"ached a bit from sheer love of it all, feeling perhaps, deep down, that he had not very much longer to enjoy it. The thought that some day - perhaps not ten years hence, perhaps not five - all this world would be taken away from him, before he had exhausted his powers of loving it, seemed to him in the nature of an injustice, brooding over his horizon. If anything came after this life, it wouldn't be what he wanted."

Commentary on the last years of Coxeter's life in "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry", by Siobhan Roberts.

What happens at dimensions higher than 4?

The only regular polytopes are the Simplexes, the Hypercubes and their duals, the Orthoplexes, or cross polytopes. These polytope families are infinite, extending to any number of dimensions. All Orthoplexes and Hypercubes form dual pairs. Apart from them, there are no equivalents of Icosahedra, there are no regular star polyhedra either.

In this way, one comes to appreciate the fact that a few of the regular polytopes shown above, like those with Icosahedral, Hexacosichoric and Icositetrachoric symmetries, are

There is one more infinite family of uniform polytopes that has been thus far hidden: the Demi-hypercubes. These are obtained by removing alternate vertices of a Hypercube. Although they generate interesting phenomena in 4 dimensions, like triality, this family is thus far hidden because all polyhedra and polychora built with the symmetry of the Demi-cube and Demi-tesseract already exist as members of other families: the Demi-cube is the Tetrahedron and the Demi-tesseract is the 16-cell. It is only in 5 dimensions that it has polytopes that are unique to it. One of these is the semi-regular Demipenteract, which has either 16-cells or 5-cells as 4-faces. But here ends the list of Demi-hypercubes that are either regular or semi-regular.

The Demipenteract is interesting because it also belongs to the finite semi-regular k

In 6 dimensions, the k

In seven dimensions, the semi-regular 3

Finally, in eight dimensions, the semi-regular 4

That's it! The family ends, because its next member (still in eight dimensions!) is the infinite 5

Fig. 41a: A Zometool model of the 4_{21} polytope.
The 240 vertices project here as the vertices of two concentric 600-cells, with one of them being
larger than the other by a factor of φ. It is not possible to project all the edges into a 3-D
Zometool model, so we must choose a subset. In this case, we chose the edges of the projections of
the two concentric 600-cells, exactly as in
David Richter's model.
Here we can see that projecting objects from dimensions higher than four in three dimensions leads
to a concentration of vertices and edges near the center.

This model required many old very long struts, no longer manufactured by Zometool: 120 B3s, 60 R3s
and 120 Y3s. Alternatively, the model can be made on a smaller scale.

For more on the Zometool models of the k_{21} polytopes, check this
vZome page.

Fig. 41b: An orthographic projection of the 4_{21} polytope in 3-D space, now with all the edges represented. This model is laser-engraved in glass.

This was presented to me by one of my students, Jose Martinez.

Diagram III: Print on my door: The orthographic projection of the 8-dimensional
4_{21} polytope onto the
Coxeter plane
where its projection has maximal symmetry.
This 2-D projection highlights even more the concentration of vertices
towards the center.

Image by Claudio Rocchini (Own work), via Wikimedia Commons.

That's not the end of the fun, though. That has just started...

Consider the 24-dimensional LeechLattice, where each sphere "kisses" 196560 neighbors, and provides the most compact sphere packing in that dimension! The number of symmetries in this lattice is:

8 315 553 613 086 720 000.

Or consider the Monster group, which can be represented by a polytope in 196883 dimensions! The number of its symmetries is:

808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000,

which is about 6000 times the number of atoms on Earth. As John H. Conway has said, we've only started scratching the surface of the laws of symmetry!

Next: References