proof. In December more than 50 experts gathered at the University of Oxford to try to make progress on breaking the impasse. Brian Conrad of Stanford University, one of the workshop participants, wrote up his notes as a guest post in Cathy O’Neil’s Mathbabe blog. This is an insider’s account, and parts of the discussion will not make much sense unless you have fairly deep background in modern number theory. (I don’t.) But that inaccesibility illustrates the point, in a way. Number theorists themselves are in the same situation with regard to the Mochizuchi papers, which are clotted with idiosyncratic concepts such as Inter-universal Teichmuller Theory (IUT). Conrad writes:
After [Kiran] Kedlaya’s lectures, the remaining ones devoted to the IUT papers were impossible to follow without already knowing the material: there was a heavy amount of rapid-fire new notation and language and terminology, and everyone not already somewhat experienced with IUT got totally lost…. Persistent questions from the audience didn’t help to remove the cloud of fog that overcame many lectures in the final two days. The audience kept asking for examples (in some instructive sense, even if entirely about mathematical structures), but nothing satisfactory to much of the audience along such lines was provided.
It’s still not clear when or how the status of the proof will be resolved. O’Neil herself has taken a stern position on the issue of inpenetrable purported proofs. When the Mochizuchi papers were first released, she wrote: “If I claim to have proved something, it is my responsibility to convince others I’ve done so; it’s not their responsibility to try to understand it (although it would be very nice of them to try).”
Anthony Bonato, the Intrepid Mathematician, offers a friendlier introduction to the abc conjecture and its consequences. Also see comments on the conjecture and the workshop by Evelyn Lamb in the Scientific American blog Roots of Unity and by Anna Haensch in an American Mathematical Society blog.
Another number-theory event that attracted wide notice in recent weeks was the successful defense and publication of a doctoral thesis at Princeton University by Piper Harron. The title is “The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist’s Rendering,” and it’s a great read (PDF). Here is the abstract:
A fascinating tale of mayhem, mystery, and mathematics. Attached to each degree \(n\) number field is a rank \(n\, - 1\) lattice called its shape. This thesis shows that the shapes of \(S_n\)-number fields (of degree \(n = 3, 4,\) or \(5)\) become equidistributed as the absolute discriminant of the number field goes to infinity. The result for \(n = 3\) is due to David Terr. Here, we provide a unified proof for \(n = 3, 4,\) and \(5\) based on the parametrizations of low rank rings due to Bhargava and Delone–Faddeev. We do not assume any of those words make any kind of sense, though we do make certain assumptions about how much time the reader has on her hands and what kind of sense of humor she has.
This is not your grandmother’s doctoral thesis! Harron interleaves sections of “laysplaining” and “mathsplaining,” and illustrates her work with an abundance of metaphors, jokes, asides to the reader, cartoons, and commentary on the course of her life during the 10 years she spent writing the thesis (e.g., a brief interruption to give birth). In terms of expository style, Mochizuki and Harron stand at opposite poles—a fact noted by at least two bloggers. Both of the posts by Evelyn Lamb and by Anna Haensch mentioned above in connection with Mochizuki also discuss Harron’s work.
Harron has a blog of her own with a post titled “Why I Do Not Talk About Math,” and she has written a guest post for Mathbabe with this description of her thesis:
My thesis is this thing that was initially going to be a grenade launched at my ex-prison, for better or for worse, and instead turned into some kind of positive seed bomb where flowers have sprouted beside the foundations I thought I wanted to crumble.
Reading her accounts of life as “an escaped graduate student,” I am in equal measures amused and horrified (not a comfortable combination). If nothing else, Harron’s “thesis grenade” promises to broaden—to diversify—the discussion of diversity in mathematical culture. It’s not just about race and gender (although Harron cares passionately about those issues). Human diversity also ranges across many other dimensions: modes of reasoning, approaches to learning, cultural contexts, styles of explaining, and ways of living. Harron’s thesis is a declaration that one can do original research-level mathematics without adopting the vocabulary, the styles, the attitudes, and the mental apparatus of one established academic community.
