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Title: In search for a perfect shape of polyhedra: Buffon transformation
Authors: Schreiber, Veronika
Veselov, A.P.
Ward, Joseph P.
Keywords: Affine polyhedra
Spectral graph theory
Issue Date: 2015
Publisher: © European Mathematical Society
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.
Abstract: 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.
Description: 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
Version: Accepted for publication
DOI: 10.4171/LEM/61-3/4-1
URI: https://dspace.lboro.ac.uk/2134/22558
Publisher Link: http://dx.doi.org/10.4171/LEM/61-3/4-1
ISSN: 0013-8584
Appears in Collections:Published Articles (Maths)

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