stellated polyhedra

The plane of each face of a polyhedron often intersect the planes of the other non adjacent faces. Thus we obtain new polyhedra, usually non convex, by successive "stellations". More precisely, two polyhedra are stellations of each other if their faces lie in the same set of planes. Different stellations may appear identical, with hidden parts of the faces, but only one consists of all visible parts.
The three successive stellations of the regular dodecahedron lead first to the small stellated dodecahedron (12 stellated pentagons, but 60 visible triangles), then to the great dodecahedron (12 convex pentagons, and still 60 visible triangles), and last to the great stellated dodecahedron (again 12 stellated pentagons, and always 60 visible triangles).
Concerning the great icosahedron (20 equilateral triangles, 120+60 visible triangles), Cauchy proved that it is one of the 59 stellations of the regular icosahedron, among which we find also the compounds of five tetrahedra, of ten tetrahedra, and of five octahedra.
The stellation of the regular octahedron is the Kepler star (anticube), compound of two tetrahedra (24 visible equilateral triangles).
The first stellation of the regular icosahedron has twenty non regular hexagonal faces (60 visible triangles).
The first stellation of the rhombic dodecahedron is a compound of three identical non regular octahedra "flatted" alongside a diagonal (48 visible triangles); it is surprising to discover that it paves the space.

the only stellation
of the regular octahedron

first stellation
of the regular icosahedron

first stellation
of the rhombic dodecahedron

The first stellation of the cuboctahedron (6x4+8x3=48 visible triangles) is the compound of two dual polyhedra (cube and regular octahedron); the same holds for the first stellation of the icosidodecahedron (20x3+12x5=120 visible triangles) which is the compound of a regular dodecahedron and a regular icosahedron.

If you wish to experiment other stellations and print their nets, then "Great STELLA" is the program you need!

You can also easily create nice "polyhedral stars" by raising "spikes" (regular pyramids) on the faces of convex polyhedra, but note: these polyhedra are not stellations of a polyhedron, only assemblies of pyramids on a convex polyhedron. Here are two examples:

cuboctahedron with 6+8 = 14 spikes

icosidodecahedron with 12+20 = 32 spikes

Many examples can be found at the tops of religious buildings:
Star polyhedra on churches by Tibor Tarnai, János Krähling and Sándor Kabai (with many illustrations).

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects July 1999
updated 16-07-2022