Patches with minimal boundary length

Abstract In which way must n hexagons and k pentagons be glued together so that the resulting patch P n,k has the shortest possible boundary ? In [1], it is proved that for k = 0 a minimal boundary length is obtained if the n hexagons are arranged in a spiral fashion. This result can be generalized as follows: For any 0 ≤k≤6 the patch P n,k has the shortest possible boundary if the faces are arranged in a spiral way starting with the k pentagons. In this talk we will sketch the proof of this result. The details can be found in [2].