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 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.
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 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!
Paulo's polytope site / Next: References