The Mathematical Tourist
By Ivars Peterson
Polyhedrons consist of triangles, squares, pentagons, hexagons, and other polygons that are joined together to form closed, three-dimensional objects. Different rules for linking various polygons generate different types of polyhedrons.
The Platonic solids, known to the ancient Greeks, consist entirely of identical regular polygons, which are defined as having equal sides and equal angles. There are precisely five such three-dimensional objects: the tetrahedron (made up of four equilateral triangles), cube (six squares), octahedron (eight equilateral triangles), dodecahedron (12 regular pentagons), and icosahedron (20 equilateral triangles).
These objects are all convex; in other words, they have no indentations. Their extreme regularity also gives them a high degree of symmetry.
Euclid's Elements contains a lengthy mathematical description of the Platonic solidsand a proof that there are no more than five.
"It is hard to overstate how profoundly amazing this proof must have beenand remains," Jaron Lanier writes in the April Discover. "The identities of the five shapes, and the certainty that there can be no more than five, [are] absolute and universal."
You can extend the notion of a Platonic solid to four dimensions, though the results can be very difficult to visualize. There are six such analogs, having 5, 8, 16, 24, 120, and 600 cells, where a cell is the three-dimensional analog of a polygonal face, or side.
But there's another four-dimensional curiosity that can also be considered analogous to a Platonic solid: a four-dimensional shape made up of 11 identical cells, or 11-cell. This curious form was discovered in 1976 by Branko Grünbaum, then later rediscovered and analyzed by H.S.M. Coxeter.
The fact that this shape consists of 11 (a prime number) faces might suggest that the form would lack the high degree of symmetry expected of a Platonic solid. Indeed, Freeman Dyson once asked, after hearing of Coxeter's discovery, "Can you imagine a regular polyhedron, a body composed of perfectly symmetrical cells arranged in a perfectly symmetrical structure, having a total of eleven faces?"
But the 11-cell has an unusual form. Its identical cells, when separated, aren't conventional, three-dimensional objects. And their sides actual pierce or coincide with each other in the combined form.
Plato would have been delighted to know about it, Dyson commented to Coxeter.
Lanier, with the help of computer scientist Carlo Séquin of the University of California, Berkeley, set out visualize this odd object. Each component of an 11-cell is a special shape called a hemi-icosahedron. You can visualize it as half an icosahedron that is folded into an octahedron, with some missing faces and a few others that coincide or interpenetrate. Yet, in four dimensions, these strange cells fit together in a perfectly regular manner.
The following image, created by Séquin and Lanier, shows a partial 11-cellone consisting of a symmetrical arrangement of five hemi-icosahedrons.
The 11-cell is also self-dual. If you draw lines between the centers of every facet of the 11-cell, you get another 11-cell.
Lanier originally dubbed this shape a hendecatope, meaning "11-related place" in Greek. The name was later changed to hendecachoron to differentiate it from certain other objects. Lanier and Séquin have prepared a presentation on the hendecachoron for the upcoming ISAMA '07 conference in College Station, Texas.
Last year, Dimitri Leemans and Egon Schulte showed that there can be only two shapes like the 11-cell. The other is the 57-cell, which had been discovered by Coxeter. However, 57 is not a prime number, so the 11-cell retains a unique place in the pantheon of polytopes.
Comments are welcome. Ivars Peterson is Director of Publications for Journals and Communications at the Mathematical Association of America. You can reach him at firstname.lastname@example.org.
Dyson, F. 1983. Unfashionable pursuits. Mathematical Intelligencer 5(No. 3):47-54.
Lanier, J. 2007. Jaron's world: Shapes in other dimensions. Discover 28(April):28-29.
Leemans, D., and E. Schulte. In press. Groups of type L2(q) acting on polytopes. Advances in Geometry. Abstract.Peterson, I. 2001. Polyhedron man. Science News 160(Dec. 22&29):396-398.
Roberts, S. 2006. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company.
Séquin, C.H., and J. Lanier. Preprint. Hyperseeing the regular hendecachoron.
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