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?

Regular polytopes

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 other regular polytopes. 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 mentioned before, a n-dimensional simplex is self-dual. It has n − 1-dimensional simplices as facets and vertex figures. The n-dimensional cross polytope also has n − 1-dimensional simplices as facets, which implies that the n-dimensional hypercube has n − 1 dimensional simplices as vertex figures. The same cross polytope has n − 1-dimensional cross polytopes as vertex figures, which implies that the same hypercube has the dual n − 1 dimensional hypercubes as facets. This will be important for understanding what follows.

Interestingly, several projections of higher-dimensional hypercubes are not only Zomable, but also Zomable in a single colour and have a high degree of symmetry. The reason for this can be understood from the study of the models of the Cube and Tesseract. Since an orthographic projection is an affine projection, two parallel edges of any n-dimensional hypercube are still parallel in the 3-d projection. This means that for a n-dimensional hypercube, where there are n distinct edge directions (which can be made to be the n distinct orthogonal axes defining that space), the projection will also have n distinct edge directions.

The three axes of the Cube can be represented in 3 dimensions without distortion, in three orthogonal directions, all with blue struts. For the Tesseract, we cannot represent the four axes as orthogonal directions, but we can choose to represent them equally, which implies a symmetric - in this case Tetrahedral - arrangement in three dimensions. This is the vertex-first projection of the Tesseract in Fig. 5.4. All edges are represented with the same length, and all vertices connect to four edges.

For the six-dimensional hypercube, we can use the six red axes of symmetry of the Icosahedron (represented in the Zometool by the R struts) as the six projected directions of the axes. A model of this projection is shown in Fig. 9.1. The projection has full Icosahedral symmetry. As in Figs. 5.2 and 5.4, all edges are represented with the same length (in this case R3 struts) and all vertices (in white) connect to the same number of edges (in this case six). However, there are strut intersections, represented by the red balls, for this reason shorter R struts (R1s and R2s) are necessary. Note also how much larger the projection is relative to the size of the edge, a phenomenon we had already see in Figs. 5.2 and 5.4. However, unlike those projections, this is not a vertex-first projection.

Fig. 9.1: An Icosahedral projection of the 6-dimensional hypercube. (ACTUAL FIGURE TO BE INSERTED SOON)

Any n-dimensional hypercubic element (including the Square faces) that is projected to a plane will be circumscribed by a polygon with central symmetry, where each edge has an opposite edge that is parallel to it. Any orthographic projection of any n-dimensional hypercube will therefore be bound by a polyhedron with these types of faces: A zonohedron. The characteristics of their faces imply that all faces are part of face rings, each containing a particular edge direction that does not exist outside that ring, each of these corresponds to one of the directions of the edges of the hypercube.

In the case of the projection of the 6-dimensional hypercube in Fig. 9.1., the envelope is a zonohedron with Icosahedral symmetry that is constructible with R struts. We have seen this object before, it is the Rhombic triacontahedron. This situation is analogous to the envelope of the vertex-first projection of the Tesseract, a Zonohedron with Octahedral symmetry that can be represented with Y struts, the Rhombic dodecahedron. There is a magnetic toy puzzle that can be arranged as a Rhombic triacontahedron, the Big ball of whacks. The pieces come in six colours. At first I was wondering why six: with five colours, it would have been easier to create symmetrical patterns. However, I could find a pattern where each of the six colours corresponds to a particular equatorial ring of faces. For each of these rings there is a single edge direction that is common to all the faces in the same ring, and perpendicular to the ring itself (see Fig. 9.2); this edge direction is absent outside this ring. This is a very nice general illustration of the concept of a Zonohedron, and in this case of the directions of the edges of the 6-dimensional hypercube in Fig. 9.1.

Fig. 9.2: My Big Ball of Whacks (6-color edition), with each colour showing five of the ten faces in each equatorial ring. Each colour is associated with a single edge direction, which is perpendicular to that ring (ACTUAL FIGURE TO BE INSERTED SOON).

The same logic can be applied to projections of any higher-dimensional hypercubes. An an example, if we treat all 10 axes of the 10-dimensional hypercube in the same way, we can represent them as the 10 3-fold axes of symmetry of the Dodecahedron, resulting in a yellow projection of the 10-cube with full icosahedral symmetry. Its envelope is the largest yellow zonohedron that can be represented in Zometool, the rhombic enneacontahedron (see Zometool model in George Hart's list of polyhedron models with the Zometool).

