In search for a perfect shape of polyhedra: Buffon transformation
Spectral Theory
2014-04-29 v2
Abstract
For an arbitrary polygon consider 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 Buffon (1707-1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine -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 Verdiere and Lovasz.
Keywords
Cite
@article{arxiv.1402.5354,
title = {In search for a perfect shape of polyhedra: Buffon transformation},
author = {V. Schreiber and A. P. Veselov and J. P. Ward},
journal= {arXiv preprint arXiv:1402.5354},
year = {2014}
}
Comments
Slightly revised version with added example of pentakis dodecahedron