Johannes Kepler

Visualizing the fourth dimension is hard enough. To go beyond that becomes very difficult indeed. For this reason, we start this by going a bit deeper into some general concepts on polytopes and groups. These allow a deeper understanding of polytopes and their symmetries, and will be very helpful moving further.

The diagrams with only white balls represent a Coxeter group, which is akin to a set of mirrors in a Kaleidoscope. In the case of polyhedra, these can be thought as dividing a spherical surface in a set of identical Schwarz triangles. In the case of polychora, these can be thought as dividing a hyper-spherical surface in a set of identical Goursat tetrahedra. Identical diagrams with only red balls at opposite ends represent dual polytopes.

Regular polytopes and their rectifications can then be thought as reflections of one of the

Fig. 36: These are the C-D diagrams of the regular polygons
used in the polychora in this site, plus their generating groups.
All these graphs are left-right symmetric, this means that
all regular polygons are self-dual. The number of such diagrams
is, of course, infinite.

Fig. 37: These are the C-D diagrams
of the regular and quasi-regular polyhedra in this site, plus their generating
groups. On the left column we have the graphs for the groups,
in the middle column we have the convex polyhedra, and
in the right column we have the star polyhedra.

These diagrams include the polygonal diagrams in Fig. 36,
such polygons are thus a part of the polyhedron.
For instance, for the Cube, the only polygon graph
that can be built with the red ball is the Square.

However,
for the Icosidodecahedron, we can form the graphs of the
Pentagon and the Equilateral triangle with the red ball (since the graph can
be read in both directions); thus the polyhedron is bound
by both types of polygons.

Two neighboring white nodes
indicate a regular vertex figure: from this, we can see immediately
that the vertex figure of the Cube is a regular triangle.
The vertex figure of the Cuboctahedron or Icosidodecahedron
are more interesting. They are both rectangles. We must imagine here
that the diagram forms a loop, with the invisible line connecting
the faraway nodes being a
``2-fold'' symmetry axis. This corresponds to the vertex of the
Schwarz triangle that is a right angle, such angles are not represented
in Coxeter-Dynkin diagrams. The red ball is opposite to this
node, this means that to generate the quasi-regular polyhedra
we are reflecting the point at the right angle (hence the term ``rectified'').
The vertex figure of a point in a 2-fold symmetry axis will therefore
be a polygon formed by the two white balls in the circuit, which
are connected by this invisible ``2-fold'' line. This has 2-fold
symmetry - a Rectangle.

In the polyhedra page, we said that regular and quasi-regular
polyhedra share the same Kaleidoscopic Wythoff construction.
This fact is evident in the C-D diagrams by the fact that all
polyhedron graphs have only one red ball, i.e., they are reflections
of vertices of their Schwarz triangles. Other Wythoff constructions, which reflect points in the edges
or in the face of the Schwarz triangles to create
many uniform polyhedra,
would require more than one red ball.

An important point to re-emphasize is that a particular polyhedron
(like the Octahedron) can be represented by different
diagrams. This is a reflection of the fact that the same polyhedron
can be produced from different symmetries. These polyhedra have
several other C-D diagrams, we present here those where only
one red ball is needed; which directly implies regularity or
quasi-regularity.

_{3} graph is symmetric, therefore the Tetrahedron can be
generated with the red ball at either end. This means that
the Tetrahedron is delf-dual.

Fig. 38: These are the C-D diagrams of the
convex polychora presented in the polychora page, plus their generating groups.

On the left we have most of the group graphs.
In the middle we have the convex polychora displayed in the polychora page.
These include as sub-sets the polyhedral graphs in the
middle column of Fig. 37.
For instance, in the 16-cell, the only polyhedral graph
that can be built with the red ball is the Tetrahedron,
therefore the 16-cell only has Tetrahedra, and is therefore
regular. In the case of the rectified 5-cell, however,
we can build the graphs of the Octahedron and the Tetrahedron
from the red ball; this means that that polychoron
is bound by both types of polyhedra.
Three neighboring white nodes indicate a regular
vertex figure (which happens, of course, only for regular
polychora), with the equivalent of the red ball next to the
actual red ball. Doing this for the 600-cell, for instance,
we see that the vertex figure is an Icosahedron.
Doing this for a the Rectified 600-cell, we must imagine
a 2-fold symmetry line merging the two outer nodes.
The white balls then form a polyhedral figure with 5-fold
symmetry along one direction and 2-fold symmetry along the other
- a (non-uniform) Pentagonal prism.

Most, but not all, of the polytope graphs presented here
(and corresponding polytopes in this site) have the same
Wythoff construction,
this is indicated in the graphs by the single red ball (this indicates reflection of
a vertex of the Goursat tetrahedron).
The exceptions are the diminished 600-cells:
the Snub 24-cell (see Fig. 17), which has two green balls (indicating rotational
but not reflexive symmetry) and the Grand antiprism. The latter does not appear
in this Figure because (as mentioned in the caption to Fig. 22) it has no Wythoff construction.

Many other convex uniform polychora are possible with the use of more red balls,
but building all of them requires many parts, as one can see in
David Richter's H4 page.
Just one example here: the truncated 600-cell.
Its graph has two red balls, indicating that the vertex is a reflection of a point on an edge of the
Goursat Tetrahedron.

Fig. 39: The truncated 600-cell, one of the ``Archimedean'' polychora.
This Zometool model has a lot of parts, and it is quite flimsy.

