(This post was originally published on the Wolfram Blog.)
A few times a year they would arrive. Email dispatches from an adventurous explorer in the world of geometry. Sometimes with subject lines like “Phenomenal discoveries!!!” Usually with images attached. And stories of how Russell Towle had just used Mathematica to discover yet another strange and wonderful geometrical object.
Then, this August, another email arrived, this time from Russell Towle’s son: “…last night, my father died in a car accident”.
I first heard from Russell Towle thirteen years ago, when he wrote to me suggesting that Mathematica’s graphics language be extended to have primitives not just for polygons and cubes, but also for “polar zonohedra”.
I do not now recall, but I strongly suspect that at that time I had never heard of zonohedra. But Russell Towle’s letter included some intriguing pictures, and we wrote back encouragingly.
There soon emerged more information. That Russell Towle lived in a hexagonal house of his own design, in a remote part of the Sierra Nevada mountains of California. That he was a fan of Archimedes, and had learned Greek to be able to understand his work better. And that he was not only an independent mathematician, but also a musician and an accomplished local historian.
In the years that followed, Russell Towle wrote countless Mathematica programs, published his work in The Mathematica Journal, created videos (“Regular Polytopes: The Movie”, “Joyfully Bitten Zonotope”, …) and recently began publishing on The Wolfram Demonstrations Project.
In 1996, he sent me probably the first image I had ever seen of digital elevation data rendered in Mathematica.
His last letter to me was this May, where he explained:
“Along the way it occurred to me that it would be interesting to treat the zonogons of a plane tiling as pixels; an example is attached, in which I mapped a photo of a butterfly onto a patch of Penrose tiling.”
Throughout the years, though, zonohedra were Russell Towle’s greatest passion.
Everyone knows about the five Platonic solids, where every face and every vertex configuration has the same regular form. Then there are the 13 Archimedean solids (PolyhedronData["Archimedean"]; the cuboctahedron, icosidodecahedron, truncated cube, etc.), constructed by requiring the same configuration at each vertex, but allowing more than one kind of regular face.
The zonohedra are based on a different approach to constructing polyhedra. They start from a collection of vectors vi at the origin, then simply consist of those regions of space corresponding to ∑ ai vi where the 0 < ai < 1.
With two vectors, this construction always gives a parallelogram. And in 3D, with three vectors, it gives a parallelepiped. And as one increases the number of vectors, one sees a lot of familiar (and not so familiar) polyhedra.
I’m not sure how the “known” polyhedra (included for example in PolyhedronData) are distributed in zonohedron space. That would have been a good question to ask Russell Towle.
My impression is that many of the “famous” polyhedra have simple minimal representations as zonohedra. But the complete space of zonohedra contains all sorts of remarkable forms that, for whatever reason, have never arisen in the traditional historical development of polyhedra.
Russell Towle identified some particular families of zonohedra, with interesting mathematical and aesthetic properties.
Zonohedra not only have all sorts of mathematical connections (they are, for example, the figures formed by projections of higher-dimensional cubes), but may, as Russell Towle suggested in his first letter to me, have practical importance as convenient parametrizations of symmetrical geometric forms.
In recent years, zonohedra have for example begun to find their way into architecture. Indeed, a 600-foot approximation to a zonohedron now graces the London skyline as the Swiss Re building (“Gherkin”).
There is a certain wonderful timelessness to polyhedra. We see ancient Egyptian dice that are dodecahedra. We see polyhedra in Leonardo’s illustrations. But somehow all these polyhedra, whenever they are from, look modern.
That there is more to explore in the world of polyhedra after two thousand or more years might seem remarkable. Part of it is that we live in a time of new tools—when we can use Mathematica to explore the universe of geometrical forms. And part of it is that there have been rather few individuals who have the kind of passion, intuition and technical skill about polyhedra of a Russell Towle.
It’s good that Russell Towle had the opportunity to show us a little more about the world of zonohedra; it’s sad that he left us so soon, no doubt with so many fascinating kinds of zonohedra still left to discover.