- all the edges are tangent to the unit sphere,
- the origin is the center of gravity of the points at which the edges touch the sphere,
- the faces are flat (i.e. the vertices of each face lie in some plane), but are not necessarily regular.

However, it is possible to construct a topologically equivalent polyhedron which is in canonical form, shown at right. The faces are no longer regular, but all edges are a unit distance from the origin and tangent to a unit sphere. In addition, all the symmetry of the original is retained, i.e., three mirror planes and three 2-fold axes.

The proof of the theorem does not provide a construction of the canonical form, it only tells of its existence. The paper presents an experimental algorithm which constructs the canonical form for a given input polyhedron. To illustrate the algorithm, three new self-dual polyhedra are generated. To understand their properties, we first review a few aspects of duality.

The above is a geometric notion of duality. There is also a combinatoric or topological notion of duality, in which the geometry is ignored. Any geometrically distorted cube, e.g., a parallelepiped, is still combinatorially dual to an octahedron (or any geometrically distorted octahedron). All that is required is that we can make a mapping that associates faces of one with vertices of the other, and preserve the property that two vertices in one are connected with an edge iff the two corresponding faces of the other are adjacent. Thus geometric duality implies combinatoric duality, but not the converse.

Duality is an operation of order two --- taking the dual of the dual always gives back the original polyhedron --- so we usually speak of dual pairs of polyhedra. In a few special cases however, a polyhedron is dual to itself, i.e., it is a fixed point of the duality operator. For example, any pyramid is combinatorially self-dual, and if it is a regular right pyramid of the proper height, it is also geometrically self-dual.

If we process the hermaphrodite through our canonicalization algorithm, we get a heptagonal hermaphrodite which is geometrically self-dual, so it makes a nice compound with itself, shown at left at the top of this page.

Canonicalizing this with our algorithm gives a tetrahedrally
stellated icosahedron in canonical form. Notice it has a very different
shape for its quadrilateral faces. Being geometrically self-dual, it makes
a nice compound with itself,
as illustrated at the top of this page, on the right. If you travel inside
it you will see that the interior, i.e., the intersection of the two, is
an attractive polyhedron composed of twenty-four
trapezoids and eight equilateral triangles. This led to one of
my sculptures, titled Yin
and Yang.

copyright 1997, George W. Hart