interesting theorem states that there exists a "canonical form" of any
given convex polyhedron. This canonical form is a possibly distorted version
of the given polyhedron in which the vertices are positioned in space to
satisfy the following properties:
It follows that a dual to the canonical polyhedron can be constructed which
has the above three properties as well, and the edges of the canonical
polyhedron and its dual cross at right angles. The representation is unique
except for its rotations and reflections. It is somewhat wonderful that
all these properties can be satisfied at once no matter what polyhedron
one begins with.
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.
The familiar symmetric polyhedra, e.g., Platonic, semi-regular, or Archimedan
duals, already satisfy the three properties above, and so are already in
their canonical form. But an arbitrary "random" polyhedron needs to be
adjusted to get into this form. One application of this is that given a
new irregular-looking polyhedron, one can find its canonical form and look
at it to see if it is the same as a previously familiar canonical form.
To appreciate what this is all about, we can look at some examples of
a not too regular polyhedron and its canonical form. The Johnson
solids are good polyhedra to start with. The following examples show
an original version and its canonical form. Be sure to notice that if the
original has any planes or axes of symmetry, then the canonical form preserves
some additional examples:
A statement of the theorem, and its history, is given in Ziegler's book
Lectures on Polytpes (p. 118), listed in the references.
Versions of the theorem were proved independently by P. Koebe and W. Thurston.
Oded Schramm recently proved that one can specify any smooth strictly convex
body instead of the sphere. I learned of the theorem (and the "hermaphrodites")
from John Conway. The theorem doesn't describe how to find the canonical
form of a given polyhedron, so I have written an algorithm to do so.
start with this random seven-zone
zonohedron (as seen on the zonohedron
page) to get this canonical
form (shown at the top of this page). Observe that the result has central
symmetry just as the original zonohedron does.
start with a tetrahedrally
stellated icosahedron to get this canonical
form with chiral tetrahedral symmetry, i.e., no planes
of symmetry. It is particularly interesting because it is self-dual.
Here is it compounded
with its dual, as illustrated at right. You can also get here starting
from the tetrahedrally
truncated dodecahedron. (More
about these polyhedra.)
check out this 22-sided polyhedron
which I cooked up to see something else with tetrahedral symmetry, but
chiral. It began as a truncated octahedron with four hexagons trisected.
start with a cube from which one corner has been maximally truncated, to
get this heptahedron in canonical
form. The 4-sided faces are like half a cube and the 3-sided faces
are like half an octahedron, so its dual is the same polyhedron, but "turned
over." When we combine
it with its dual, the result contains two half-cubes and two half-octahedra,
like yet unlike the compound
of a cube and octahedron.
start with the net shown here (Figure 1.5A of Coxeter's Regular
Polytopes, p.8), to get this nice
enneahedron. It is interesting because it is the simplest polyhedron
for which there is no Hamiltonian cycle (a round trip via edges which visits
each vertex exactly once). What is more interesting is that from the net
it is not at all apparent that the solid has 3-fold symmetry, but the canonical
form brings it out immediately. (I was surprised !) I believe it is also
the simplest polyhedron with an odd number of faces that each have the
same number of edges.
next, here is a canonical form of a "heptagonal
hermaphrodite." It is composed of half a prism
and half a dipyramid. The result
is self-dual, so it makes a nice compound
and finally, here is the canonical form of a "heptagonal
antihermaphrodite." It is composed of half an antiprism
and half a trapezohedron.
The result is again self-dual, so it also makes a nice compound
My canonicalizing algorithm inputs a polyhedron (or "3-connected" planar
graph) and computes its canonical form numerically. I used it to produce
the canonical forms above (and many others, for testing purposes.) The
algorithm inputs a polyhedron (or maps a graph around the unit sphere to
create a set of vertices and edges) and then iteratively modifies it to
improve its conformance with the three conditions at the top of this page.
After a couple of hundred iterations it typically finds that the conditions
are satisfied within a very small tolerance, and stops. Three simple operations
are all that is needed; they correspond to the three conditions:
Iterating, it sometimes takes many steps for a correction at one part of
the polyhedron to percolate its way around and equalize the conditions
everywhere. On the examples above, between 50 and 500 iterations were sufficient
to have all conditions satisfied within a tolerance of 10^-7. As the individual
steps are quite simple, only seconds are required.
For each edge, the closest point to the origin is determined, call it x.
If x lies at unit distance from the origin, the first condition
is satisfied for that edge. If not, a small fraction of x is added
to the two vertices which define the edge, (in proportion to the sign and
amount of the error) so that at the next iteration the edge will be closer
to tangency with the unit circle.
The center of gravity of all the points x is determined. If it is
zero, the second condition is satisfied. If not, it is subtracted from
all the vertices.
For each face, if the vertices lie in a plane in space, the third condition
is satisfied. If not, a plane which approximates it is computed. (Rather
than a least-squares fit, a quick method is used: the average of the cross
products of the face angles defines a normal direction and the centroid
of the face's vertices defines a distance from the origin.) Each vertex
of the face is then projected on to the plane.
Mathematica code is available in the article Hart  listed in the