Beyond the fourth dimension

Coxeter, like Forsyte's character "Old Jolyon", "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 Simplices, the Hypercubes and their duals, the cross polytopes. These polytope families are infinite, extending to any number of dimensions. 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 exceptional objects.

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, as we've seen, all polyhedra and polychora built from these symmetries using the Wythoff construnction already exist as members of other families. All are Zomable, and two are regular: the Demi-cube is the Tetrahedron and the Demi-tesseract is the 16-cell.

It is only in 5 dimensions that this symmetry generates, by the Wythoff construnction, new polytopes that are unique to it. One of these is the semi-regular Demipenteract, which has 16-cells and 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 k₂₁ polytope family. In this family the facets of each polytope are Simplices and cross polytopes and the vertex polytope is the (k-1)₂₁ polytope. In that family the Demipenteract is the 1₂₁ polytope. Its vertex polytope is the 0₂₁ polytope; as mentioned in the polychora page, this is the rectified 5-cell.

In 6 dimensions, the k₂₁ symmetry starts generating polytopes with symmetries unique to it. Of particular interest is the 2₂₁ semi-regular polytope. With its symmetry - associated with the E6 group - we can use the Wythoff construction to build and 38 other uniform polytopes.

In seven dimensions, the semi-regular 3₂₁ polytope has a symmetry related to the E7 group, using the Wythoff construction we can build 126 other uniform polytopes from this symmetry.

Finally, in eight dimensions, the semi-regular 4₂₁ polytope (see Figs. 9.1 and 9.2, and Diagram III below) has a symmetry related to the E8 group, we can use the Wythoff construction to build 254 other uniform polytopes with this symmetry. The largest of these, the Omni-truncated 4₂₁ polytope, has 696,729,600 vertices, the number of symmetries of this group.

That's it! The family ends, because its next member (still in eight dimensions!) is the infinite 5₂₁ honeycomb. Its vertex polytope is the E8 polytope and its vertex arrangement is the E8 lattice; this provides the most compact sphere packing in 8 dimensions. In 8-dimensional space, each hypersphere can "kiss" 240 other hyperspheres (note that 240 is the number of vertices of the 4₂₁ polytope). Thus, the E6, E7 and E8 are also exceptional objects.

Fig. 9.1: A Zometool model of the 4₂₁ 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 centre.
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.

Fig. 9.2: An orthographic projection of the 4₂₁ 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 4₂₁ polytope onto the Coxeter plane where its projection has maximal symmetry. This 2-D projection highlights even more the concentration of vertices towards the centre.
Image by Claudio Rocchini (Own work), via Wikimedia Commons.

For more on the Zometool models of the k₂₁ polytopes, especially their Zomable projections, check this vZome page.

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

Consider the 24-dimensional Leech lattice, where each sphere centered on each point of the lattice "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!

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