faceted polyhedra

Faceting is an other powerful but less known tool; it is the dual process of stellation.
Just as stellation may be defined as two polyhedra with faces in the same planes, faceting may be defined as two polyhedra with the same vertices. So, the small stellated dodecahedron, the great dodecahedron and the great icosahedron are facetings of the icosahedron, but the great stellated dodecahedron is a faceting of the dodecahedron. The convex hull of a polyhedron is a special case of a faceting.

Here is a simple example: Kepler's star (stella octangula) is a stellation of the regular octahedron, but also a faceting of the cube. This compound of two regular tetrahedra has eight self intersecting triangular faces (the midpoint triangles, faces of the octahedron, are not visible).

stellation of the regular octahedron
(a layer of eight regular tetrahedra added to the octahedron)

faceting of the cube: eight triangular facets
(twelve non regular tetrahedra removed from the cube)

Two other examples: the two Poinsot polyhedra are two facetings of the regular icosahedron:
  •   the great dodecahedron (a stellation of the dodecahedron) is build by 12 regular pentagonal facets,
  •   the great icosahedron (a stellation of the icosahedron) is build by 20 regular triangular facets.

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects February 2004
updated 04-07-2005