Polyhedron Compound

A polyhedron compound is an arrangement of a number of interpenetrating polyhedra, either all the same or of several distinct types, usually having visually attractive symmetric properties. Compounds of multiple Platonic and Archimedean solids can be especially attractive, as can compounds of these solids and their duals. For example, the compound of the tetrahedron and its dual gives a tetrahedron 2-compound whose hull is known as the stella octangula.


Particularly nice compounds are produced by rotating copies of a regular solid with n-gonal faces about axes from the origin through the center of each face by an angle of pi/n radians. Such compounds are illustrated above for the Platonic solids and Kepler-Poinsot polyhedra

Other attractive compounds can be obtained by rotating n copies of a solid about a C_k rotational axis by 2pii/(kn) for i=0, ..., n-1.

While there is no standard notation for polyhedral compounds, in Coxeter's notation, d distinct polyhedron vertices of {m,n} taken c times are denoted


or faces of {s,t} e times


or both


See also


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Cundy, H. and Rollett, A. "Regular Compounds." §3.10 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 129-142, 1989.Hart, G. "Compounds of Cubes.", A. C. and Smith, A. "Computer Drawings of Compounds of Star Polyhedra." Math. Gaz. 57, 303-306, 1973.Skilling, J. "Uniform Compounds of Uniform Polyhedra." Math. Proc. Cambridge Philos. Soc. 79, 447-457, 1976.Smith, A. "Uniform Compounds and the Group A_4." Proc. Cambridge Philos. Soc. 75, 115-117, 1974.Verheyen, H. F. Symmetry Orbits. Boston, MA: Birkhäuser, 2007.Webb, R. "Miscellaneous Polyhedra: Compounds.", D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 37-38, 1991.Wenninger, M. J. "Some Interesting Polyhedral Compounds." Ch. 5 in Dual Models. Cambridge, England: Cambridge University Press, pp. 143-148, 1983.

Referenced on Wolfram|Alpha

Polyhedron Compound

Cite this as:

Weisstein, Eric W. "Polyhedron Compound." From MathWorld--A Wolfram Web Resource.

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