Regular Polytopes (Dover Books on Mathematics) by Coxeter H. S. M

Author:Coxeter, H. S. M. [Coxeter, H. S. M.]
Language: eng
Format: azw3
ISBN: 9780486141589
Publisher: Dover Publications
Published: 2012-05-22T16:00:00+00:00

having 96 vertices, 288+144 edges, 96+96+288 triangular faces, 24+96 tetrahedra, and 24 icosahedra. One type of edge is surrounded by one tetrahedron and two icosahedra, the other by three tetrahedra and one icosahedron.

8·5. Gosset’s construction for {3, 3, 5}. Since the circum-radius of an icosahedron is less than its edge-length, we can construct, in four dimensions, a pyramid with an icosahedron for base and twenty regular tetrahedra for its remaining cells. Let us place such a pyramid on each icosahedron of s{3, 4, 3}. The effect is to replace each icosahedron by a cluster of twenty tetrahedra, involving one new vertex, twelve new edges, and thirty new triangles. Thus we obtain a polytope with 96+24 vertices, 288+144+288 edges, 96+96+288+720 triangular faces, and 24+96+480 tetrahedral cells. For the purpose of counting the number of cells that surround an edge, each icosahedron of s{3, 4, 3} counts for two tetrahedra. Thus an edge of any of the three types is surrounded by just five tetrahedra. This, therefore, is the regular polytope

{3 3, 5},

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