# 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 beyond 10 faces. The results are tabulated below.

 Number of Faces Number of Polyhedra N(f)/N(f-1) 4 1 ---- 5 2 2 6 7 3.5 7 34 4.9 8 257 7.6 9 2606 10.1 10 32300 12.4 11 440,564 13.6 12 6,384,634 14.5 13 96,262,938 15.1 14 1,496,225,352 15.5 15 23,833,988,129 15.9 16 387,591,510,244 16.3 17 6,415,851,530,241 16.6 18 107,854,282,197,058 16.8

Beyond 10 faces, there are only formulas for estimating the number. The values are from The Online Encyclopedia of Integer Sequences ID Number: A000944. Surprisingly enough, the increase doesn't keep accelerating but levels off. The increase factor from adding a face seems to converge to about 17. Extrapolating to 20 faces, it would take about 30 million years to view them all at movie speed (32 per second).

Plantri is a graph generating program by Gunnar Brinkmann of the University of Ghent and Brendan McKay of the Australian National University. It is very fast and is claimed to be able to generate over a million graphs per second. It's not clear how many of the counts above are known exactly and which are approximated by a formula, but plantri could easily tally the results up to 16 faces. The program downloads as a C program and must be compiled to run. A compiled version (plantri.exe) is available at http://www.squaring.net./downloads/

## Topological varieties of 8- and 9-hedra

These tables were generated using plantri.exe. A list of graphs (equivalent to face adjacency lists) was created, then sorted by numbers of faces. Symbol refers to the number of faces of each type. Thus 33334456 means four triangles, three quadrilaterals, and one each pentagon and hexagon. Ntypes lists the number of topologically distinct polyhedra for each type.

### Octahedra

 Vertices Edges Symbol Ntypes 6 12 33333333 2 7 13 33333335 2 7 13 33333344 9 8 14 33333337 1 8 14 33333346 2 8 14 33333355 3 8 14 33333445 19 8 14 33334444 17 9 15 33333447 2 9 15 33333456 6 9 15 33333555 3 9 15 33334446 11 9 15 33334455 25 9 15 33344445 21 9 15 33444444 6 10 16 33334457 4 10 16 33334466 4 10 16 33334556 11 10 16 33335555 3 10 16 33344447 2 10 16 33344456 15 10 16 33344555 13 10 16 33444446 5 10 16 33444455 16 10 16 34444445 2 10 16 44444444 1 11 17 33335557 1 11 17 33335566 2 11 17 33344467 2 11 17 33344557 3 11 17 33344566 6 11 17 33345556 4 11 17 33444457 1 11 17 33444466 2 11 17 33444556 8 11 17 33445555 4 11 17 34444456 2 11 17 34444555 2 11 17 44444455 1 12 18 33336666 1 12 18 33345567 1 12 18 33345666 1 12 18 33444477 1 12 18 33444567 1 12 18 33445557 1 12 18 33445566 2 12 18 33455556 1 12 18 33555555 1 12 18 34444566 1 12 18 34445556 1 12 18 44444466 1

### Enneahedra

 Vertices Edges Symbol Ntypes 7 14 333333334 8 8 15 333333336 3 8 15 333333345 24 8 15 333333444 47 9 16 333333338 1 9 16 333333347 3 9 16 333333356 6 9 16 333333446 38 9 16 333333455 50 9 16 333334445 143 9 16 333344444 55 10 17 333333448 3 10 17 333333457 10 10 17 333333466 9 10 17 333333556 14 10 17 333334447 20 10 17 333334456 125 10 17 333334555 64 10 17 333344446 73 10 17 333344455 198 10 17 333444445 103 10 17 334444444 14 11 18 333334458 6 11 18 333334467 16 11 18 333334557 24 11 18 333334566 35 11 18 333335556 22 11 18 333344448 5 11 18 333344457 52 11 18 333344466 43 11 18 333344556 164 11 18 333345555 35 11 18 333444447 15 11 18 333444456 134 11 18 333444555 124 11 18 334444446 17 11 18 334444455 66 11 18 344444445 9 11 18 444444444 1 12 19 333335558 2 12 19 333335567 8 12 19 333335666 5 12 19 333344468 6 12 19 333344477 6 12 19 333344558 9 12 19 333344567 43 12 19 333344666 13 12 19 333345557 19 12 19 333345566 43 12 19 333355556 8 12 19 333444458 6 12 19 333444467 21 12 19 333444557 48 12 19 333444566 66 12 19 333445556 80 12 19 333455555 11 12 19 334444448 1 12 19 334444457 15 12 19 334444466 15 12 19 334444556 72 12 19 334445555 31 12 19 344444447 1 12 19 344444456 10 12 19 344444555 15 12 19 444444446 1 12 19 444444455 3 13 20 333336667 1 13 20 333345568 4 13 20 333345577 4 13 20 333345667 11 13 20 333346666 2 13 20 333355558 1 13 20 333355567 3 13 20 333355666 2 13 20 333444478 3 13 20 333444568 7 13 20 333444577 8 13 20 333444667 6 13 20 333445558 5 13 20 333445567 29 13 20 333445666 13 13 20 333455557 5 13 20 333455566 9 13 20 333555556 4 13 20 334444468 1 13 20 334444477 2 13 20 334444558 3 13 20 334444567 14 13 20 334444666 4 13 20 334445557 12 13 20 334445566 25 13 20 334455556 13 13 20 334555555 1 13 20 344444467 2 13 20 344444557 3 13 20 344444566 6 13 20 344445556 9 13 20 344455555 3 13 20 444444556 2 13 20 444445555 2 14 21 333346677 1 14 21 333355668 1 14 21 333355677 1 14 21 333445578 2 14 21 333445668 2 14 21 333445677 2 14 21 333446667 2 14 21 333455568 1 14 21 333455577 1 14 21 333455667 2 14 21 333555666 2 14 21 334444488 1 14 21 334444578 1 14 21 334444668 1 14 21 334445568 2 14 21 334445577 2 14 21 334445667 3 14 21 334446666 1 14 21 334455558 1 14 21 334455567 3 14 21 334455666 3 14 21 334555557 1 14 21 334555566 2 14 21 344444577 1 14 21 344445567 3 14 21 344445666 1 14 21 344455566 1 14 21 344555556 1 14 21 444444477 1 14 21 444444666 1 14 21 444445566 1 14 21 444455556 1

## References

Use a search engine to locate current links.

• 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 000944
• plantri and fullgen; Gunnar Brinkmann (University of Ghent) and Brendan McKay (Australian National University)