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# Counting Polyhedra

Total (row) 4 5 6 7 8 9 10 11 12 13 Vertex Count in row and Face Count in column, or vice-versa 1 1 2 1 1 7 1 2 2 2 34 2 8 11 8 5 257 2 11 42 74 76 38 14 2606 8 74 296 633 768 558 219 32300 5 76 633 2635 6134 8822 7916 440564 38 768 6134 25626 64439 104213 6384634 14 558 8822 64439 268394 709302 96262938 219 7916 104213 709302 2937495 1496225352 50 4442 112082 1263032 8085725 23833988129 1404 79773 1556952 15535572 387591510244 233 36528 1338853 21395274 6415851530241 9714 789749 21317178 107854282197058 1249 306470 15287112 70454 7706577 7595 2599554 © 2000  Gérard P. Michon, Ph.D.  A larger table is available ! 527235 49566 The number of edges is the sum of the row and column indices minus 2.
 6 1 0 1 2 2 4 12 22 58 158 448 1342 4199 13384 43708 144810 485704 1645576 5623571 19358410 67078828 233800162 819267086 2884908430 10204782956
• The tabulated counts given here pertain to polyhedra that are topologically different.  Because a permissible topological transformation is a symmetry with respect to a plane (or a point), the two different mirror images of a chiral polyhedron are not counted as distinct.  For example, there is one (and only one) chiral hexahedron (the tetragonal antiwedge with 4 triangular faces and two tetragonal ones).  If both chiralities of the tetragonal antiwedge were counted as distinct, there would be 3 polyhedra with 6 faces and 6 vertices (instead of 2, including the pentagonal pyramid), and there would be 8 hexahedra (instead of 7).  Similarly, there are 11 chiral heptahedra: 3 with 7 vertices, 5 with 8, 2 with 9, and a single chiral heptahedron with 10 vertices (if different chiralities were counted as distinct polyhedra, there would be 45 heptahedra, not 34).

• The  duality  of polyhedra makes this table symmetrical. The edges of any polyhedron may be considered to connect two faces instead of two vertices.  Interchanging the role of faces and vertices in this way gives two polyhedra which are said to be dual of each other. (The dual of a chiral polyhedron is chiral, because if it was not, its dual --namely the original polyhedron-- would not be chiral.) For example, the  convex  polyhedra with  equivalent  vertices whose faces consist of two or more different types of regular polygons are called Archimedean solids. Their duals are called Catalan solids.  (Two nodes of a polyhedron are said to be equivalent when one matches the other in some symmetry of the polyhedron.)

• Each node (face or vertex) is connected by at least 3 edges and each edge connects exactly 2 nodes. This means that twice the number E of edges must be at least equal to three times the number V (or F) of nodes.  The Descartes-Euler formula V-E+F=2 then implies that 2(V+F-2) is at least equal to 3V or 3F.  This can be rewritten in terms of the two inequalities:  (V-4) £ 2(F-4)  and  (F-4) £ 2(V-4).  This explains why the above table only lists nonzero entries between two "lines" of slope 2 and 1/2...

• When either of the above inequalities is an equality, we have a polyhedron where either all nodes have degree 3  (the polyhedron is then called a simplicial polyhedron)  or all faces are triangles, such a polyhedron is sometimes called an (irregular) deltahedron (the term deltahedron by itself usually implies equilateral faces; there are only 8 convex regular deltahedra).  The number of simplicial polyhedra with n nodes is known for a few more values beyond the range of the above table.
• Starting at n=4, the sequence is: 1, 1, 2, 5, 14, 50, 233, 1249, 7595, 49566, 339722, 2406841, 17490241, 129664753, 977526957, 7475907149, 57896349553, 453382272049 ...
• The sequence adjacent to the above (the inner border) corresponds to n-hedra with 2n-5 vertices (the vertices are all of degree 3, except one which is of degree 4).  The "other" inner border corresponds to the dual polyhedra, which have n vertices and 2n-5 faces (one quadrilateral and 2n-6 triangles).
• Starting at n=5, the sequence is: 1, 2, 8, 38, 219, 1404, 9714, 70454, 527235, 4037671, 31477887, 249026400, 1994599707, 16147744792, ...
• The number of polyhedra with n nodes and n faces is also known beyond what is shown in the table.
• Starting at n=4, the sequence is: 1, 1, 2, 8, 42, 296, 2635, 25626, 268394, 2937495, 33310550, 388431688, 4637550072, 56493493990, 700335433295, ...
• So is the number of n-hedra with n+1 nodes.
• Starting at n=4, the sequence is: 0, 1, 2, 11, 74, 633, 6134, 64439, 709302, 8085725, 94713809, 1134914458, 13865916560, 172301697581, 2173270387051, ...
• The number of polyhedra with n edges is given approximately by the empirical formula   (n-6) (n-8)/3.
• Starting at n=6, the sequence is: 1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, 1342, 4199, 13384, 43708, 144810, 485704, 1645576, 5623571, 19358410, 67078828, 233800162, 819267086, 2884908430, 10204782956, 36249143676, 129267865144, 462669746182, 1661652306539, 5986979643542, ...
• Finally, the total number of n-hedra is very roughly  2/3(n-3)!(n-5)(n-8).
• Starting at n=4, the sequence is: 1, 2, 7, 34, 257, 2606, 32300, 440564, 6384634, 96262938, 1496225352, 23833988129, 387591510244, 6415851530241, 107854282197058, ...

## Online References:

### From Sloane's On-Line Encyclopedia of Integer Sequences:

• Tough ones:
• A000109 The number of deltahedra with n vertices: The outer border(s) of the table.
• A058786 The number of n-hedra with 2n-5 vertices (one of degree 4, others of degree 3): The inner border.
• A002856 The number of polyhedra with n vertices and n faces: The diagonal.
• A058789 The number of n-hedra with n+1 vertices: The off-diagonal(s).
• A000944 The number of polyhedra with n faces (or n vertices): The file totals.
• A002840 The number of polyhedra with n edges: The antidiagonal totals.
• A058787 The number of polyhedra with n faces and k edges: The whole table.
• A058788 The number of polyhedra with n edges and k vertices: Another viewpoint.
• Easy ones:
• A001651 The distinct vertex counts an n-hedron may have, namely ë3n/2û-5, which is never a multiple of 3 . The lengths of the rows or columns in our table: 1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26,28...
• A008611 The distinct vertex (or face) counts that are possible with n edges, namely n-1-2ë(n+2)/3û . The lengths of the table's antidiagonals: 1,0,1,2,1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,7,8,9...