If I show you a cube, you can easily place it in a three-dimensional cartesian coordinate system in such a way that all the vertices have rational \(x,y,z\) coordinates. By scaling the cube, you can make all the coordinates integers. The same is true of the regular tetrahedron and the regular octahedron, but for these objects the scaling factor includes an irrational number, \(\sqrt{2}\). For the dodecahedron and the icosahedron, some of the vertex coordinates are themselves irrational no matter how the figure is scaled; the irrational number \(\varphi = (1 + \sqrt{5})\, /\, 2\) plays an essential role in the geometry.
This intrusion of irrationality into geometry troubled the ancients, but we seem to have gotten used to it by now. However, David Eppstein, a.k.a \(0\)xDE, writes about a class of polyhedra I still find deeply disconcerting: “Polyhedra whose vertex coordinates have no closed form formula.” In this context a closed-form formula is a mathematical expression that can be evaluated in a finite number of operations. There’s no universal agreement on exactly what operations are allowed; Eppstein works with computational models that allow the familiar operations \(+, -, \times, \div\) as well as finding roots of polynomials. He constructs a polyhedron—it looks something like the teepee his children used to play with—whose vertex coordinates cannot all be calculated by a finite sequence of these operations.
With this development we land in territory even stranger than that of the irrational numbers. I cannot draw an exact equilateral triangle on a computer screen because at least one of the coordinates is irrational; nevertheless, I can tell you that the vertex ought to have the coordinate \(y = 1 / \sqrt{3}\). In Eppstein’s polyhedron I can’t give you any such compact, digestible description of where the vertex belongs; the best anyone can offer is a program for approximating it.
Brent Yorgey in The Math Less Traveled offers an unlikely question: What’s the best way to read a manuscript printed on three-sided paper? If you have a sheaf of unbound pages printed on just one side, the obvious procedure is to read the top sheet, then shuffle it to the bottom of the heap, and continue until you come back to page 1. If the pages are printed on two sides, life gets a little more complicated. You can read the top page, flip it over and read the back, then flip it again and move it to the bottom of the sheaf. An alternative (attributed to John Horton Conway) is to read the front page, move it to the bottom of the heap, then flip over the entire stack to read the back side; then flip the stack again to read the front of the following page. In either case, you must alternate two distinct operations, which means you must somehow keep track of which move comes next.
The unsolved problem is to find the best algorithm when the pages are printed on three sides. “You may not be familiar with triple-sided sheets of paper,” Yorgey writes, “so here’s how they work: they stack nicely, just like regular sheets of paper, but you have to flip one over three times before you get back to the original side.”
Yorgey gives some criteria that a successful algorithm ought to satisfy, but the question remains open.
Mr. Honner takes up a problem encountered at a Math for America banquet:
Suppose you are standing several miles from the Pentagon. What is the probability you can see three sides of the building?
He discovers that it’s one of those cases where interpreting the problem is as much of a challenge as finding the answer. “In particular, it’s a reminder of how the different ways we model random selection can make for big differences in our solutions!”
Long-tailed data distributions—where extreme values are more common than they would be in, say, a normal distribution—are notoriously tricky. John D. Cook points out that long tails become even more treacherous when the variables are discrete (e.g., integers) rather than continuous values.
Suppose \(n\) data points are drawn at random from a distribution where each possible value \(x\) appears with frequency proportional to \(x^{-\alpha}\). Working from the data sample, we want to estimate the value of the exponent \(\alpha\). If \(x\) is a continuous variable, there’s a maximum-likelihood formula that generally works well: \(\hat{\alpha} = 1 + n\, / \sum \log x\). But with discrete variables, the same method leads to disastrous errors. In a test case with \(\alpha = 3\), the formula yields \(\hat{\alpha} = 6.87\). But we needn’t lose hope. Cook presents another approach that gives quite reliable results.
