ILLUSTRATING MATH TOGETHER

In 2016, the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, Rhode Island hosted a workshop called Illustrating Mathematics with the hope of bringing together researchers who, like me, study mathematical abstractions that can be brought to life by appropriate visuals. The workshop spawned a community that has held meetings at ICERM from time to time and has been running a webinar series since 2023.

I’ve spoken twice at the webinar. Back in 2024, I gave a brief mathematical eulogy for the brilliant mathematical explorer Roger Antonsen, now sadly deceased (though you can’t tell that he’s deceased from his website), who had a unique knack for coming up with cool visuals related to every topic we ever discussed. The striking figure below, arising from a deterministic model of a one-dimensional gas I’d proposed, is just one instance among many dozens he created as part of our email conversations.

On October 10th, 2025, I spoke at the webinar for a second time, even more briefly: I gave a five-minute “show-and-ask” pitch as a warmup-act for the phenomenal math explainer/animator Grant Sanderson (aka 3Blue1Brown). My lightning talk was entitled “Evolving cross sections of Ford spheres”, and it was my way of testing the waters of the webinar crowd. I wondered: if I described a compelling mathematical object that nobody has illustrated yet in a fully satisfying manner, or at least not in a way that I find fully satisfying, and I shared with other webinar attendees my vision of how one could make that mathematical object more available to the brain by way of the eye, then could I convince others, more skilled than I in the art of computer-assisted illustration, to bring my vision into reality?

The answer proved to be a resounding “Yes!” Roice Nelson (with whom I’ve corresponded in the past) was one of several people who expressed interest, and Roice and I have moved forward with this project. Arguably I shouldn’t be spending my time this way—I don’t plan to write any research articles on the Ford spheres. I just think that they’re cool things that other people would find interesting if they were better publicized. And they got stuck in my head like a catchy tune.

AN 87-YEAR-OLD FRACTAL

I’m sure you’ve heard of fractals—they had a moment back in the 1980s that basically never ended, with fractals penetrating not just the sciences and geek culture but popular culture as well, culminating in a line about frozen fractals in a stirring power ballad in the 2013 Disney movie Frozen. The Ford spheres form a three-dimensional fractal that not enough mathematicians know about, even though Lester Ford described it in a charming article called, simply, “Fractions”, back in 1938—thirty-seven years before Benoit Mandelbrot coined the term “fractal”.

There are actually many arrangements that are called Ford sphere arrangements nowadays, but the one Ford himself described looks like this:

This image is a still from a video made by Sam Wells and Aidan Donahue. The video gives some intuition for the fractal, but (to quote another Disney movie heroine) I want more.

What makes the Ford spheres worthy of study? From a research perspective, they’re descendants of a more famous two-dimensional fractal Ford wrote about in his 1938 article. The Ford circles are geometrical surrogates for the rational numbers, and the way the circles nestle against one another turns out to reflect important facts in number theory.

It stands to reason that the analogous three-dimensional fractals would have secrets to teach us as well.

ALL OVER THE PLACE BUT ALMOST NOWHERE

Another thing that makes the Ford spheres worthy of illustration is the way they offer math-loving non-mathematicians the chance to have their minds blown by the counterintuitive behavior of countable dense sets. The primordial example of such a set is the set of rational numbers: as elements of the real line, the rational numbers are all over the place but they’re also almost nowhere. I’ve chosen my phrasing to be provocative and a little paradoxical, but in a certain mathematical sense, it’s true: hardly any real numbers are rational, but no tiniest stretch of the real line is free of them. If you zoom in on (say) the square root of two, no matter how far in you zoom, you’ll keep on seeing rational numbers with ever-bigger numerators and denominators. Ford circles give geometric meaning to that bigness: the bigger those numerators and denominators are, the tinier the corresponding circles are.

All the Ford circles are tangent to a single horizontal line. One way to think about the Ford circles is as what you get when you try to pack together as many circles as you can above that line. You start by drawing evenly-spaced circles tangent to the number line at the points . . . , −2, −1, 0, 1, 2, . . . I’ll just show the two circles that touch the line at 0 and 1 and hereafter ignore all the circles to the left or right of them:

Then you add a circle to fill the gap between the 0-circle and the 1-circle, tangent to the line at 1/2):

Then you add more circles to fill the new gaps with tangencies at 1/3 and 2/3:

Then you add even more circles to fill the newer gaps with tangencies at 1/4, 2/5, 3/5, and 3/4:

If you continue this process, the circles you’ll draw are precisely the Ford circles, all tangent to the line, and the points of tangency will be all the rational numbers and nothing else.