Studying these diagrams, we can see that the process of
rectification of a regular polychoron with X-type of cell
and Y-type of vertex figure generates a new polychoron
for which the cells are rectified Xs (as many as the previous
number of cells) and Ys (as many as the previous number of vertices).
The number of vertices will be the same as the number of
edges in the polychoron before.

Again, almost all of these polychora can be represented by
multiple C-D diagrams; those presented here are those that require
only one red ball.
Note that the 24-cell is represented by 3 different
diagrams in the figure, and the 16-cell by two. Some of these extra
diagrams appear in the right column, we have, on top,
the Demi-hypercubic
group D_{4}.

In 4 dimensions, this type of symmetry does not yet produce
new polytopes, this starts happening only in 5 dimensions.
However, it has some interesting consequences, one of them
being Triality.

As mentioned in the polychoron compounds page,
in the case of duality in 3 dimensions,
one can, by the definition of the meaning of duality,
scale two concentric dual polyhedra in such a way that each edge from
a polyhedron meets another (perpendicular) edge from the dual polyhedron
at a point that is in the middle of both (the midpoint).
Those two edges form a plane that is perpendicular
to a third line joining the midpoint to the center of those
polyhedra. Rectifying those two dual polyhedra results in the same rectified
polyhedron.

In the case of Triality in 4 dimensions, the situation is very similar,
except that the perpendicular edges of *three* concentric ``trial'' polychora
(16-cells for the only regular case) meet at
their midpoints. These edges are in a 3-D ``plane'' that is itself perpendicular to
a fourth direction, the line joining those midpoints to the centers of the trial polychora.
Rectifying all three polychora results in the same rectified polychoron,
the regular case being the 24-cell.

This possibility is a consequence of the symmetry of the
D_{4} graph. In this graph, we can imagine all outer nodes
to be connected by an invisible ``2-fold'' line, this is what gives
us this possibility of having three perpendicular directions intersecting
along a direction that is orthogonal to the central direction.

Fig. 40: These are the C-D diagrams of the
regular star polychora presented in the polychora page (on the left column),
plus those of their rectifications (on the right column).
I have built models for all polychora on
the left, but not for their rectifications (for lack of space, time and money...),
except for one or two (see below).

These also include the star polyhedra diagrams in the right
column of Fig. 37. From these diagrams, we can see, for instance,
that the Small stellated 120-cell
is bound by Small stellated dodecahedron cells (the only polyhedral
graph that can be built from the red ball) and that this
has a Dodecahedral vertex figure. Thus its dual, the Icosahedral 120-cell,
has Icosahedra as cells, and Great dodecahedra as vertex figures.
We can also see that, following the logic of rectification,
the Rectified small stellated 120-cell has Dodecadodecahedra
and Dodecahedra as cells, and that the Rectified icosahedral 120-cell
has Icosidodecahedra and Great dodecahedra as cells.

Note that the edges of the latter rectified polychoron are
identical to those of the Rectified 600-cell; thus
both can be represented by the same Zometool model. The identical rectification is a
consequence of the conservation laws in Fig. 26c: the rectifications of
two polytopes that share the same edge arrangement will necessarily share the
same vertex arrangement, since the vertex of the rectification is the
middle point of the edge of the original polytope.
The rectifications of two polytopes that
additionally share the same face arrangement (like the 600-cell and
the Icosahedral 120-cell) will necessarily share the
same edge arrangement.

These graphs also tell us something interesting: the rectification
of the Grand 600-cell is, like the rectification of the 600-cell,
a semi-regular polychoron, the only such case
among these ten rectifications. The cells are Great icosahedra and
Octahedra. By the rule we have mentioned in the previous paragraph,
that Zometool model will be the same as the model of the
Rectified great Icosahedral 120-cell.

The symmetry of graphs for the Great 120-cell and the Grand-stellated
120-cell indicates that these are self-dual.

The only regular polytopes are the Simplexes (generated by the A

In this way, one comes to appreciate the fact that a few of the symmetries shown above, like the Icosahedral (H

There are other interesting symmetries in higher dimensions. There is one more infinite family of uniform polytopes that has been thus far hidden: the Demi-hypercubes, generated by the D

Fig. 41: C-D diagrams for some of the higher-dimensional polytopes.
In the center we see the infinite Demi-hypercubic family. All polytopes of
this family in 3 and 4 dimensions are also generated by the B_{3} and B_{4} groups.
In 5 dimensions, the D_{5} family starts generating unique symmetries that
are not present in the A_{n} or B_{n} families.

We also see the k_{21} family evolving along the diagonal.
This ends at 8 dimensions, with the 4_{21} polytope.
The next member of the family is an infinite tesselation filling
8-dimensional space.

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

Thus, one comes to the realization that the E6, E7 and E8 are also exceptional objects - rare jewels among the infinite geometric treasures we have just contemplated.

Fig. 42a: 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 sub-set. 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 higher-dimensional figures 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 very long yellows, 120 very long blues, and 60 very long reds.
Alternatively, the model can be made on a smaller scale.

Fig. 42b: 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 a present given to me by one of my students, Jose Martinez.

Fig. 43: 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,
As its 240 vertices represent the
root vectors of the
simple
Lie group
E_{8},
the polytope is sometimes referred to as the ``E_{8} polytope''.
Using a complex number coordinate system, it can also be constructed as a 4-dimensional
regular
complex polytope known as
the Witting polytope.

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 ``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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