Enumeration of polyhedra

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

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How Many polyhedra are There?
polyhedra with 4-7 Faces (1, 2, 7 and 34, respectively)
Non-Convex polyhedra
Explanation of Topological Data for polyhedra

Octahedra

polyhedra with 8 Faces (257 in all). For convenience of access, these are broken down into several files:

Octahedral Nets

• Octahedra with 6 to 8 Vertices
• Octahedra with 9 Vertices
• Octahedra with 10 Vertices
• Octahedra with 11 or 12 Vertices

Topological Data for Octahedra

• Topological Data for Octahedra with 6-9 vertices
• Topological Data for Octahedra with 10-12 vertices

Enneahedra

polyhedra with 9 Faces (2606 in all). This set was finally completely enumerated on 2 November 2016. For convenience of access, these are broken down into numerous files:

Enneahedral Nets (7-9 Vertices) 378 total

• Enneahedra with 7 or 8 vertices (8 with 7 vertices, 74 with 8 vertices)

• Enneahedra with 9 vertices and 6- 7- or 8-sided faces (296 total)
• Enneahedra with 9 vertices: 333 333 455
• Enneahedra with 9 vertices: 333 334 445
• Enneahedra with 9 vertices: 3333 44444

Topological Data for Enneahedra with 7-9 Vertices

• Topological Data for Enneahedra with 9 vertices: 333333455 and 333334445
• Topological Data for Enneahedra with 7-9 vertices: All Other Classes

Enneahedral Nets (10 Vertices) 633 total

• Enneahedra with 10 vertices: 33 444 4444 (0-13)
• Enneahedra with 10 vertices: 33333 4 555 (a:) (14-77)
• Enneahedra with 10 vertices: 33333 4 555 (b)
• Enneahedra with 10 vertices: 33333 4 555 (c)
• Enneahedra with 10 vertices: 33333 4 555 (d)
• Enneahedra with 10 vertices: 3333 444 55(a); Pentagons Share Edge (78-275)
• Enneahedra with 10 vertices: 3333 444 55(b); Pentagons Share Edge
• Enneahedra with 10 vertices: 3333 444 55(c); Pentagons Share Edge
• Enneahedra with 10 vertices: 3333 444 55(d); Pentagons Share Edge
• Enneahedra with 10 vertices: 3333 444 55; Pentagons Joined at One Vertex
• Enneahedra with 10 vertices: 3333 444 55; Pentagons Non-Adjacent
• Enneahedra with 10 vertices: 333 44444 5(a) (276-323)
• Enneahedra with 10 vertices: 333 44444 5(b) (324-378)
• Enneahedra with 10 vertices: 333333 4 66 and 333333 55 6 (379-401)
• Enneahedra with 10 vertices: 333 33 44 5 6 (403-527)
• Enneahedra with 10 vertices: 3333 4444 6 (528-599)
• Enneahedra with 10 vertices: Heptagon and Octagon Faces (600-632)

Topological Data for Enneahedra with 10 Vertices

• Topological Data for Enneahedra with 10 vertices: 33444444, 333334555, and 333444445
• Topological Data for Enneahedra: 333333466, 333333556, 333333457, 333334447, and 333333448
• Topological Data for Enneahedra with 10 vertices: 333344455
• Topological Data for Enneahedra with 10 vertices: 333334456 and 333344446
• Topological Data for Enneahedra with 10 vertices: Heptagon and Octagon faces

Enneahedral Nets (11 Vertices) 768 total

• Enneahedra with 11 vertices: 444444444 and 333345555(a) (0-35)
• Enneahedra with 11 vertices: 333345555(b)
• Enneahedra with 11 vertices: 333 444 555: All Three Pentagons Share an Edge (36-159)
• Enneahedra with 11 vertices: 333 444 555: Two 5-5 Edges, One 5-5 Vertex
• Enneahedra with 11 vertices: 333 444 555: Two 5-5 Edges Only, or One 5-5 Edge and Two 5-5 Vertices
• Enneahedra with 11 vertices: 33 44444 55: No 5-5 Edges or One 5-5 Vertex (160-225)
• Enneahedra with 11 vertices: 33 44444 55: One 5-5 Edge
• Enneahedra with 11 vertices: 3 4444444 5 and 33333 4 5 66  (226-269)
• Enneahedra with 11 vertices: 333335556 (270-291)
• Enneahedra with 11 vertices: 333344466 (292-334)
• Enneahedra with 11 vertices: 333344556 (a) (335-498)
• Enneahedra with 11 vertices: 333344556 (b)
• Enneahedra with 11 vertices: 333444456 (a) (499-632)
• Enneahedra with 11 vertices: 333444456 (b)
• Enneahedra with 11 vertices: 334444446 (633-649)
• Enneahedra with 11 vertices: 333334467 and 333334557 (650-689)
• Enneahedra with 11 vertices: 333344457 (690-741)
• Enneahedra with 11 vertices: 333444447, 333334458 and 333344448 (742-767)

