Maybe Wolfram? Or Omohundro and Skiena?

Back in 1988, a contest sponsored by a major computer company asked for visions of a future personal computer. The winning team came from the University of Illinois at Urbana-Champaign. The faculty advisers were Stephen Wolfram and Stephen Omohundro (at far left in the photo above); the student members (from left to right) were Arch Robison, Steven Skiena, Bartlett Mel, Luke Young and Kurt Thearling. Their proposal of a device called the TABLET had eerie anticipations of the contraption that goes on sale this weekend.

Some quotes from the description of the TABLET:

This rectangular slab will weigh but a few pounds, and have no buttons or knobs to play with. The front surface will be a touch-sensitive display screen and will blink to life upon touching two corners.

The front surface of our computer is a high-resolution touchscreen that yields slightly to the touch. With this single input device, we can get a tremendous range of flexibility and options. We can use it to create an entirely soft interface.

A high-resolution color display can do more than just imitate a notebook page. It will be fast enough to support video…. It takes only a little more courage to predict a Global Positioning System (GPS) receiver on our machine, either as a clip-on or a built-in component.

If we can take our computer anywhere, we need to be able to use it anywhere. This brings us to communications capabilities. Through our national telephone network, we can access any person or machine within seconds.

It seems the UIUC prophets even saw Facebook coming over the horizon:

In addition to communicating with peripherals… we can also talk to other computers. Each machine can continually broadcast personal facts that users may wish the world to know: perhaps their name, image, interests, and marital status for openers.

And YouTube:

One interesting problem is who will appreciate all this new art? Some form of “shareware video” might arise.

Of course they didn’t get everything right. There’s the matter of the stylus and handwriting recognition (suggesting a foreknowledge of the Newton rather than the iPad). And it was all supposed to happen in the year 2000, so we’re a decade behind schedule.

Which computer company sponsored this contest? It was Apple, of course. The judges were Ray Bradbury, Alan Kay, Diane Ravitch, Alvin Toffler and Stephen Wozniak. (Steve Jobs was in exile at the time.) All of the prizes were paid in Macintosh merchandise.

The story of the 1988 predecessor of the iPad is told in Communications of the ACM, Vol. 31, No. 6, pp. 639–646. Included there is the first teardown photograph of an iPad (right). Note the wafer-scale integration. For those who can’t get past the ACM pay-wall, a slightly different telling of the story appears on Kurt Thearling’s web site. There’s also a brief account from the UIUC computer science department.

[Just to be clear: I am aware that there's a considerable difference between imagining an innovative device and actually building it.]

A few weeks ago I committed an act of deception. I needed a graph showing a few sets of data points, with curves fitted to them. This was strictly for show; neither the points nor the curves would mean anything; it just had to look good. So, out of laziness, I did it backwards: I drew a few arbitrary curves, chose some points on the curves, and then perturbed those points by small random displacements. The result is Figure 2 in Home Baked Graphics.

Apparently I have gotten away with this fakery, since no one has complained. But it nags at my conscience. The curves shown are not fitted to the given data points, but the other way around. And barring an extraordinary coincidence, we can’t expect the curves to give the best possible trend through the points.

I began to wonder: What if you iterate this process? In other words, you start out by generating a curve, say a simple quadratic:

You select a few points on the curve, in this case the five points with x coordinates equal to {1, 2, 3, 4, 5}. You perturb the y coordinates of these points by choosing random variates from a normal distribution with unit standard deviation:

Now you do a least-squares fit to the five selected points, generating a new quadratic:

At this stage you can start over, selecting five points on the new curve and randomly displacing them:

And a least-squares fit to those five purple points yields yet another quadratic curve:

So what happens if we continue in this way, repeatedly jiggling a few points on a curve and then fitting a new curve to the jiggled points? There’s a seeming hint of convergence in the three curves we’ve seen so far. They look like they might be approaching some fixed trajectory. Could that possibly be true?

I’ll post an update in a few days to say what more I’ve been able to learn about this question. If you just can’t wait, click here for a sneak peek.

The promised update (2010-04-02): You can’t fool anybody in this crowd, not even circa April 1. Of course there is no convergence to a limiting curve.

The butterfly above shows 100 quadratic curves, selected from 100,000 iterations of the least-squares fitting process, with that process applied at points whose x values are in the set {–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5}. Clearly enough, the coefficients of the quadratic curves are wandering widely; in particular note that the x² term can be either positive or negative, yielding curves that are concave upward or downward.

