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 (Formula presented.). 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.

Zero-sum cycles in flexible polyhedra

Gallet M.;Schicho J.
2022-01-01

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 (Formula presented.). 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.
2022
4-mar-2022
Pubblicato
https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12562
File in questo prodotto:
File Dimensione Formato  
Bulletin of London Math Soc - 2022 - Gallet - Zero‐sum cycles in flexible polyhedra.pdf

Accesso chiuso

Tipologia: Documento in Versione Editoriale
Licenza: Copyright Editore
Dimensione 409.93 kB
Formato Adobe PDF
409.93 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Bulletin+of+London+Math+Soc+-+2022+-+Gallet+-+Zero‐sum+cycles+in+flexible+polyhedra-Post_print.pdf

accesso aperto

Tipologia: Bozza finale post-referaggio (post-print)
Licenza: Digital Rights Management non definito
Dimensione 983.08 kB
Formato Adobe PDF
983.08 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/3037702
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact