Descartes Circle Theorem, Steiner Porism, and Spherical Designs

Richard Evan Schwartz, Serge Tabachnikov

Research output: Contribution to journalArticle

Abstract

A Steiner chain of length k consists of k circles tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be constructed starting with any initial circle tangent to the parent circles. What do the circles in these 1-parameter family of Steiner chains of length k have in common? We prove that the first k–1 moments of their curvatures remain constant within a 1-parameter family. For k = 3, this follows from the Descartes circle theorem. We extend our result to Steiner chains in spherical and hyperbolic geometries and present a related more general theorem involving spherical designs.

Original languageEnglish (US)
Pages (from-to)238-248
Number of pages11
JournalAmerican Mathematical Monthly
Volume127
Issue number3
DOIs
StatePublished - Mar 15 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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