Physics & Math / String Theory

(Click on image to enlarge)

In this 3-D slice of the four-dimensional
hendecatope, colored beams represent the
edges of triangles; some triangles are left
out for simplicity.

(Computer model courtesy of Carlo Sequin,
UC Berkeley, styled by Jaron Lanier)

As a little boy I would look into the bright, immediate stars of the New Mexico night and curse the immensity of space. Some of those lights must be suns that warm other living beings, I told myself. If I could only meet that other life, I would have something to compare with the singular, lonely shape of life on Earth. Then I could know a little more of my place in the universe and be a little less alone. Like many other children who had pondered the night sky, I became fascinated with the one meeting of minds we already have with the aliens who may be hiding out there, too far away for our telescopes to resolve: mathematics.

Consider the Platonic solids. These are shapes, like the cube and the tetrahedron (the regular three-sided pyramid) in which every angle, every facet, and every edge is identical. There are only five such shapes in the three-dimensional world; the other three are the octahedron (eight triangular sides), icosahedron (20 triangles), and dodecahedron (12 pentagons). This was proved in ancient times by Euclid, and it is hard to overstate how profoundly amazing this proof must have been—and remains. The identities of the five shapes, and the certainty that there can be no more than five, is absolute and universal. While it is possible that an alien would never think to ask the question, all life everywhere would indisputably agree on the answer. A mathematical proof is something anyone can do, yet it is bigger than the universe.

One reason I was such a lonely kid is that my mom died in a car accident when I was 9. My dad decided that it would be good therapy to let me design and build a house out of the Platonic shapes that possessed me. This was also the early 1970s, in the middle of the hippie obsession with geodesic domes, so I designed a house that was a mix of domes, some Platonic solids, and some other interesting geometric shapes. My bedroom was an icosahedron. Some of the house is still standing, although part of it collapsed and almost killed my dad about 15 years later. Don’t let an 11-year-old design a building!

These memories came flooding back as I read Siobhan Roberts’s new biography of Donald Coxeter, the grand geometer of the 20th century. He is sometimes said to have “saved” geometry from the wave of mathematicians who were more interested in dry abstractions than in shapes and pictures. Buckminster Fuller, the famed designer of geodesic domes, said Coxeter’s mastery of shapes was “shared by but one or two other humans in all history.”

Buried in a footnote in the book was an anecdote that electrified me. Freeman Dyson, the renowned physicist, had remarked in an essay that “Plato would have been delighted to know about” a shape that Dyson—incorrectly, it turned out—said had been discovered by Coxeter: an 11-sided, perfectly regular polytope. (Polytope is Coxeter’s word for polyhedrons—forms like the Platonic solids—that exist in higher dimensions.) Dyson suggested this was the sort of obscure mathematical object that just might turn out to be important.

The idea of an 11-sided regular polytope was so startling that the book literally fell out of my hands. In order to explain why I was so shocked I need to go over some background.