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Transactions of the American Mathematical Society

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Rigidity of circle polyhedra in the $ 2$-sphere and of hyperideal polyhedra in hyperbolic $ 3$-space


Authors: John C. Bowers, Philip L. Bowers and Kevin Pratt
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 52C26
DOI: https://doi.org/10.1090/tran/7483
Published electronically: September 25, 2018
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Abstract: We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $ 3$-space $ \mathbb{E}^{3}$ to the context of circle polyhedra in the $ 2$-sphere $ \mathbb{S}^{2}$. We prove that any two convex and proper nonunitary c-polyhedra with Möbius-congruent faces that are consistently oriented are Möbius congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere, as well as that of certain
hyperideal hyperbolic polyhedra in $ \mathbb{H}^{3}$.


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Additional Information

John C. Bowers
Affiliation: Department of Computer Science, James Madison University, Harrisonburg, Virginia 22807
Email: bowersjc@jmu.edu

Philip L. Bowers
Affiliation: Department of Mathematics, The Florida State University, Tallahassee, Florida 32306
Email: bowers@math.fsu.edu

Kevin Pratt
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: kevin.pratt@uconn.edu

DOI: https://doi.org/10.1090/tran/7483
Keywords: Circle packing, inversive distance, hyperbolic geometry, hyperideal polyhedra
Received by editor(s): June 1, 2017
Received by editor(s) in revised form: December 6, 2017
Published electronically: September 25, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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