On the other hand, these are not just any old quadratic curves. They all have their apex (the point where f′(x) = 0) fairly close to the origin. Here’s a closer view:

(When Ros glanced at this graph on the screen, her comment was: “Bad hair day.”) And here’s a tracing of the wandering of the apex (taken from a different, longer, run of the program):

The path has the look of a classic random walk, but note that it’s confined to a narrow range of x values. What attractive force holds the apex in this region? It’s the decision to fit the curves to points in the interval –5 ≤ x ≤ 5. Here’s what happens if we instead fit to points in the range 100 ≤ x ≤ 110:

And note that without the symmetry-breaking x¹ term, the curve would always be a parabola with its apex on the y axis; only the curve’s shape (or scale) and the height of the y intercept could vary.

This whole process of repeatedly fitting curves to data and then deriving new data from the fitted curves is weird and useless enough that it seems unlikely to appear anywhere in the universe outside my own idle mind. And yet…. Think of the Dow-Jones index, which is calculated as a weighted average of the prices of 30 selected stocks. They are not always the same 30 stocks. Indeed, recent years have seen quite a lot of turnover in the components of the Dow: AIG was replaced by Kraft Foods; General Motors was dropped in favor of Cisco Systems; Honeywell lost out to Chevron. Would it be too utterly cynical of me to suggest that what’s going on here resembles a process of fitting a curve to randomly chosen points, then choosing more points that lie near the curve?

Tom Siegfried, the editor of Science News, has published a blistering indictment of statistical methods in science and medicine. I am moved to speak for the defense.

Siegfried discusses a number of specific cases, mainly drawn from the biomedical literature, where faulty statistical reasoning has led to unreliable or erroneous conclusions. I don’t want to quibble over the particulars of those cases; I’ll concede that science provides plentiful examples of statistical analyses gone wrong. Indeed, I could add to Siegfried’s list. But I see these events mainly as failures to use the tools of statistics properly; Siegfried suggests that the problem goes deeper. If I understand him correctly, he believes that the tools themselves are defective and that science would be better off without statistics. Here is how he begins his essay:

For better or for worse, science has long been married to mathematics. Generally it has been for the better. Especially since the days of Galileo and Newton, math has nurtured science. Rigorous mathematical methods have secured science’s fidelity to fact and conferred a timeless reliability to its findings.

During the past century, though, a mutant form of math has deflected science’s heart from the modes of calculation that had long served so faithfully. Science was seduced by statistics, the math rooted in the same principles that guarantee profits for Las Vegas casinos. Supposedly, the proper use of statistics makes relying on scientific results a safe bet. But in practice, widespread misuse of statistical methods makes science more like a crapshoot.

It’s science’s dirtiest secret: The “scientific method” of testing hypotheses by statistical analysis stands on a flimsy foundation.

This argument strikes me as so totally wrong-headed that I have a hard time believing Siegfried is serious about it.

To begin with, the snide remarks about Las Vegas and crapshoots are off-target. The branch of mathematics with roots in the study of gambling is not statistics but the theory of probability. The two fields are closely allied, but they’re not identical. Early statistical ideas came out of astronomy and geodesy, with later developments in the social sciences, genetics and agriculture. If you really must find some vaguely disreputable locale for statistics, the apt choice is not the casino but the brewery (a notable Student of statistics worked for Guinness).

More disturbing than this minor historical flub is Siegfried’s vision of a lost golden age of “rigorous mathematical methods,” debased by the seductive wiles of statistics. I don’t believe there was any such fall from grace. Siegfried doesn’t tell us much about the nature of his prestatistical mathematical paradise, but since he mentions Galileo and Newton, I suppose he may be thinking of classical mechanics as an exemplar of lost innocence. It’s true that the study of planetary orbits and ballistic trajectories does offer up some pithy mathematical laws that purport to be exact descriptions of nature:

We don’t usually attach error bars to these expressions, or hedge our bets by saying “Force is equal to mass times acceleration within one standard deviation.” But where do such “exact” laws come from? When Galileo performed his experiments with balls rolling down an inclined plane, the measured data did not exactly conform to a parabolic trajectory. Likewise with Newton’s inverse-square law: No real-world observations precisely follow the form 1/r²–not unless the experiment has been fudged. Making the leap from experimental data to mathematical law requires a process of statistical inference, where we extract some plausible model from the data and attribute any departures from the model to measurement error.

