On Coxeter’s Loxodromic Sequences of Tangent Spheres

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

$$ .{\rm{ }}.{\rm{ }}.{\rm{ }},{C_{ - 1}},{C_0},{C_1},{C_2},\;.{\rm{ }}.{\rm{ }}. $$

(1)

with the property that every n + 2 consecutive members are mutually tangent.