TY - JOUR

T1 - Descartes Circle Theorem, Steiner Porism, and Spherical Designs

AU - Schwartz, Richard Evan

AU - Tabachnikov, Serge

N1 - Funding Information:
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.
Publisher Copyright:
© 2020, © THE MATHEMATICAL ASSOCIATION OF AMERICA.

PY - 2020/3/15

Y1 - 2020/3/15

N2 - 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.

AB - 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.

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U2 - 10.1080/00029890.2020.1690909

DO - 10.1080/00029890.2020.1690909

M3 - Article

AN - SCOPUS:85079755223

SN - 0002-9890

VL - 127

SP - 238

EP - 248

JO - American Mathematical Monthly

JF - American Mathematical Monthly

IS - 3

ER -