Now imagine that, having drawn in the Ford circles (or as many of them as you have the patience to draw), you add to your picture a horizontal line parallel to, but slightly above, the line we were talking about before. This new line will intersect some of the circles. If you move that new line downward slightly, it’ll intersect more of the circles. As you continue to move the new line further downward, closer and closer to the original line (which I’ll call the “limit line”), you start to intersect more and more circles.

FROM TWO TO THREE

Ford also described a similar fractal one dimension up. We have infinitely many spheres, all tangent to the x, y plane, and the points of tangency correspond exactly to the points (x, y) with x and y rational. Here’s Ford’s sketch showing four of the infinitely many spheres:

(Yeah, four is a lot less than infinity, but cut Ford some slack: this was before computers.)

I want to picture this complicated three-dimensional object by way of its two-dimensonal cross-sections. Here’s one of the animations Roice sent me a few days ago, as part of our ongoing work:

It shows what you get when you intersect the Ford spheres with a moving plane that approaches, without ever reaching, the limit plane that all the Ford spheres touch (analogous to the limit line that all the Ford circles touch). As time passes in the video and the moving plane moves on, we see a mix of growing disks and shrinking disks; the shrinking disks are cross-sections of the spheres whose centers the plane has already passed through, while the growing disks are cross-sections of the spheres whose centers still lie just a bit ahead of us. The picture becomes frothier and frothier, approaching the limit of infinite fractal frothiness.

The video is just a rough cut, but already I can see features of the image that I didn’t expect: halos and solar arches, one might call them. Perhaps one of you will find a way to make rigorous mathematics of what your eyes are telling you, but even if not, I hope the animation gives you visual pleasure.

WHY I BOTHER

If this essay inspires any of you to drop by one of the monthly meetings of the Illustrating Math webinar, visit the webinar link. Or, if you’re feeling brave and want to pitch an idea or to give a five-minute presentation of any kind, go to the show-and-ask signup sheet. Or if you just want to see what other cool visuals Roice has created, check out his website.

I realized shortly before I published this essay that there is a connection to my research, though it’s not a direct link, and that it was probably subconsciously driving me to explore the Ford spheres. Two decades ago I was looking at “rotor-router blobs” that gave rise to images like this one, generated by Tobias Friedrich and Lionel Levine:

If you’re like me, your eyes and brain see ghostly circles (or near-circles), forming bands separated by lines of orange fire. The trouble is, those near-circles are very much creations of your eyes and brain, intermediated by software called ImageMagick. The task of figuring out what details at the pixel-level create ghostly near-circles in my brain defeated me. Such are the frustrations of “digital pointillism”: when we zoom in, we tend to lose sight of what we are trying to understand! It’s the problem faced by creators of monumental paintings: you have to stand close to the canvas to paint your strokes or dots or whatever, but when you stand close it’s easy to literally lose sight of the big picture. I’d like to try looking at those blobs again sometime, once I have the tools and the skills to “interrogate” such pictures more effectively than I could in the past.

A smaller-scale version of this gap in my skill-set manifests itself for the Ford spheres. Those halos and solar arches exist in my brain (and I hope in yours), but what do they correspond to at the pixel level? I don’t know how to ask the picture to tell me, but I’m hoping I can learn.

I’ll finish by mentioning one last reason for bringing the Ford spheres from the world of fancy to the world of the senses: videos like these can convey to non-mathematicians, in a way that words and symbols can’t, what makes math so addictive to those of us who love it.

Thanks to David Jacobi and Roice Nelson.

REFERENCES

L. R. Ford, Fractions, American Mathematical Monthly, 45, 586–601.

S. Northshield, Ford circles and spheres, 2015.

C. Pickover, Beauty and Gaussian Rational Numbers, Chapter 103 (pages 243-247) in: “Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning”, Oxford University Press, 2001.

S. Wells and A. Donahue, Ford spheres, 2021.