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
- 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
- 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
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
|Number of Faces
||Number of Polyhedra
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.
- 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.
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
ID Number: A000944 (Formerly M1796 and N0709)
Name: Polyhedra (or 3-connected simple planar graphs) with n nodes.
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Created 23 Sep 1997, Last Update
14 December 2009
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