
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 halfhearted attempt to make the
page at least partially interesting to nonspecialists. 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 edgetoedge, 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 ``120cell''. 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 "hemidodecahedron", a shape that
can be drawn on the socalled "projective plane".
Another way to imagine the hemidodecahedron 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 hemicube? How many faces would it have? What about the
hemioctahedron? The hemiicosahedron? Is there a hemitetrahedron? 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 hemiicosahedron.
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 J_{1} and
the projective special linear group L_{2}(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 (s_{0}, s_{1}, s_{2} and s_{3})
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 J_{1}, and a2, b2, c2 and d2 for
L_{2}(19).
 A complete list of the conjugacy classes of subgroups of W.
 A list (cl) of the indices of those subgroups that yield welldefined
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 welldefined quotients. Various
operations may be performed on the groups in this list, if desired, to check the results
of the paper.
