JournalslemVol. 61, No. 3/4pp. 261–284

In search for a perfect shape of polyhedra: Buffon transformation

  • Veronika Schreiber

    Loughborough University, UK
  • Alexander P. Veselov

    Loughborough University, UK
  • Joseph P. Ward

    Loughborough University, UK
In search for a perfect shape of polyhedra: Buffon transformation cover
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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 Buffon (1707–1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine BB-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.

Cite this article

Veronika Schreiber, Alexander P. Veselov, Joseph P. Ward, In search for a perfect shape of polyhedra: Buffon transformation. Enseign. Math. 61 (2015), no. 3/4, pp. 261–284

DOI 10.4171/LEM/61-3/4-1