In the time of Galileo and Newton, tools for statistical inference were crude; by the time of Gauss, they were much sharper. In 1801 the newly discovered planetoid Ceres was observed for 41 days before it was lost in the glare of the sun. Astronomers hustled to predict where and when it would reappear in the sky. Among all the attempts, the clear winner was the prediction of Gauss, whose advantage in this competition was not so much superior astronomy as superior statistics. His secret was the method of least squares, which he later backed up with a comprehensive theory of measurement error, introducing the idea of the normal distribution.

Later still, statistics had a role in showing that the “exact” mathematics of Newtonian celestial mechanics is not exact after all. It took careful observations–and careful statistical analysis of those observations–to quantify a tiny anomalous precession in the perihelion of Mercury, explained by general relativity but not by classical gravitation.

Statistics is no “mutant form of math”; it’s the way that science answers the fundamental and inescapable question, “How do we know what is true?” I really can’t imagine how science could survive without statistics. What would replace it? Divination?

Siegfried complains that statistical tools offer no certainty–that when a result is reported as statistically significant at the 1-sigma 2-sigma level (or in other words with a P value of 0.05), there’s still a 1-in-20 chance that it’s a meaningless fluke. Quite so; that’s essentially the definition of a 2-sigma P value. But the uncertainty is not some methodological malfunction. It reflects the true limits of our knowledge. The strength of statistical reasoning is that it makes those limits explicit.

Again, I’ll readily agree that standards of statistical practice should be strengthened, and that weak or faulty conclusions are too common in some areas of the published literature. But the claim that “any single scientific study alone is quite likely to be incorrect, thanks largely to the fact that the standard statistical system for drawing conclusions is, in essence, illogical” is, in essence, illogical. Yes, we have lies, damn lies and statistics. But we also have lies and damn lies about statistics.

All my closest friends know about my strange obsession with the mathematics of mattress flipping. A few thousand other people also know my secret, since I have written about it in an American Scientist column (HTML, PDF), which later became a chapter in a book.

To recapitulate: Fussy housekeepers rotate their mattress twice a year to ensure even wear. But a mattress has three axes of twofold symmetry (roll, pitch and yaw). Rotating around the same axis over and over will not cycle through all four possible states of the mattress, so you need to vary the procedure. Perhaps you do a roll one time and a yaw the next. But in the spring you may have forgotten which way you turned last fall. Thus the quest for a “golden rule” of mattress flipping:

A golden rule of mattress flipping would be some set of geometric maneuvers that you could perform in the same way every time in order to cycle through all the configurations of the mattress. Following this algorithm might entail extra physical labor on each occasion–perhaps making multiple flips or turns–but at least it would eliminate the mental effort of remembering.

The rules of this peculiar game forbid putting any marks on the mattress (which would break the symmetry). If we abide by this constraint, the multiplication table for the group of flip operations looks like this:

R, P and Y indicate half-turns around the roll, pitch and yaw axes; I is the identity or do-nothing operation. This table is bad news for golden rules. We already know that no single operation will cycle through all four states of the system, and the table shows that every combination of two operations is equivalent to some single operation. Hence there is no golden rule.

There the matter stood until a few days ago, when I received a letter from Tim Knoll:

I just completed reading your collection of columns entitled “Group Theory in the Bedroom” and I wanted to offer up an alternative solution to the mattress flipping problem. I believe if you add one more operation to the mattresses you can achieve your “golden rule.” What you need to add is a second bed to allow a shift operation in addition to the rotations. If you have the second bed oriented in the opposite manner as the first bed (with the first having the headboard facing north, the second facing south) you can simply have an operation of a non-rotating shift from the first (north-facing) bed to the second (south-facing) and then do a shift in addition to a rotate about the pitch axis from the south-facing bed to the north-facing bed. This mattress juggling should also achieve the desired results using the roll axis in the second step. I unfortunately can’t implement this method in my own apartment since it is equipped with one queen-sized bed and one full-sized bed, but my tests using an index card seemed accurate.

Ingenious, no? But is it truly a golden rule? I’m going to leave that question for readers to decide.

Knoll also points out that even if the new “shift” operation doesn’t yield a golden rule, it does succeed in getting two mattresses flipped with the same mental effort that would otherwise be needed for one mattress. This raises a further question: Why stop at two beds? What about mattress flipping in a barracks, where many beds are lined up in a row? And then there’s the job of flipping mattresses in the Hilbert Hotel, which has infinitely many beds. Does the mental effort per mattress go to zero in this limit?

A couple of commenters have asked what software package I use to create the graphs that appear in bit-player posts–illustrations like the one below, which is a slightly improved version of something I posted last week. Let’s call it Figure 1.

