%0 Journal Article %A Schreiber, Veronika %A Veselov, Alexander %A Ward, Joseph P. %D 2016 %T In search for a perfect shape of polyhedra: Buffon transformation %U https://repository.lboro.ac.uk/articles/journal_contribution/In_search_for_a_perfect_shape_of_polyhedra_Buffon_transformation/9385688 %2 https://repository.lboro.ac.uk/ndownloader/files/16998884 %K Affine polyhedra %K Spectral graph theory %K Mathematical Sciences not elsewhere classified %X 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. %I Loughborough University