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Napoleon in isolation

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arxiv math/9909106 v2 pith:Q4BBN2RR submitted 1999-09-18 math.GT

classification math.GT
keywords certaincollectioncuspshyperbolicisolationnapoleonpolygonssome
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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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  1. Expansion joints in hyperbolic manifolds

    math.GT 2025-11 conditional novelty 7.0 of 10

    Cone-deforming an ideal arc through 'expansion joints' interpolates between stacked Borromean ring complements and lantern manifolds, and yields cone deformations of highly twisted 2-bridge unknotting tunnels.

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