Bruce L. Chilton

MY INTEREST in polyhedron modelmaking and, more generally, in “Coxeterian” geometry began when I first met Bruce Chilton more than 40 years ago, in 1957. I was 11 years old, in the seventh grade of elementary school, and becoming interested in mathematics. He was a mathematics undergraduate at the University of Buffalo (now the State University of New York at Buffalo), paying his way by working part-time as a bottle washer in the virus laboratory where my mother was a technician. At the time, I had been pestering my parents to show me “higher math,” such as how to manually extract square roots and the ruler-and-compass construction for a regular pentagon, and my mother decided to introduce me to Bruce, probably to have a real math student deal with my questions. She invited him over for dinner.

Bruce proved a fountain of information. Extracting square roots (and cube roots!) manually proved too tedious to hold my interest for long, but his geometric sketches and models fascinated me. One evening—it might have been as early as his second visit—Bruce showed me how to construct a polyhedron model. He brought with him a sharp pencil, a few pins, a little steel protractor, a single-edged safety razor, polystyrene glue (we now use a white glue, such as Elmer’s), and some Manila file folders. With these tools, he quickly drafted a template for the face of the rhombic dodecahedron. He copied the template 12 times into a net for the solid—in two sections because the complete net would not fit on one sheet—then slashed it out with the razor. Minutes later, after folding the net along the scored edges and working the final face into place with a pin, a virtually perfect rhombic dodecahedron stood on the table. I was mesmerized.

This established a visitation pattern. Every few weeks—perhaps once or twice a month—my parents would arrange for Bruce to have dinner with us, and he and I would spend a couple of hours doing “hands-on” geometry. Because he had no car, my father drove him to and from his home for these visits.

Bruce left his copy of Cundy & Rollett’s Mathematical Models—the first edition, with the light blue dust jacket and red great icosidodecahedron on the front cover—for me to read. Its illustrations of the Archimedean solids and duals, the Kepler-Poinsot star polyhedra, and the regular compounds were my first exposure to these fascinating objects. They all looked frightfully complicated, and I imagined that their construction required great patience. My first attempts to model the simpler of these were pretty crude, but after a while I assembled enough of a collection for a little display for my eighth-grade math class. I included Bruce’s rhombic dodecahedron in the display, and it was obvious by simple inspection that somebody else—not me—had put it together.

Bruce’s technique of using a razor blade against the straight edge of a steel protractor gave him a perfectly straight cut and score, something I’ve never managed with scissors. Unfortunately, I never accustomed to using a razor and steel straight-edge, and so my polyhedron models often showed this little “scissor wobble” inaccuracy. Now I use a little paper cutter to get a straight edge, and this has largely cured the problem. Nevertheless, although the models I make now are quite serviceable, I’ve never mastered Bruce’s delicacy of touch and confidence in his own draftsmanship. Nor have I yet acquired his patience for assembling hundreds or thousands of pieces into a single polyhedron model. I like mine to go together quickly whenever possible, and I balk at building models that require lots of work. His models, regardless of their intricacy, are virtually perfect.

Though Bruce showed me a few of his more complicated models (such as the compound of five tetrahedra—which I now consider a rather simple model to assemble), I had no idea how extensive his own collection was until early in my first year of high school (1959). My math teacher arranged for me and a few other interested students to attend a small mathematics teachers’ conference organized, I think by the MAA, in downtown Buffalo. There, on a large table, was an exhibit of dozens of colorfully painted polyhedron models, including many not illustrated in Cundy & Rollett. Each was accompanied by a card listing its properties. Then and there I resolved to make mathematics my career: The idea of making polyhedron models and studying them all day struck me as a marvelous way to spend one’s time.

By the time Bruce graduated from UB, he had introduced me to stellations, faceting, the geometry of higher-dimensional spaces, hyperbolic tessellations, and a rich array of other geometric topics. He showed me the works of M. C. Escher well before Escher became a well-known artist in the world at large. Besides being a mathematician and a draftsman par excellence, he is also an accomplished photographer and cartoonist. Many evenings we spent not doing math but creating hilarious parodies of newspapers and magazines, which he illustrated with his cartoons. Other times he would arrive with an envelope full of his latest stereo photographs of polyhedron models. He also created stereo pairs from paintings of unsupported stellations, of which proper models are not constructible.

One of his projects was to enumerate and draw the stellations of the rhombic triacontahedron. This he abandoned once he realized there were literally millions of them. He also embarked on stellating a simpler Archimedean dual, the triakis tetrahedron. This solid has 12 congruent isosceles triangular faces (making it a “triangular dodecahedron”), and its set of stellations turned out to be pretty manageable—just a few hundred. Bruce compiled a notebook of colored drawings of them. Many resemble those four-pointed spikes strewn across roads by the military during wartime to flatten the tires of enemy vehicles. Another of his projects was to animate the passage of the simpler regular four-dimensional polytopes through 3-space, by hand-drawing their sections on notepads and rapidly flipping the pages. This project, years later, inspired my own M.Sc. thesis, for which I wrote a computer program to generate 16mm movies of arbitrary convex polychora passing through our 3-space.

