Quotients of a Universal Locally Projective Polytope of Type {5,3,5} (Auxiliary Information)


This webpage summarises the results of work by M. I. Hartley and D. Leemans completed in 2003. The first few paragraphs are a half-hearted attempt to make the page at least partially interesting to non-specialists. If you have come to this page after reading the article by Hartley and Leemans, you can skip almost to the end of the article.
If one joins hexagons edge-to-edge, with three around each vertex, one finds that one can keep adding hexagons forever and fill the entire plane with them. If one uses pentagons instead of hexagons, then the process cannot continue forever. Instead, the twelfth hexagon will neatly close the shape formed by the other eleven into what is called a dodecahedron. We say the dodecahedron is of "type" {5,3}, since it has pentagons ({5}) arranged three per vertex in the same pattern as the edges of a triangle ({3}). The arrangement of hexagons is of type {6,3}.

Puzzle: What "type" is the cube? What about the other platonic solids? Does a prism have a "type"?

Similar phenomena occur in higher dimensions. Abstractly, if one joins dodecahedra together, four per vertex arranged in the pattern of the faces of a tetrahedron, the arrangement of dodecahedra slowly closes in on itself, and the 120th dodecahedron forms the final cap on a shape known as the ``120-cell''. This shape is of ``type {5,3,3}'', since the "faces" have type {5,3}, and are arranged around the vertices like the faces of a tetrahedron (type {3,3}).

If the dodecahedra were arranged around the vertices like the faces of an octahedron (type {3,4}), one would obtain an infinite polytope of type {5,3,4}.

Puzzle - what goes wrong when you try to arrange the dodecahedra around the vertices like the faces of a cube?

The dodecahedron is obtained by joining pentagons together. It takes 12 before the shape closes up. However, we could always try to close it up earlier. For example we could place the initial pentagon, then the five adjoining ones (to make 6), and then say "now it's closed", by gluing the loose edges together in a neat way, edge to opposite edge. Then we would obtain a "hemi-dodecahedron", a shape that can be drawn on the so-called "projective plane".

Another way to imagine the hemi-dodecahedron is to take a dodecahedron, and proclaim "I will now imagine that opposite faces, edges and vertices are actually the same". This would give a shape with 6 pentagonal faces, arranged in threes around the ten vertices - a "quotient" of the dodecahedron.

Puzzle - is there a hemi-cube? How many faces would it have? What about the hemi-octahedron? The hemi-icosahedron? Is there a hemi-tetrahedron? Can you find a finite shape of type {4,4}?

In the article by Hartley and Leemans, we tried to find all possible shapes with dodecahedra for faces, arranged around each vertex like the faces of a hemi-icosahedron. That is, we were looking for "locally projective" polytopes of type {5,3,5}. ("Locally projective" because the dodecahedra are arranged around the vertices in the pattern of a shape that can best be drawn on the projective plane.)

It was important to discover:

  • Whether the biggest such shape closes in on itself, or whether one can keep adding dodecahedra forever?
  • What kind of symmetry the shape has? (Technically, what is its symmetry group?)

In the article mentioned, it is discovered that the symmetry group W is finite, but very large. It is a direct product of the Janko group J1 and the projective special linear group L2(19), and so has order 600415200. This means that the largest locally projective polytope of type {5,3,5} has 5003460 dodecahedra, arranged in groups of 10 around its 10006920 vertices.

Using the subgroups of the symmetry group, and using earlier results by the first author, it is possible to characterise all the quotients of this polytope. It turns out there are 145 quotients, a few of which were known earlier.

The (>5Mb) file g.txt gives:

  • Standard generators (s0, s1, s2 and s3) for the symmetry group $W$ as permutations on 286 points. In the file these are denoted as a,b,c,d, and expressed a := a1*a2, b := b1*b2, c := c1*c2 and d:=d1*d2, where a1,b1,c1 and d1 are generators for J1, and a2, b2, c2 and d2 for L2(19).
  • A complete list of the conjugacy classes of subgroups of W.
  • A list (cl) of the indices of those subgroups that yield well-defined quotients of the polytope.

The file is written in the GAP scripting language (version 4.3). If it is imported into a GAP session (using the Read command), this will produce a list (geo) of the permutation presentations of all groups that yield well-defined quotients. Various operations may be performed on the groups in this list, if desired, to check the results of the paper.