Icosahedron Stellations

Applying the Stellation process to the Icosahedron gives

$\begin{displaymath} 20+30+60+20+60+120+12+30+60+60 \end{displaymath}$

cells of ten different shapes and sizes in addition to the Icosahedron itself. After application of five restrictions due to J. C. P. Miller to define which forms should be considered distinct, 59 stellations are found to be possible. Miller's restrictions are

: 1. The faces must lie in the twenty bounding planes of the icosahedron.
: 2. The parts of the faces in the twenty planes must be congruent, but those parts lying in one place may be disconnected.
: 3. The parts lying in one plane must have threefold rotational symmetry with or without reflections.
: 4. All parts must be accessible, i.e., lie on the outside of the solid.
: 5. Compounds are excluded that can be divided into two sets, each of which has the full symmetry of the whole.

Of these, 32 have full icosahedral symmetry and 27 are Enantiomeric forms. Four are Polyhedron Compounds, one is a Kepler-Poinsot Solid, and one is the Dual Polyhedron of an Archimedean Solid. The only Stellations of Platonic Solids which are Uniform Polyhedra are the three Dodecahedron Stellations the Great Icosahedron (stellation # 11).

	name
1	Icosahedron
2	Triakis Icosahedron
3	Octahedron 5-Compound
4	Echidnahedron
11	Great Icosahedron
18	Tetrahedron 10-Compound
20	Deltahedron-60
36	Tetrahedron 5-Compound

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 146-147, 1987.

Bulatov, V. ``Stellations of Icosahedron.'' http://www.physics.orst.edu/~bulatov/polyhedra/icosahedron/.

Coxeter, H. S. M. The Fifty-Nine Icosahedra. New York: Springer-Verlag, 1982.

Hart, G. W. ``59 Stellations of the Icosahedron.'' http://www.li.net/~george/virtual-polyhedra/stellations-icosahedron-index.html.

Maeder, R. E. Icosahedra.m notebook. http://www.inf.ethz.ch/department/TI/rm/programs.html.

Maeder, R. E. ``The Stellated Icosahedra.'' Mathematica in Education 3, 1994. ftp://ftp.inf.ethz.ch/doc/papers/ti/scs/icosahedra94.ps.gz.

Maeder, R. E. ``Stellated Icosahedra.'' http://www.mathconsult.ch/showroom/icosahedra/.

Wang, P. ``Polyhedra.'' http://www.ugcs.caltech.edu/~peterw/portfolio/polyhedra/.

Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, pp. 41-65, 1989.

Wheeler, A. H. ``Certain Forms of the Icosahedron and a Method for Deriving and Designating Higher Polyhedra.'' Proc. Internat. Math. Congress 1, 701-708, 1924.