Prompted by these inquiries, I immodestly ask myself: Why do my graphs look so darn good? I immodestly answer: It’s not because of any packaged software! I don’t need a cake mix, or even a recipe. These are home-baked graphs, made from scratch out of locally grown organic pixels.

I have strong opinions about the aesthetics of scientific illustrations, and I could certainly spout off about the design elements of Figure 1, such as that putty-colored background, just dark enough to allow drop-out white grid lines, yet neutral enough to avoid competing with the data curves, which also have a distinctive color scheme on which I could discourse at length. Yes, I can talk the Tufte talk. But I think the commenters were really asking how I create the graphs rather than why they’re so elegant, and so I’m going to focus here on the practical programming problem.

Most of my experience in drawing pictures with a computer comes from the world of print publishing, where the final product is ink on paper rather than pixels on a screen. Compared with the online environment, print has some advantages, notably higher resolution (up to 1,000 dots per centimeter) and precise control over typography and color. But print also has obvious limitations: On a magazine page, there are no mouseovers or clickable buttons, and you can’t make a square knot twirl in 3D.

Thirty years ago, the big challenge for computer-generated illustrations was not how to draw the picture but how to get it out of the computer and onto the printing press. You couldn’t just export a PDF and place it in a Quark or InDesign document; none of those things existed. The only practical option was to print out the artwork, photograph it, and “strip” the negative into the page-size film that would be used to make the press plate. Because of this emphasis on printouts, most of the effort went into programming the printer rather than the computer.

The figure below is the first published computer-generated illustration I had a hand in creating. It appeared in Scientific American in 1983.

The array of 28² tiny bar graphs was produced with an Epson MX-80 dot-matrix printer, using escape codes to fire combinations of the eight pins in the printhead. Of course the MX-80 was a black-and-white device. The two-color illustration was created from two separate printouts. Also, the Epson letterforms were replaced with typeset characters.

The world of computer-generated illustrations changed dramatically with the arrival of PostScript, the “page description language” created by John Warnock and his colleagues at Adobe Systems (based in part on earlier work at Evans and Sutherland and Xerox PARC). PostScript was designed as a complete programming language rather than just a file format or a set of drawing commands. And something else set it apart as well: attention to details of graphic design. With most earlier software (such as programs based on the Apple Quickdraw library), trying to create publishable figures was an exercise in frustration. For example, the apparent weight of a line would vary depending on its orientation: lighter when vertical or horizontal, heavier when diagonal. PostScript allows very precise control over such niceties of presentation. To take another example, where lines meet the edge of a graph, you don’t want to have to choose between falling short and overshooting; PostScript provides the tools needed to make it look right.

(The version in the rightmost panel is created by allowing the colored lines to extend outside the background box, and then applying a clipping mask that cuts off all objects at the boundary of the box.)

Obsessing over minute details like these may seem comically fussy, but I believe that neatness counts in these matters. To some extent, illustration is an art of illusion. Graphs and diagrams work best when you can look through them rather than at them. The viewer should be seeing the underlying information or abstraction–the array of correlation coefficients, the function y = f(x), or whatever–rather than noticing the mechanics of how the drawing was constructed. A ragged edge is the kind of distraction that destroys the illusion.

Although PostScript was a giant step forward from the MX-80 command set, in the early years it was still just another printer language, not a computer language. The only way I could execute a PostScript program was to send it to a laser printer and wait to see what came out. Sometimes it was a long wait. I had no way of running a PostScript program on the computer itself. (Ghostscript came later.)

My first PostScript illustrations were created as hand-written PostScript programs; the same language was used both for doing the computations and for presenting the results. The faces at right were created in this way. (They were inspired by the work of Herman Chernoff and drawn to illustrate an American Scientist article by Robert Levine in 1990.) The dual role of the language caused me a moment of disorientation just now when I went looking for my records of this project. I found an EPS (encapsulated PostScript) file, which I knew was the finished illustration, but where was the source code? And then I remembered: It’s the same file! Open it up in Ghostscript or Adobe Illustrator and you see those silly faces smiling or scowling at you; open the same file in a text editor, and you see procedures for drawing elements of the faces:

   /draweyes
     { newpath
       dx dy eyewidth eyeheight 0 360 ellipse stroke
       ex ey eyewidth eyeheight 0 360 ellipse stroke
     } bind def
   /drawpupils
     { fx fy pupilsize pupilsize 0 360 ellipse fill
       gx gy pupilsize pupilsize 0 360 ellipse fill
     } bind def

Bill Casselman, the graphics editor of the Notices of the American Mathematical Society, still favors this direct-to-PostScript methodology. He has written an excellent guidebook, taking you from the basics of PostScript through an elaborate library for rendering three-dimensional objects.

