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
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
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
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.