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.
First of all, about the higher dimensions: You, as a three-dimensional being, can look down on a two-dimensional world, like a drawing on a piece of paper. A creature living in the “flatland” of the paper could also look around and see the same drawing, but only from the side, as a collection of lines lined up end to end. (There’s a famous novel called Flatland about life in a 2-D world—a worthwhile and surprisingly entertaining read.) If you put a coffee cup down on a piece of paper, the Flatland creature would see only the circle of the base, not the whole structure, and even that circle would look like a line since it could be seen only from the side.
In much the same way, it is possible to imagine shapes in higher dimensions, but we 3-D creatures can see only 3-D objects that represent small slivers of complete four-dimensional objects. For instance, the 4-D version of the cube, called a tesseract or a hypercube, can look like two cubes with interconnecting lines to us—that is, one 3-D intersection, just as a circle is one intersection of a cup within a 2-D world.
As it happens, there are six analogues of Platonic solids in the fourth dimension. They have 5, 8, 16, 24, 120, and 600 “sides” (although in the fourth dimension each side is 3-D, so the sides are called cells). You might think that this fact is of little more than academic interest, but actually these 4-D shapes are incredibly important. They represent some of the most fundamental symmetries in nature.
The concept of symmetry is so simple that it is actually difficult to capture. Symmetry is the order that exists when different things are related in consistent ways. The image in a mirror has symmetry because it is identical except for having the left and right directions flopped. A starfish has rotational symmetry in that you can give it one-fifth of a turn and it looks the same as before. Theoretical physicists spend much of their time contemplating other, more complicated symmetries that help explain the patterns seen in nature. The common thread among symmetries is that they are all governed by mathematics.
Many a fine mathematician has had the same initial reaction when hearing about the 11-cell shape: Impossible. Well, it’s not.
Which brings me back to the Platonic solids, whose regularity of shape is a rigorous form of symmetry. The idea of an 11-sided Platonic shape with a prime number of sides initially sounds wrong. The essence of symmetry is that one part mirrors the other, so you ought to be able to break a symmetrical object into similar pieces, the very thing a prime number refuses to do. (Before you ask: The 5-cell shape—also called a simplex—is too simple even to think about breaking apart. It is the simplest possible 4-D polytope. The simplest Platonic solid in three dimensions is the tetrahedron, the three-sided pyramid, which has four points. To move it into four dimensions, you need to add a point to take up room in the extra dimension, hence five points.)
Many a fine mathematician has had the same initial reaction when hearing about the 11-cell shape: Impossible. Well, it’s not impossible, it’s true. To dispose of one obvious objection, yes, it’s proved we already know all six of the 4-D regular polytopes—but the 11-cell evaded attention by having an unusual form. It is therefore designated an “abstract” polytope, as if the fourth dimension weren’t abstract enough. What makes the 11-cell abstract is that if the cells were separated, they could not serve as conventional 3-D objects, because they have some odd qualities, such as the fact that their sides can pierce or coincide with each other.
To untangle the mystery of the 11-cell, I called on my friend Carlo Séquin, a professor at the University of California at Berkeley. Carlo is another sufferer of Plato’s disease. His office is filled with amazing sculptures of strange shapes, including various 3-D projections of 4-D objects. Many of these have never been realized in physical form before. Carlo often has to create his own programs to direct robots to build these shapes or to guide lasers to form them out of chemical baths.
After convincing himself that the 11-cell is real, Carlo caught my obsession with seeing it. I contacted every mathematician I could find who had worked with the 11-cell, including Branko Grünbaum at the University of Washington, who turned out to have discovered it in the 1970s, before Coxeter described it more thoroughly. Amazingly, it seemed no one had ever tried to create a picture of the thing.
Carlo and I set to work, first visualizing a single cell. Each “side” of an 11-cell is a shape called a hemicosahedron (a.k.a. a hemi-icosahedron). You can visualize it as half an icosahedron that is folded into an octahedron with some missing outer faces, plus some extra internally coinciding and interpenetrating extra faces. (Words don’t really suffice here.) A hemicosahedron has 10 cells. Glue more hemicosahedrons on each of these and you get 11 cells total.
Amazingly, in 4-D space these forms connect to each other in a perfectly regular symmetry. Furthermore, the form is self-dual, meaning that if you draw lines between the centers of every facet in the 11-cell, you get another 11-cell. If you do that to a cube, you get an octahedron. So, in an important sense, the 11-cell is more elegantly symmetrical than a cube.
On these pages, I am thrilled to present to you the first published picture of the wondrous 11-cell. There are 11 colors in this image, one for each of the hemicosahedron cells. Now that we can see it, I would like to give the 11-cell a nickname. I suggest hendecatope, meaning “11-related place” in Greek.
Adding a final twist to the story, Dimitri Leemans of the Free University of Brussels and Egon Schulte of Northeastern University showed last year that there can be only two shapes like the 11-cell. The other is a 57-cell shape (discovered by none other than Coxeter), but 57 is not prime. So the 11-cell is truly the only one of its kind.
Of what use is all this? Maybe nature will have found some use for the symmetries of the hendecatope. In theoretical physics? Perhaps something in the life cycle of a living cell? Sooner or later, as Freeman Dyson suggested, the 11-cell might turn out to be important.
Beyond that, though, is the certainty that somewhere up there in the sky, some form of life that might be otherwise incomprehensible has had the same thought on the same magic occasion.
I dedicate this piece to Rich Newton, a former dean at UC Berkeley, who died last January. He brought me into my new position at the university, which enabled me to work with Carlo, making this whole chain of ideas possible.
