I’m a great believer in low-tech math. I don’t like to rely on things a computer tells me; what if there’s a bug in the code? I prefer trusting things that I can check for myself. At the same time, I’m keenly aware of the limits of my imagination even when it’s aided by paper and pencil. Sometimes I need a computer to show me things I can imagine myself imagining but don’t yet know how to imagine.
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:
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