How Many Polyhedra are There?

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Special Classes of Polyhedra

• Platonic Solids (convex solids with identical regular polygon faces and identical vertices): 5
  • Tetrahedron: 4 triangles
  • Cube: 6 squares
  • Octahedron: 8 triangles
  • Dodecahedron: 12 pentagons
  • Icosahedron: 20 triangles
• Archimedean Solids (solids with several types of regular polygon faces and identical vertices): 13
• Johnson Solids (solids with regular polygon faces): 91
• Kepler-Poinsot Solids (solids with identical regular polygon faces and identical vertices but with interpenetrating faces): 4. Thus there are nine solids altogether with identical regular polygon faces and identical vertices.
• Uniform (Coxeter-Skilling) Polyhedra: 81

Generic Polyhedra

A polyhedron can be represented as a graph, and the effort on enumerating graphs in mathematics has been enormous because graphs apply to networks of all kinds. Thus, the number of polyhedra of each type is known exactly up through 10 faces. The results are tabulated below.

Beyond 10 faces, there are only formulas for estimating the number. Note that the number of faces for 9 and 10 faces is increasing faster than factorially. We can estimate half a million solids with 11 faces and about six million for 12. It is safe to say nobody will ever enumerate all the solids with 20 faces. Viewing them at movie speed (32 per second) it would probably take more than the age of the Universe to see them all.

References 

• B. Grunbaum, Convex Polytopes. Wiley, NY, 1967, p. 424.
• Duijvestijn, A. J. W.; Federico, P. J.; The number of polyhedral (3-connected planar) graphs. Math. Comp. 37 (1981), no. 156, 523-532.
• M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
• H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, B15.

The Online Encyclopedia of Integer Sequences gives a listing of numbers

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000944

ID Number: A000944 (Formerly M1796 and N0709)
Sequence: 0,0,0,1,2,7,34,257,2606,32300,440564,6384634,96262938,
1496225352,23833988129,387591510244,6415851530241
Name: Polyhedra (or 3-connected simple planar graphs) with n nodes.

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Created 23 Sep 1997, Last Update 31 December 2002

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