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 centre 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. Two dual canonical 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).



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



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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).



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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.



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reference: Canonical Polyhedra by George W. Hart


January 2004
updated 03-02-2004