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 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 DescartesEuler formula VE+F=2 then implies that 2(V+F2) is at least equal to 3V or 3F. This can be rewritten in terms of the two inequalities: (V4) £ 2(F4) and (F4) £ 2(V4). 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 nhedra with 2n5 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 2n5 faces (one quadrilateral and 2n6 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 nhedra 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 (n6)^{ (n8)/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 nhedra is very roughly 2/3(n3)!(n5)(n8).
 Starting at n=4, the sequence is: 1, 2, 7, 34, 257, 2606, 32300, 440564, 6384634, 96262938, 1496225352, 23833988129, 387591510244, 6415851530241, 107854282197058, ...