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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).
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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.)
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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...
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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 ...
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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, ...
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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, ...
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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, ...
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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, ...
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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, ...
