This is an online electronic supplement to
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
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:
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.
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 faces are flat (i.e. the vertices of each face lie in some plane),
but are not necessarily regular.
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.
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.
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
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
copyright 1997, George W. Hart