This logic continues for higher dimensions. Any Zomable zonohedron corresponds to a Zomable projection of a Hypercube. Given that the Zometool connector has 31 axes of symmetry (6 red, 10 yellow and 15 blue), it can theoretically represent the projection of a 31-dimensional hypercube. While that projection itself is practically impossible to make, its zonohedral envelope is relatively simple.

Semi-regular polytopes

As we remarked when discussing the partially regular polychora, associated with the cross polytopes are the non-convex demi-crosses. The facets of the n - dimensional demicrosses are n − 1-dimensional simplices and cross polytopes, the vertex figure is the n − 1-dimensional demi-cross. This works because, as mentioned above, the vertex figures of the n − 1-dimensional simplices and cross polytopes are, respectively, n − 2-dimensional simplices and cross polytopes.

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 a dual operation, 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 that have these symmetries 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, among the polytopes built from the octahedral, tesseractic and icositetrachoric symmetries, only the uniform polytopes that can also be built with demi-cubic and demi-tesseractic symmetries - and only those plus the Cube and Tesseract - are Zomable. Furthermore, as also mentioned before, the demi-tesseractic 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 we find new polytopes that have the demihypercubic symmetry as their highest symmetry. Of special interest 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, 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 with the same behaviour, the k₂₁ polytope family. Like the demi-crosses, this family exists because, as mentioned above, n-dimensional simplices and cross-polytopes have, respectively, n − 1-dimensional simplices and cross polytopes as vertex figures. This implies that the Rectified 5-cell - a semi-regular polychoron with simplices (Tetrahedra) and cross-polytopes (Octahedra) as cells - is the vertex polychoron of a 5-dimensional object with 4-dimensional simplices (5-cells) and 4-D cross polytopes (16-cells) as facets: the demipenteract. The cells of the Rectified 5-cell are the vertex figures of the facets of the demi-penteract. As mentioned in the polychora page, the Rectified 5-cell is the 0₂₁ polytope and the demipenteract is the 1₂₁ polytope.

There is another semi-regular ``polychoron'' with Tetrahedral and Octahedral cells, the Tetrahedral-Octahedral honeycomb of the 3-D Euclidean space, which is the dual of the aforementioned Rhombic dodecahedral honeycomb. Its vertex figure is the dual of the Rhombic dodecahedron, the Cuboctahedron. Because this is not finite (its angular defect around each vertex is zero), it cannot serve as a vertex figure for a 5-dimensional polytope.

It is only in 6 dimensions that the k₂₁ symmetries start generating polytopes unique to them. The 2₂₁ semi-regular polytope has the demipenteract as vertex figure and is one of 39 uniform polytopes with E6 symmetry. In seven dimensions, the 3₂₁ semi-regular polytope has the 2₂₁ polytope as vertex figure and is one of 127 uniform polytopes with E7 symmetry. Finally, in eight dimensions, the 4₂₁ semi-regular polytope (see Figs. 9.3 and 9.4) has the 3₂₁ polytope as its vertex figure and is one of 255 uniform polytopes E8 symmetry. The E8 polytopes have 696,729,600 symmetries!

Fig. 9.3: 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.

That's it! Unlike in the case of the demi-crosses, the family ends, because its next member has an angular defect around each vertex of zero, being therefore a tesselation of the 8-dimensional Euclidean space, the 5₂₁ honeycomb. Like the Tetrahedral-Octahedral honeycomb, the fact that it is infinite implies that it cannot serve as a vertex figure for any higher-dimensional polytope.

From this, we conclude that the k₂₁ polytopes, and the E6, E7 and E8 symmetries are also exceptional objects.

Fig. 9.4: 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.

This 5₂₁ honeycomb has other properties that make it analogous to the Tetrahedral-Octahedral honeycomb. The vertex arrangement of the Tetrahedral-Octahedral honeycomb is the face centered cubic (FCC) lattice. Centering spheres at these points of this lattice provides one of the maximally compact sphere packings in 3-dimensional space, where each sphere kisses the maximum possible number of other spheres around it (12), with the latter's centres arranged as the vertices of a Cuboctahedron.

Similarly, the vertex arrangement of the 5₂₁ honeycomb is the E8 lattice. Centering 7-spheres in this lattice provides the most compact 7-sphere packing in 8-dimensional space. Each 7-sphere kisses the maximum possible number of 7-spheres around it (240), with the latter's centers arranged as the 240 vertices of a 4₂₁ polytope (the vertex figure of 5₂₁ honeycomb).

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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