Descartes Circle Theorem, Steiner Porism, and Spherical Designs

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.

• MSC: Primary 51B10
• Secondary 05B30

Acknowledgments

We thank A. Akopyan and J. Lagarias for their interest and encouragement. RES and ST were supported by NSF Research Grants DMS-1204471 and DMS-1510055, respectively. Part of this material is based upon work supported by the National Science Foundation under Grant DMS-1440140 while ST was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester. We are grateful to the referees for their suggestions and criticisms.

Additional information

Notes on contributors

Richard Evan Schwartz

RICHARD EVAN SCHWARTZ grew up in Los Angeles, where he enjoyed playing tennis and video games. He got a B.S. in math from UCLA in 1987 and then a Ph.D. in math from Princeton in 1991. He is currently the Chancellor’s Professor of Mathematics at Brown. In his spare time, he enjoys cycling, working out, yoga, drawing, listening to music, and computer programming.

Serge Tabachnikov

SERGE TABACHNIKOV grew up in Moscow, USSR; he received Ph.D. in mathematics from Moscow State University in 1987. Since 1990, he has taught in American universities. He is serving as Editor-in-Chief of Experimental Mathematics and associate editor of Arnold Mathematical Journal, The Mathematical Intelligencer, and this Monthly. His favorite pastime is to walk his dog while thinking about mathematics.