This is an online electronic supplement to the article: George W. Hart, "Calculating Canonical Polyhedra", Mathematica in Education and Research, Vol 6 No. 3, Summer 1997, pp. 5-10. The article contains Mathematica code for the algorithm that produces these polyhedra.

Calculating Canonical Polyhedra

George W. Hart

Canonical Polyhedra

The paper concerns an interesting theorem that there exists a "canonical form" of any given convex polyhedron. This canonical form is a distorted version of the given polyhedron in which the vertices are positioned in space to satisfy the following properties:

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.

To illustrate, we take an arbitrary polyhedron, the bilunabirotunda, which consists of four regular pentagons, two squares, and eight equilateral triangles. It is shown at left. Click on the image to view it in 3D. Clearly its edges are not all tangent to a single sphere.

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.

Duality

The dual to a given polyhedron has a face for each vertex of the original and a vertex for each face of the original. For example, the cube and the octahedron are mutually dual. Their edges cross at right angles and the intersection points are on the unit sphere. For each face of one, the other has a vertex positioned as follows: the distance of the vertex from the origin is the reciprocal of the distance of the face from the origin. Two vertices in the dual are connected with an edge if and only if the two corresponding faces of the other polyhedron are adjacent. However, the edges of a polyhedron and its dual do not generally cross as they do here; it is only because they are tangent to the unit sphere that this occurs.

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.

Examples

The connection between the canonical form and duality is that it follows from the theorem that the dual to the canonical polyhedron also has the above three properties, i.e., it is also in canonical form. To see this, imagine the circle where the face plane intersects the unit sphere; it is inscribed in the face and the dual edges lie on a cone tangent to the sphere at that circle. So starting with any combinatorially self-dual polyhedron, its canonical form is geometrically self-dual. The paper contains the mathematica code to construct three examples.

1. The Hermaphrodite

If we adjoin a prism and a pyramid we get a structure which is combinatorially, but not geometrically self-dual. John Conway has called this construction a hermaphrodite. To understand it, it is better to think of it as half a prism adjoined to half of a dipyramid. Because the prism and the dipyramid are mutually dual, taking half of each gives a construction with its top half dual to its bottom half, making it self-dual combinatorially.

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.

2. The Antihermaphrodite

An analogous construction gives the canonical form of a heptagonal antihermaphrodite. It is composed of half an antiprism and half a trapezohedron. Because the antiprism and the trapezohedron are mutually dual, the result is again self-dual, so it also makes a nice compound with itself, illustrated at right.

3. The Tetrahedrally Stellated Icosahedron

It is posible to color an icosahedron with five colors so that any four faces of one color are in the planes of a tetrahedron. If we choose four such faces and put a low pyramid on each, of just the right height to blend in with the adjacent triangles, we get a tetrahedrally stellated icosahedron. It consists of twelve kite-shaped faces and four equilateral triangles. They come about because the pyramids have covered four faces of the icosahedron's twenty, have extended twelve faces to quadrilaterals, and have left four triangles remaining. It is combinatorially, but not geometrically self-dual.. Notice it has chiral tetrahedral symmetry, i.e., four 3-fold axes, three 2-fold axes, but no planes of symmetry.

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.