Beyond the Descartes circle theorem

2001/01/09 by Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks · 1 citation
Mathematics · #math.MG #msc:52C26 #msc:11H55 #msc:51M10 #msc:53A35

paper · pdf

published as Amer. Math. Monthly 109 (2002), 338--361. · 25 pages, 6 figures, Latex

arxiv created 2001/01/09 · arxiv updated 2009/11/30

Abstract

The Descartes circle theorem states that if four circles are mutually tangent with disjoint intersion, then their curvatures (or "bends) b_j = 1/r_j satisfy the relation (b₁ + b₂ + b₃ + b₄)² = 2(b₁² + b₂² + b₃² + b₄²). We show that similar relations hold involving the centers of the circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes theorem. These relations have matrix generalizations to the n-dimensional case, in each of Euclidean, spherical and hyperbolic geometries, and they include a Descartes circle theorem for spherical and hyperbolic space.

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• The Local-Global Conjecture is False for Generalized Circle Packings

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