Bruce’s visits continued until he graduated from UB and left to pursue his doctorate under Coxeter at the University of Toronto. The frequency of his visits necessarily diminished, but they did not cease entirely. For his doctoral dissertation (1962), on symmetric polytopes, Bruce extended the concept of stellation from three dimensions to four, and enumerated the stellations of the regular 24-cell (icositetrachoron). Here, instead of a “complete face,” there is a “complete cell,” whose various bits and pieces bound the four-dimensional units (I call them “stellachunks”) from which the stellations are constructed. As I recall, he found 25 distinct stellations of the 24-cell. The dissertation was illustrated, à la Coxeter, Duval, Flather & Petrie’s booklet of the 59 icosahedra, with the 25 different stellation hyperfaces. (These all have cuboctahedral symmetry, of course.) In addition, his dissertation included plane Petrie-polygon projections of the 120-cell (hecatonicosachoron) and great grand stellated 120-cell, which he drafted by hand. (This was in the days before computer graphics. Bruce is the only person I know who could actually sketch a plane Petrie-polygon projection of the 120-cell from memory, without even a ruler and compass; he drew about one-quarter of one at our kitchen table one evening.) The drawing of the 120-cell was published by Coxeter in his textbook Introduction to Geometry. There was also a study of the limiting surface of a polar zonohedron as the number of faces increases to infinity.

Once Bruce had earned his doctorate, he took a position as mathematics professor at the State University of New York at Fredonia. He continued to visit my family at random times until I graduated from high school in 1963 and afterward, during the summer vacations between my own college semesters and later, when I attended the University of Toronto myself as a graduate student with Coxeter as my thesis advisor. Bruce continued to build ever more intricate polyhedron models, including orthogonal shadows of all the regular 4-dimensional polytopes. Some of these he published in an article in Leonardo (I’ve forgotten the exact reference). He also modeled the dual of the final stellation of the icosahedron, which has 60 isosceles triangles meeting 9 at each vertex. It’s a ridiculously intricate polyhedron, considering it only has the 20 vertices of a regular dodecahedron, with 1320 facelets that have to be assembled to make the model. (Since it is the dual of H in Coxeter, Duval, Flather & Petrie, Bruce called it a huitzilopochtli. This started our game of naming intricate polyhedra after mythological deities. Neither we nor anyone else knew it at the time, but it has since turned out that this very huitzilopochtli is the vertex figure of the Great Prismosaurus, the uniform polychoron bounded by 720 pentagonal prisms profiled elsewhere at my Web site.)

Bruce and I collaborated on one project after I went to work for the University of Toronto Computer Centre. I acquired a copy of Magnus Wenninger’s Polyhedron Models and, after successfully building “Miller’s Monster” (the non-Wythoffian snub polyhedron) according to the plans therein, I decided to try my hand at the most complicated uniform polyhedron. (We called it a “yog-sothoth”; we have similar Lovecraftian names for the other Wythoffian snub polyhedra.) When I found the plans in the book were inaccurate, I started corresponding with Wenninger and eventually wound up redrawing the faces with computer graphics on a CalComp 30" plotter. Bruce more or less gave up modeling uniform polyhedra after Wenninger’s book appeared, because the adventure of constructing a polyhedron he had never before seen “in the flesh” was lost. The idea of building the world’s first accurate model of a yog-sothoth, however, picqued his interest—particularly since he didn’t have to go through the tedium of calculating the shapes of the faces—so he finished the model from my plans. (A somewhat more detailed account of this project was recently published in the Spring 1996 issue of 21st Science & Technology.) At about this time, Wenninger himself visited Bruce and me in Buffalo for a few days, I believe after the yog-sothoth was finished. (Alas, the yog-sothoth is no more. Bruce had to dispose of it for lack of space.)

Once I moved from Toronto across the continent to San Diego, California (in 1979), my occasional meetings with Bruce were confined largely to when I returned to Buffalo for the holidays. He visited me one summer, when we went to the Palomar Observatory together and he spent a day in the Mojave Desert bird watching (another of his hobbies). In 1984, we attended the Shaping Space conference at Smith College, where I presented my M.Sc. polychoron-sectioning movies and the yog-sothoth model and Bruce displayed an array of his polyhedron models. Thereafter, I seldom returned to Buffalo. My work—no longer in mathematics and computer science but in self-publishing—and other obligations consumed so much of my time that polyhedron modeling and playing with geometry had become a luxury. Indeed, after 1984, Bruce and I didn’t meet again face-to-face until January, 1996, when he attended my mother’s funeral. He told me he had taken an early retirement offer from Fredonia and was free of his “teaching chores.”

Bruce and I finally got together for dinner on one of my subsequent Buffalo trips in 1997. As a result, he says, his interest in four-dimensional figures has been rekindled, and he has written a program in Visual Basic that creates sections of the regular and other interesting polychora. The sections are input as data structures to Imagine, a commercial three-dimensional display system, which renders them as pictures using a ray-tracing algorithm. Bruce has also started using the Bryce package to create “polyhedral art,” such as the Stone Small Stellated Dodecahedron at the head of this Web page. As far as I know, he is the first person in the world to have sectioned all ten regular star-polychora using computer graphics.

Here is a picture from his 4-D sectioning program. Others will appear at my Web site as I continue to update it.

Above: An “edge-first” section of {3,3,5/2}, the grand hexacosichoron (600-cell), with the sectioning realm at a distance 0.1 of the circumradius from the center. The section has the symmetry of a pentagonal prism; the view is directly down its five-fold symmetry axis. The grand hexacosichoron is the closest analogue of the great icosahedron among the star-polychora; note the multitude of “deformed great icosahedra” about the vertices. Also note that you’re not seeing the polychoron itself, just a 3-D section of it (and a 2-D projection of that, to boot). All of its faces are sections of tetrahedra—either triangles or quadrilaterals, interpenetrating one another many times over.

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Text for this page ©1998 George Olshevsky. The Stone Small Stellated Dodecahedron art was produced by Bruce L. Chilton and appears here by permission. The section of the grand hexacosichoron was produced by George Olshevsky using Bruce Chilton’s 4-D sectioning program.

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This page was activated September 14, 1998. Latest modification was 9/24/98.