Date:           Tue, 19 Dec 1995 15:12:42 -0500 (EST)
From:           John Conway <conway@math.princeton.edu>
Subject:        Re: polygons
To:             Brian Hutchings <brihut@pro-palmtree.cts.com>

On 17 Dec 1995, Brian Hutchings wrote:

> hey, I've seen this, before ... anyway,
> what's the simplest hecatohedron (to describe) ??...  I mean,
> aside from stuff like a pentacontagonal dipyramid -- or
> is that about as good of a symmetry as can be found?
> 
> "Time is the only dimension." -RBFuller ... Conjecture on "FG#s":
> Non Compes Mentes!
> We return you to your regular channel.

     This is a nice question, if we interpret "simplest" as
"nicest".  Let's ask for one whose group contains the icosahedral
rotation group.  Then unless the center of a face is at one
end of an axis of symmetry, that face is one of an orbit of size 60.
If it IS, the face is one of an orbit of size  12,20, or 30,  but
we can have at most one each of these.  This gives 8 residue-classes
modulo 60, namely

   60N +  0, 12, 20, 30, 32, 42, 50, 62

for icosahedrally symmetric N-hedra - note that 100 isn't in any of these.

     If we took the rotational cube group, we'd get

   24N + 0, 6, 8, 12, 14, 18, 20, 26

and again 100 isn't present.

    The rotational tetrahedral group gives

   12N + 0, 4, 4, 6, 8, 10, 10, 14

and here we can get 100, but only as 12N + 4  for  N = 8.

     Here's a hecatohedron with full tetrahedral symmetry:

Form the "16-reticulated cube", by dividing each face of a
cube into 18 smaller "square" faces in the obvious way,
giving a 96-hedron.  Then tetrahedrally truncate this.

I suppose I'd call it the

     "semi-trivalently-truncated 16-reticulated cube" !

    Otherwise, the group of rotational symmetries is
cyclic or dihedral, and it follows that at most 2
faces can be fixed by a subgroup of size more than 2 (of
ANY symmetries, rotational or not).

   If the full group has order N, it must have an orbit of
size at least N/2,  whence  N/2 is at most 100, and N at most 200.

A bit of thought shows that the only two possibilities with a group
of order 200 are the pentacontagonal bipyramid and antibipyramid.

    John Conway