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In search for a perfect shape of polyhedra: Buffon transformation

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posted on 23.09.2016 by Veronika Schreiber, Alexander Veselov, Joseph P. Ward
For an arbitrary polygon generate a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buff on (1707–1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine B -regular polyhedra. The main result is the proof of existence of star-shaped affine $$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

L’Enseignement Mathématique

Volume

61

Issue

3

Pages

261 - 284

Citation

SCHREIBER, V., VESELOV, A.P. and WARD, J.P., 2015. In search for a perfect shape of polyhedra: Buffon transformation. L’Enseignement Mathématique, 61 (3/4), pp. 261 - 284.

Publisher

© European Mathematical Society

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2015

Notes

This article was published in the journal, L’Enseignement Mathématique [© © European Mathematical Society] and the definitive version is available at: http://dx.doi.org/10.4171/LEM/61-3/4-1

ISSN

0013-8584

Language

en

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