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Published online by Cambridge University Press: 20 November 2018
H. S. M. Coxeter has shown the existence of three infinite regular polyhedra, and has proved that there are no infinite regular polyhedra other than these. In his paper he gives the definition of regularity of a polyhedron :
A polyhedron is said to be regular if it possesses two particular symmetries: one which cyclically permutes the vertices ox any face c, and one which cyclically permutes the faces that meet at a vertex C, C being a vertex of c.
1 Regular skew polyhedra in three and four dimensions, Proc. London Math. Soc. (2) 43 (1937), 33-62.
2 This definition is adequate for polyhedra in which faces are allowed to intersect only at edges; otherwise a proviso must be added to exclude compounds such as the five cubes with the vertices of a dodecahedron.
3 I must thank the referee for pointing this out.
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