Napoleon in isolation

Details Napoleon in isolation Napoleon in isolation

1999-09-18

arxiv.org/abs/math/9909106v2

Proc. Amer. Math. Soc. 129 (2001), no. 10, 3109-3119

Mathematics

Geometric Topology

10 pages, 5 figures; minor changes. Accepted for publication in PAMS

Scientific paper

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 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 leaves the geometric structure at some other collection of cusps unchanged.

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