Topological Data for Enneahedra with 11 Vertices

• Topological Data for Enneahedra with 11 vertices: 444444444, 333345555 and 333444555
• Topological Data for Enneahedra with 11 vertices: 334444455, 344444445, 333334566, 333335556, and 33334466
• Topological Data for Enneahedra with 11 vertices: 333344556
• Topological Data for Enneahedra with 11 vertices: 333444456
• Topological Data for Enneahedra with 11 vertices: 334444446, 333334467, 333334557, 333344457, 333444447, 333334458 and 333344448

Enneahedral Nets (12 vertices) 558 total

Here it gets ugly. Lots of classes with no unique faces and many requiring multiple views.

• Enneahedra with 12 vertices: 333455555 (0-10)
• Enneahedra with 12 vertices: 334445555 (11-25)
• Enneahedra with 12 vertices: 334445555 (16-41)
• Enneahedra with 12 vertices: 334445555 and 444444455 (42-59)
• Enneahedra with 12 vertices: 333335666 and 333344666 (60-77)
• Enneahedra with 12 vertices: 333345566 (78-120)
• Enneahedra with 12 vertices: 333355556 and 333444566 (a) (121-194)
• Enneahedra with 12 vertices: 333444566 (b)
• Enneahedra with 12 vertices: 333445556 (195-274)
• Enneahedra with 12 vertices: 334444466 (275-289)
• Enneahedra with 12 vertices: 334444556 (290-361)
• Enneahedra with 12 vertices: 344444456 and 444444446 (362-372)
• Enneahedra with 12 vertices: 333335567 and 333344477 (373-386)
• Enneahedra with 12 vertices: 333344567 (387-429)
• Enneahedra with 12 vertices: 333345557 (430-448)
• Enneahedra with 12 vertices: 333444467 (449-469)
• Enneahedra with 12 vertices: 333444557 (470-517)
• Enneahedra with 12 vertices: 334444457, 344444447, 333335558, 333344468, 333344558, 333444458 and 334444448 (518-557)

Topological Data for Enneahedra with 12 Vertices

• Topological Data for Enneahedra with 12 Vertices: 333455555, 334445555, 334445555, 444444455, 333335666, 333344666, 333345566, 333355556 and 333444566
• Topological Data for Enneahedra with 12 Vertices: 333445556, 334444466, 334444556, 344444456, 444444446 and 333335567
• Topological Data for Enneahedra with 12 Vertices: 333344477, 333344567, 333345557, 333444467, 333444557, 334444457, 344444447, 333335558, 333344468, 333344558 333444458 and 334444448

Enneahedral Nets (13 vertices) 219 total

• Enneahedra with 13 vertices: 334555555, 344455555, 444445555, 333346666 and 333355666 (0-9)
• Enneahedra with 13 vertices: 333445666 (10-22)
• Enneahedra with 13 vertices: 333455566, 333555556 and 334444666 (23-39)
• Enneahedra with 13 vertices: 334445566 (40-64)
• Enneahedra with 13 vertices: 334455556, 344444566,344445556 and 444444566 (65-94)
• Enneahedra with 13 vertices: 333336667, 333345577, 333345667, 333355567 and 333444577 (95-121)
• Enneahedra with 13 vertices: 333444667, 333445567, 333455557 and 334444477 (122-163)
• Enneahedra with 13 vertices: 334444567, 334445557, 344444467, and 344444557 (164-194)
• Enneahedra with 13 vertices: 333345568, 333355558, 333444478, 333444568, 333445558, 334444468, and 334444558 (195-218)

Topological Data for Enneahedra with 13 Vertices

• Topological Data for Enneahedra with 13 Vertices: No Heptagon or Octagon Faces
• Topological Data for Enneahedra with 13 Vertices: With Heptagon or Octagon Faces

Enneahedral Nets (14 vertices) 50 total

Enneahedra with 14 vertices (0-49)

Topological Data for Enneahedra with 14 Vertices

Topological Data for Enneahedra with 14 Vertices

Listing of 10-hedra

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Created 23 Sep 1997, Last updated 31 May 2020