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 analogues of Icosahedra/Dodecahedra or 600-cell/120-cell, there are no regular star polytopes either, and nothing like the 24-cell, which exists only in 4-dimensional Euclidean space. In this way, one comes to appreciate the fact the latter are exceptional objects.

As we remarked when discussing the partially regular polychora, associated with the cross polytopes are the non-convex demi-crosses. There is a related infinite family of convex uniform polytopes, the aforementioned demi-hypercubes. The demi-crosses are obtained by removing alternate facets of cross polytopes and using their equatorial n − 1 cross polytopes as new facets; the demi-hypercubes are obtained by removing alternate vertices of the duals of the cross polytopes, the hypercubes. This implies that the demi-crosses share the demi-hypercubic symmetries.

The demi-hypercubic family is thus far hidden because, as we've seen here, all uniform polyhedra and polychora built from these symmetries using the Wythoff construnction already exist as members of other families, two of them being regular: the demi-cube is the Tetrahedron and the demi-tesseract is the 16-cell. However, as also discussed in detail here, there is a special relation between these symmetries and the Zometool, because, unlike the case for the cubic and tesseractic symmetries, all uniform polyhedra and polychora built with demi-cubic and demi-tesseractic symmetries are Zomable. Furthermore, this 4-dimensional symmetry generates an interesting phenomenon, triality: this means that these polychora have either 3 or 9 Zomable projections each!

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 facets. But here ends the list of demi-hypercubes that are either regular or semi-regular! Interestingly, like the demipenteract, all n-dimensional demi-crosses (which have the same symmetry) are not only semi-regular, with simplices and cross polytopes as facets, but also (as defined in this page) partially regular: a faceting of a regular polytope where all elements but the facets are identical.

This similarity of the demipenteract with the demi-crosses marks it as a member of a family of semi-regular convex polytopes that behaves like the non-convex demi-crosses: the facets of the n-dimensional polytope are simplices and cross polytopes and the vertex polytope is the n − 1 polytope in the same family: this is the k₂₁ polytope family. Unlike the demi-crosses, this family is finite. In this 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.

It is only in 6 dimensions that the k₂₁ symmetries start 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 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 model of the 4₂₁ polytope built with the Zometool. 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 the Zometool company: 120 B3s, 60 R3s and 120 Y3s. Alternatively, the model can be made on a smaller scale.

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

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 a present from 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.

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!

Paulo's polytope site / Next: References