Icosahedron Stellations

Applying the stellation process to the icosahedron gives

cells of 11 different shapes and sizes (including the icosahedron itself).

The icosahedron has 18 fully supported stellations, 16 of them reflexible and 2 of them chiral (Webb; where, as usual, the original icosahedron itself is included in this count).

After application of five restrictions known as Miller's rules to define which forms should be considered distinct, 59 stellations (including the original icosahedron itself) are possible (Coxeter et al. 1999).

Of the 59, 32 have full icosahedral symmetry and 27 are enantiomeric forms. One is a Platonic solid (the icosahedron itself), one is a Kepler-Poinsot polyhedron, four are polyhedron compounds, and one is the dual polyhedron of an Archimedean solid. Note that the first real stellation (stellation #2 in Coxeter's counting) is that obtained by cumulating the icosahedron until the faces of each triangular pyramid lie parallel to the surrounding original faces. This gives fairly small spikes, and results in a solid known as the small triambic icosahedron. Note also that the great stellated dodecahedron is not an icosahedron stellation, since the faces of its groups of five triangular pyramids do not lie in the same plane even though they appear very close to it.

The stellations illustrated above are given in the ordering of Maeder (1994); Rogers uses a different ordering. Special cases are summarized in the following table.

	name
1	icosahedron
2	small triambic icosahedron
3	octahedron 5-compound
4	echidnahedron
11	great icosahedron
13	medial triambic icosahedron
13	great triambic icosahedron
18	tetrahedron 10-compound
36	tetrahedron 5-compound