Abstract

Pentagonal subdivision turns any tiling on oriented surface into a pentagonal tiling. Geometrically, we know the edge-to-edge tilings of the sphere by congruent pentagons with edge length combination a 2 b 2 c are exactly the pentagonal subdivisions of the platonic solids. This talk is about the combinatorial aspects of the pentagonal subdivision, which is motivated by the question of evenly distributed high degree (i.e., degree > 3) vertices in a pentagonal tiling of the sphere. Underlying the pentagonal subdivision is certain pentagonal subdivision of quadrilateral tilings, which we call simple pentagonal subdivision. We develop a theory of when a quadrilateral tiling admits simple pentagonal subdivision, first in terms of orientations at vertices, and second in terms of the degeneracy of quadrilateral tiles. The theory is carried out over all the surfaces. Then we apply the theory to answer the problem of evenly distributed high degree vertices, and also answer the other related problems. There are two further research directions. The first is the characterisation of the other subdivisions, such as barycentric subdivision. The second is the further technical studies of the simple pentagonal subdivision. I may discuss the work of SUSTech student Ceng Si-Yu in the second direction.