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Zero‐sum cycles in flexible polyhedra

Zero‐sum cycles in flexible polyhedra We show that if a polyhedron in the three‐dimensional affine space with triangular faces is flexible, that is, can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and −1$-1$. We do this via elementary combinatorial considerations, made possible by a well‐known compactification of the three‐dimensional affine space as a quadric in the four‐dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one‐dimensional analog, which is trivial to solve. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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References (41)

Publisher
Wiley
Copyright
© 2022 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms.12562
Publisher site
See Article on Publisher Site

Abstract

We show that if a polyhedron in the three‐dimensional affine space with triangular faces is flexible, that is, can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and −1$-1$. We do this via elementary combinatorial considerations, made possible by a well‐known compactification of the three‐dimensional affine space as a quadric in the four‐dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one‐dimensional analog, which is trivial to solve.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Feb 1, 2022

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