canonical polyhedra

an interesting theorem

Any convex polyhedron has an unique "canonical form" which is generally a distorted version of the polyhedron, with all its edges tangent to the unit sphere and the origin at the center of gravity of the tangency points.
The faces of a canonical polyhedron are not necessary regular. The canonical form of a polyhedron has the maximum symmetry, thus the original planes and axes of symmetry are preserved on the canonical form. If we reciprocate a canonical polyhedron about its midsphere (the unit sphere), the dual polyhedron will share the same edge-tangency points: it is the canonical dual. Two canonical dual polyhedra have their edges perpendicular at the tangency points with the unit sphere.
The regular and semi-regular polyhedra are canonical. For a "very irregular" polyhedron it is interesting, and sometimes surprising, to observe its canonical form.
John Conway popularized this nice theorem which unfortunately doesn't give a method to find the canonical form. Fortunately George W. Hart wrote an algorithm to do the job; his Mathematica code is available on his web site (see reference) and has been used in the following examples.

a few examples

These two pentahedra (half tetrahedron and half cube) have the same canonical form (regular triangular prism).

These two heptahedra derive from the cube: there are no right angles on their canonical form.

The canonical forms of the "hermaphrodites" (composed of a half prism and a half diamond) and the "antihermaphrodites" (composed of a half antiprism and a half antidiamond) are their own duals. Here are the pentagonal models (11 faces and 11 vertices).

The enneahedron on the left is the simplest polyhedron with an odd number of n-faces (n=4, thus its 9 faces are quadrilaterals).
Thus its dual (on the right) has 9 vertices of order 4 and 11 faces (3 quadrilaterals and 8 triangles).
It is surprising to discover a 3-fold symmetry on the canonical forms.

reference: Canonical Polyhedra by George W. Hart

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