How Many polyhedra are There?

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

Special Classes of polyhedra

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
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

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Created 23 Sep 1997, Last Update 31 May 2020