A post on DRHagen.com tackles a mystery found in an xkcd cartoon:
The size of each date is proportional to its frequency of occurrence in the Google Books ngram database for English-language books published since 2000. September 11 is clearly an outlier. That’s not the mystery. The mystery is that the 11th of most other months is noticeably less common than other dates. The cartoonist, Randall Munroe, was puzzled by this; hover over the image to see his message. Hagen convincingly solves the mystery. I’m not going to give away the solution, and I urge you to try to come up with a hypothesis of your own before following the link to DRHagen.com. And if you get that one right, you might work on another calendrical anomaly—that the 2nd, 3rd, 22nd, and 23rd are underrepresented in books printed before the 20th century—which Hagen solves in a followup post.
A shiny new blog called Off the Convex Path, with contributions from Sanjeev Arora, Moritz Hardt, and Nisheeth Vishnoi, “is dedicated to the idea that optimization methods—whether created by humans or nature, whether convex or nonconvex—are exciting objects of study and often lead to useful algorithms and insights into nature.” A December 21 post by Vishnoi looks at optimization mthods in the guise of dynamical systems—specifically, systems that converge to some set of fixed points. Vishnoi gives two examples, both from the life sciences: a version of Darwinian evolution, and a biological computer in which slime molds solve linear programming problems.
In the Comfortably Numbered blog, hardmath123 writes engagingly and entertainingly about numerical coincidences, like this one:
\(1337^{47168026} \approx \pi \cdot 10^{147453447}\) to within \(0.00000001\%\). It begins with the digits \(31415926 \ldots\).
The existence of such coincidences should not be a big surprise. As hardmath123 writes, “Since the rationals are dense in the reals, we can find a rational number arbitrarily close to any real number.” The question is how to find them. The answer involves continued fractions and the Dirichlet approximation theorem.
For the final acts of this carnival, we have a few items on mathematics in the arts, crafts, games, and other recreations.
In the Huffington Post arts and culture department, Dan Rockmore takes a mathematical gallery walk in New York. The first stop is the Whitney Museum of American Art, showing a retrospective of the paintings of Frank Stella, some of whose canvases are nonrectilinear, and even nonconvex. In another exhibit mathematics itself becomes art, with 10 equations and other expressions calligraphically rendered by 10 noted mathematicians and scientists.
Katherine, writing on the blog Will Knit for Math, tells how “Being a mathematician improves my knitting.” It’s not just a matter of counting stitches (although a scheme for counting modulo 18 does enter into the story). I hope we’ll someday get a followup post explaining how “Being a knitter improves my mathematics.”
Chelsea VanderZwaag, a student majoring in mathematics and elementary education, visits an arts-focused school school, where a lesson on paper-folding and fractions blends seamlessly into the curriculum. (An earlier post on the problem of creating a shape by folding paper and making a single cut with scissors provides a little more background.)
On Medium, A Woman in Technology writes about Dobble, a combinatorial card game. “My children got this game for Christmas. We haven’t played it yet. We got as far as me opening the tin and reading the rules, at which point I got distracted by the maths and forgot about the game.”
There are 55 cards, each bearing eight symbols, and each card shares exactly one symbol with each of the other cards. How many distinct symbols do you need to construct such a deck of cards. The game instructions say there are 50, but the author determines through hard work, scribbling, and spreadsheets that the instructions are in error. It all comes to a happy end with the formula \(n^2 - n + 1\) and a suggested improvement to the game that the manufacturer should heed.
That’s it for Carnival of Math 130. My apologies to a few contributors whose interesting work I just couldn’t squeeze in here.
Next month the carnival makes a hometown stop with Katie at The Aperiodical.
Hypertwistoploppic pseudotheomorphisms. (The last word is actually a thing.)
Thank you! The link to Comfortably Numbered blog by hardmath123 seems to be missing. Found it by searching Google for the quoted sentence; I guess it’s https://hardmath123.github.io/a-balance-of-powers.html
Thanks very much for letting me know. Fixed now.
This was a particularly excellent CoM, thanks for the fun read!
We want our kids to think deeply about mathematics. But what about ourselves? NCTM ( @NCTM ) President Robert Berry encourages teachers to Start the School Year by Unpacking Your Why .