Patches with short boundaries.

In this article we prove bounds for the boundary length of patches with a given set of bounded faces. We assume that with t the number of given triangles, q the number of quadrangles, and p the number of pentagons, the curvature 3t+2q+p is at most 6 and that at an interior vertex exactly 3 faces meet. We prove that one gets a patch with shortest boundary if one arranges the faces in a spiral order and with increasing size. Furthermore we give explicit formulas that allow to determine all boundary lengths that occur for patches with given numbers p,q and t<2 and no bounded face larger than 6.