On Coxeter’s Loxodromic Sequences of Tangent Spheres

Asia Weiss
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Asia Weiss: University of Toronto, Department of Mathematics

A chapter in The Geometric Vein, 1981, pp 243-250 from Springer

Abstract: Abstract A loxodromic sequence of tangent spheres in n-space is an infinite sequence of (n − 1)-spheres having the property that every n + 2 consecutive members are mutually tangent. When considering mutually tangent spheres we’ll always suppose they have distinct points of contact. Given any ordered set of n + 2 mutually tangent (n − 1)-spheres, we can invert into n congruent (n — 1)-spheres sandwiched between two parallel hyperplanes, and hence (since the centres of these n are the vertices of a regular simplex) they are all inversively equivalent. Furthermore, any ordered set of n + 1 mutually tangent (n − 1)-spheres (C 0, C 1,..., C n can be completed to a set of n + 2 spheres in exactly two ways. Hence the spheres belong to just one sequence 1 $$ .{\rm{ }}.{\rm{ }}.{\rm{ }},{C_{ - 1}},{C_0},{C_1},{C_2},\;.{\rm{ }}.{\rm{ }}. $$ with the property that every n + 2 consecutive members are mutually tangent.

Date: 1981
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DOI: 10.1007/978-1-4612-5648-9_16

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