From: frank@Cookie.secapl.com (Frank Adams)
Subject: Re: Three-sided coin
Date: Wed, 11 Nov 1992 19:28:04 GMT
Organization: Security APL, Inc.

In article <1992Nov10.204733.25206@massey.ac.nz> news@massey.ac.nz (USENET News System) writes:
> If you want to throw a randomising device, you'd better
>stick to Platonic solids.

Actually, Platonic solids is too constrained a condition.  A Platonic solid
must have all faces isomorphic (strictly, in a single orbit under the
symmetries of the object), and likewise for edges and vertices.

All that's required for a randomizing device is that the faces are
isomorphic; the constraints on edges and vertices are not relevant.  And, as
it turns out, there other solids which work.

There are two infinite families of these, each with 2n sides for each n>2.
The easiest to visualize consists of two n-sided pyramids, with their faces
stuck together.  The regular octahedron is a special case of this, for n=4.
(We count polyhedra as equivalent if they have the same structure.
Polyhedra in this family can be deformed by changing the steepness of the
pyramids without changing the equivalence of the faces.)

The second class is a bit harder to visualize.  It can be described as two
n-sided pyramids, stuck together skew, with the faces extended to form
quadrilaterals.  The cube is a special case of this, for n=3.  For actual
randomizing devices, this form is better for n odd, since it then has a top
face; the first class has a top face for n even.  10 sided dice of this form
are commercially available.

The other polyhedra of this type can all be "built" from Platonic solids.

One way to build them is to build a short pyramid on each face of the solid
(short enough that the previous edges remain convex).  This produces five
distinct polyhedra, with respecively 12, 24, 24, 60, and 60 faces for the
tetrahedron, cube, octahedron, dodecahedron, and icosohedron.  These are all
distinct from each other and from the Platonic solids.

If the pyramids are made a bit higher, until the old edges become flat,
polyhedra with half as many sides are created: 6, 12, 12, 30, and 30.
However, there are only 2 new solids here: the one created from the
tetrahedron is a cube, and the other pairs with the same number of faces are
in fact the same.  These are called rhombic polyhedra.  The 30-sided rhombic
polyhedron is commercially available, and the 12-sided one may still be.

It is also possible to create polyhedra with 12, 24, 24, 60, and 60 faces
with quadilateral faces.  Create a shallow "bump" in the center of each
edge, add a new vertex in the middle of each face (raised), and connect
these new vertices to the "bumps".  This gives us 2 new polyhedra: the pairs
derived from dual regular polyhedra are equivalent, and the one derived
from the tetrahedron is a rhombic dodecahedron.

Yet another way to get size 12, 24, 24, 60, and 60 faces creates pentagonal
faces.  Turn each edge into a zig-zag:

  /\
    \/

and connect the centers of the faces to the "outdents" on the adjacent
edges.  This only produces 2 new polyhedra; again the duals produce
equivalent solids, and the tetrahedron produces the regular dodecahedron.
These two are the only polyhedra with equivalent faces which have a
"handedness"; they cannot be transformed into their mirror images in
3-space.

Finally, one can put a "bump" on each edge, and connect each old face to
both the bumps and the old vertices, giving polyhedra with 24, 48, 48, 120,
and 120 faces.  Again, there only 2 new polyhedra here.  The duals are again
equivalent, and the polyhedron derived from the tetrahedron is the same as
the one derived before by adding bumps to the cube or octahedron.

That these are the only possibilities can be shown by considering the
sequence of valences of the vertices around a face.  (This must, of course,
be the same for each face.)  The basic condition is that

  sum(1/2 - 1/v_i) < 1

which is required for the network to close off into a polyhedron.  The
actual proof involves checking each solution to this, showing that each
produces at most one polyhedron.

Note that tilings of the plane with all regions equivalent satisfy the above
condition, with "<" replaced by "=".  For example, 3,6,3,6 gives a tiling
into rhombuses:

     *---*     *---*     *---*     *---*
    / \   \   /   / \   / \   \   /   / \
   /   \   \ /   /   \ /   \   \ /   /   \
  *     *---*---*     *     *---*---*     *
   \   /   / \   \   / \   /   / \   \   /
    \ /   /   \   \ /   \ /   /   \   \ /
     *---*     *---*     *---*     *---*
    / \   \   /   / \   / \   \   /   / \
   /   \   \ /   /   \ /   \   \ /   /   \
  *     *---*---*     *     *---*---*     *
   \   /   / \   \   / \   /   / \   \   /
    \ /   /   \   \ /   \ /   /   \   \ /
     *---*     *---*     *---*     *---*