But here I part company from Casselman; I’d rather not do all my computing in PostScript. It’s not that I have anything against the language itself, but the development environment is not to my taste. I therefore adopted the modus operandi of writing a program in my language of choice (usually some flavor of Lisp) and having that program write a PostScript program as its output. After doing this on an ad hoc basis a few times, it became clear that I should abstract out all the graphics-generating routines into a separate module. The result was a program I named lips (for Lisp-to-PostScript).

Most of what lips does is trivial syntactic translation, converting the parenthesized prefix notation of Lisp to the bracketless postfix of PostScript. Thus when I write (lineto x y) in Lisp, it comes out x y lineto in PostScript. The lips routines also take care of chores such as opening and closing files and writing the header and trailer lines required of a well-formed PostScript program.

But the lips interface is low-level, confined to drawing individual dots, line segments, rectangles and the like. Assembling a complete graph out of these primitives is tedious. For example, the grid of white lines in Figure 1 would have to be drawn one line at a time, with each line specified by a sequence of commands such as

    (newpath)
    (moveto u v)
    (lineto x y)
    (stroke)

Before you can issue those commands, you have to calculate u, v, x and y. Clearly, a higher-level front end is needed; like everyone else, I call mine plot.

At the core of any plotting program is a simple operation: mapping points from an abstract user space to coordinates in a rectangular pane, the page space. In Figure 1, the y axis runs from 0 to 5000; values in this range have to be scaled to the dimensions of the graph, which is about 300 PostScript points, or 11 centimeters. Mathematically, the transformation is straightforward. Indeed, if I wished I could leave all the arithmetic to the PostScript interpreter, simply passing in the appropriate matrix elements for scaling and translation. This is an attractive option; it would allow plot to work entirely in user space. But a few niggling details get in the way. Consider the tick marks along the y axis in Figure 1. Their vertical positions are conveniently expressed in user coordinates: one tick every 500 units. But what about the length of the ticks–their horizontal extent? This dimension is purely concerned with the appearance of the graph and has nothing to do with the content; it ought to be expressed in unscaled units of points or pixels.

Here’s a possible solution: Let everything inside the rectangular frame of the graph–the area with the putty-colored background in Figure 1–go through the scaling engine, but define everything outside the frame, including the tick marks and the axis labels, directly in page coordinates. If you think this is the final answer, take a look at Figure 2:

In this nonsensical graph (constructed just for this occasion), data points are indicated by stars, crosses and diamonds. The positions of those glyphs ought to be defined in user space, but the drawing commands that create the shapes are properly defined in page coordinates. If we tried to draw the glyphs in user space, their size and shape would vary with position in the graph.

What’s the best way to deal with this messy situation? Is there some tidy solution that will reconcile the two coordinate systems and allow all dimensions to be treated uniformly? I don’t believe so; it’s just in the nature of graphs to mix up elements from these two disparate realms. We look through a window into a world of data or mathematical abstractions, but we also draw our own little doodles on the window itself.

Of course there are solutions; they’re just not as pretty as I would like. My own strategy for coping is to attach extra information to each geometric point, indicating whether or not the x and y coordinates are to go through the scaling transformation. This is less troublesome than it might seem; from the user’s point of view, it’s almost always invisible.

In writing the lips and plot programs, I walk a path that is already worn smooth by many earlier footsteps. I don’t know who wrote the first computer program for plotting data, but it probably came soon after the first program for producing data. Today we have hundreds of clever, comprehensive, well-designed and well-maintained programs for plotting and graphing. Gnuplot is very capable; Grace is one I’ve never used but I’ve heard good things about it; Mathematica, Sage, R, MATLAB, Octave and the like all have elaborate graphics facilities built in; the Python world, as usual, has an overabundance of options; there are a few libraries for my beloved Lisp; you can even do dataviz online.

All of which raises the question of why I bother to roll my own. I’ll never keep up–or even catch up–with the efforts of major software companies or the huge community of open-source developers. In my own program, if I want something new–treemaps? vector fields? the third dimension?–nobody is going to code it for me. And, conversely, anything useful I might come up with will never benefit anyone but me.

The trouble is, every time I try working with an external graphics package, I run into a terrible impedance mismatch that gives me a headache. Getting what I want out of other people’s code turns out to be more work than writing my own. No doubt this reveals a character flaw: Does not play well with others.

