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 solids—and a proof that there are no more than five.
"It is hard to overstate how profoundly amazing this proof must have been—and 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-cell—one consisting of a symmetrical arrangement of five hemi-icosahedra.
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.
References:
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.
This "blog entry" basically copies the points made in Jaron Lanier's column on the 11-cell that appeared in Discover, even down to mentioning what few mathematicians would consider relevant: that 11 is prime although 57 is not.
ReplyDeleteit is very hard to see what "The Fabulously Odd 11-Cell" adds to the discussion. It is more notable for what it doesn't add than what it does: It barely tells us what the 11-cell is, though anyone familiar with the 11-cell could explain this in layman's language using no more words than this author uses to inflict his lack of understanding on us.
The article also completely fails to explain on what basis the 11-cell is considered a "4-dimensional" object.
(Unlike the ordinary regular 4D polytopes -- having 5, 8, 16, 24, 120, and 600 3D cells -- the 11-cell has no pieces that are 3D in any common meaning of that term.)
Alas, this level of insight is all too typical of this author.
'no other polytopes like the 11-cell, except the 57-cell'. Well, in the sense that their symmetric group is one of a list of projective special linear groups, perhaps. To the layman, it might be more useful to ask for polychora whose 'cells' are 'like' the 11-cell's - in that they are a familiar shape folded in on itself. And then, there are many more than just the two mentioned.
ReplyDeleteAnd anyway, err, doesn't the regular pentachoron (that is, the 4D tetrahedron) have a prime number of cells too????
Are there any regular polytopes whose symmetry groups are sporadic groups (like the Mathieu groups, for instance?)
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