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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Napoleon in isolation


Author: Danny Calegari
Journal: Proc. Amer. Math. Soc. 129 (2001), 3109-3119
MSC (2000): Primary 57M50, 57M25
Published electronically: April 2, 2001
MathSciNet review: 1840118
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Abstract:

Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles $\mathbb{R} ^2$ admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation phenomena in cusped hyperbolic $3$-manifolds, where hyperbolic Dehn surgeries on some collection of cusps leave the geometric structure at some other collection of cusps unchanged.


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Additional Information

Danny Calegari
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: dannyc@math.berkeley.edu, dannyc@math.harvard.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05915-9
PII: S 0002-9939(01)05915-9
Received by editor(s): June 15, 1999
Received by editor(s) in revised form: March 6, 2000
Published electronically: April 2, 2001
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2001 American Mathematical Society