In any case, the time for change is coming. My way of working is woefully out of date and out of fashion. PostScript is a technology that even Adobe seems to regard as outmoded. And making ultraprecise PostScript graphs is quite silly when their destination is the web; before I can put them online, I have to convert them to low-res PNG images. Furthermore, a PostScript-based workflow loses out on all the interactive richness of the web. These are deathly still images. How can I expect to earn any web cred when my work is not even clickable, much less multitouch-enabled?

If I continue in my stubborn, do-it-yourself mode, I could replace the PostScript back end with one that generates SVG. This wouldn’t be a major undertaking. But is SVG the right answer? It’s been around for more than a decade and you still don’t see much of it in the wild. And there are horrid browser incompatibilities. I suspect that Javascript (and JQuery) has a brighter future. And if I can get over my abreaction to libraries, there are plenty of options. Advice anyone?

Update 2010-03-13: Many thanks for all the thoughtful comments. Herewith a few comments on comments:

Ron Renaud’s graphs using Javascript in a canvas element are really very pretty, and they give me renewed hope that web graphics can measure up to a print standard. But is the world quite ready for the canvas? This is a blog, after all. Lots of people get to it with an RSS reader, not a web browser.

Zvika requests a link to a higher-resolution PNG. I don’t know how to do that. I can make a larger PNG, but the resolution–the dots per centimeter–is really determined by the screen you’re looking at. Which is not to say that larger illustrations wouldn’t be a good idea. When I redesign these pages, I want to allow more room for bigger pictures.

Gary Reuben suggests Python Matplotlib and the ggplot2 package for R. The latter is new to me, and very impressive. I want to go read more about it.

Several other readers favor SVG. I’m okay with that. It looks easy to change my current software to generate SVG output instead of (or in addition to) PostScript. The question remaining for me is whether SVG on the web is something that browsers (and, again, RSS readers) can swallow without choking.

John Haugland mentions PDF as the successor to PostScript. I couldn’t possibly survive without PDF these days, but I don’t see it as an ideal medium for illustrations embedded in web pages. Reading PDFs within a browser requires a plugin, which some people refuse to install. (I’m one of those people.) Furthermore, because PDF is a binary format based on a directory of offsets to tables, it’s more trouble to write PDF files than either PostScript of SVG.

Nate mentions Processing, the graphics and animation language created by Casey Reas and Ben Fry. I’ve written about this before at bit-player and elsewhere. I’m a big fan, but Processing is essentially a front end to Java, and I have reservations about embedding Java applets in web pages. John Ressig’s reimplementation of the language in Javascript overcomes that problem, and one of these days I’ll get around to doing something serious with it.

Finally, Marc asks for a look at my Lisp code. I’m always shy about sharing such unfinished things, but here it is.

Update 2010-03-28: When commenter Zvika asked for higher-resolution graphics, my reply was unhelpful; let me now try to say something more useful.

Even though I can’t do anything to improve screen resolution, I can, as Zvika notes, provide a larger version of each image. The Wikipedia way of doing this, which Zvika mentions, is to link to a separate HTML page, which replaces everything on the page you’re currently reading. As a test, I’ve done this with Figure 1 above; click on that figure and you’ll be whisked away to a supersized version of the graph. To get back here, you’ll have to use your browser’s “back” button.

Another approach, adopted by, for example, the New York Times, is a pop-up window. I’ve done this with Figure 2 above; click on it and a new window with a bigger version will open in front of this page.

Personally, I dislike both of these strategies. I want to see the words and the pictures at the same time in the same context. I suppose I could implement (or swipe) some glitzy Ajaxian solution that expands the artwork box within the window. But I still feel the right solution is to redesign the pages so that there’s room for larger illustrations in the first place. (Admittedly I’ve been meaning to do that for more than a year.)

There’s also a question about how best to make use of a more spacious picture box. In the supersized Figure 1, the transformation is much like an optical enlargement, projecting the same information at larger scale. The overall dimensions are doubled, and so is the width of each line, the size of the type, and so on. This is probably the answer for those troubled by visual deficiencies (including my own presbymyopic eyes).

In Figure 2 I have enlarged the illustration by 150 percent but kept all the individual graphic elements (line weights, labels, the glyphs that mark data points) the same size. This kind of rescaling allows information to be read from the graph with greater precision.

Perhaps a middle way is sensible here. Indeed, maybe type should scale as the square root of graph size?