I don’t know the meaning of the number of these flags. But there’s an excellent group that acts on the complete flags regardless of whether the polytope is regular or not! It’s called the **cartographic group**. It has one generator for each dimension.

In the case at hand, when we’re starting with a polyhedron, these generators could be called and Their meaning is ‘change the vertex’, ‘change the edge’ and ‘change the face’.

You see, if you have a vertex-edge-face flag, there’s exactly one way to change the vertex and get a new such flag. Similarly there’s one way to change the edge, and one way to change the face.

In the cartographic group, these generators obey only the relations that hold *regardless* of which polyhedron we’re dealing with:

This presentation defines the cartographic group.

This story extends to vertex-edge flags, or edge-face flags, or vertex-face flags. Various quotient groups of the cartographic group act on these.

All this generalizes to arbitrary dimensions. In general, each generator of the cartographic group. squares to 1, and generators that aren’t of neighboring dimensions commute.

]]>For other, non-regular polyhedra these numbers seem a bit more mysterious to me.

]]>One is that Gauss showed it’s possible to construct a regular 17-gon using a ruler and compass, because

and 17 is prime. The other is that there are 17 wallpaper groups.

Plutarch wrote that the Pythagoreans “utterly abominate” the number 17.

]]>https://johncarlosbaez.wordpress.com/2017/04/06/periodic-patterns-in-peptide-masses/

except that it seems to come down to counting unordered sums instead of ordered sums.

Two actions of a finite group on a finite set are isomorphic if and only if they have the same number of orbits of the same length. The length of an orbit is the index of a point stabilizer in the full group.

The subgroups of have orders 1,2,3,4,12 so the orbits can have lengths 12, 6, 4, 3, 1. The number of equivalence classes of actions of on a set of 12 elements comes down to the number of ways 12 can be written as an unordered sum of 1, 3, 4, 6, 12. I wrote the cases out as Young diagrams here:

There appear to be 17 equivalence classes of actions of on a 12 element set.

In a solid tetrahedron there are no points with orbits of length 3 under the action of . So it appears that for any 12 point subset of a solid tetrahedron there are only 7 equivalence classes of actions by .

But if acts on collections of subsets of the tetrahedron then we can get an orbit of length 3. For instance there are 3 pairs of opposite edges.

We can reduce this to finite sets. Denote the vertices by 1,2,3,4. acts transitively on the 3 element “class of collections of finite subsets”:

{{{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}}}

Other orbits of length 3 could be obtained this way by selecting other points from the tetrahedron’s medians. So it seems that all possible equivalence classes of actions of on a 12 element set would be possible with a regular tetrahedron.

]]>The tetrahedron has 12 rotational symmetries, giving

Using the double cover

we get a 24-element subgroup of This is quite famous:

• Wikipedia, Binary tetrahedral group.

Since is the unit sphere in the quaternions the binary tetrahedral group forms the vertices of a 4d polytope, called the 24-cell:

• Wikipedia, 24-cell.

The 24-cell is a 4-dimensional regular polytope. The vertices of the 24-cell can be broken up into 3 sets of 8, each set being the vertices of *another* 4-dimensional polytope, which is called the ‘hyperoctahedron’ or ‘cross-polytope’ or ‘4-dimensional orthoplex’ or ’16-cell’:

• Wikipedia, 16-cell.

If you set up things nicely, these 3 cross-polytopes inside the 24-cell get permuted when we permute the quaternions i, j, and k. This is called ‘triality’.

Now the idea is this: since the 3 cross-polytopes inside the 24-cell are a very nice way to think about the identity

perhaps they can be used to help understand how we build the Leech lattice out of 3 copies of the E_{8} lattice, or build the Golay code out of 3 copies of the 8-bit Hamming code!

We (in some sense of ‘we’) already know how to build the Leech lattice out of 3 copies of the E_{8} lattice, or build the Golay code out of 3 copies of the 8-bit Hamming code. It’s called the ‘Turyn construction’. In the case of building the Leech from 3 E_{8}‘s, Greg Egan and I described it here:

• Integral octonions (part 9).

The question is whether this is at all illuminated by the geometry I just described.

It’s worth noting that Greg and I already *did* connect triality to the Turyn construction of the Leech lattice. The three E_{8} lattices used to build the Leech lattice can be thought of as lying in the vector, left-handed spinor, and right-handed spinor representations of . These representations are permuted by the triality automorphisms of . Furthermore, the weight lattice of is a 4-dimensional lattice generated by the points in the 24-cell, and the weights of the vector, left-handed spinor and right-handed spinor representations lie in the 3 cross-polytopes I mentioned!

So, all this stuff *does* fit together. I guess the question is whether the Turyn construction makes a bit more sense if we examine it in this larger context.

Whenever we’ve got a set of things that a tetrahedron has 12 of, we should get an action of the tetrahedron’s rotational symmetry group A_{4} on this 12-element set . A set on which a group acts is called a -set, and an **isomorphism** between -sets and is a bijection

such that

for all

**Puzzle 6.** How many isomorphism classes of 12-element A_{4}-sets are there?

In simple rough terms: how many ‘fundamentally different ways’ are there to find 12 things in a tetrahedron?

For example, the set of face medians is a 12-element A_{4}-set, and so is the set of ‘vertex-face flags’ (defined in the obvious way). But these are isomorphic as A_{4}-sets, since a face median touches one vertex and one face, and these give a vertex-face flag.