Planar Hexagonal Meshing for Architecture

Meshes with planar faces, i.e. polyhedral meshes, have attracted a great deal of attention recently due to the increasing demand in architecture for modeling freeform surfaces with planar panels [1], [2], [3], [4]. Tiling a surface with planar faces is an extension of the classical plane tiling problem, which has been well studied and reviewed [5]. To tile a plane with only congruent copies of a regular polygon, there are only three possible shape choices: equilateral triangle, square, or regular hexagon (or their affine copies). However, the problem of tiling a surface with the same type of polygons is more challenging. In general, the mesh faces can no longer have identical shapes as they are constrained by the surface geometry. For example, as we shall demonstrate, a negatively curved surface cannot be tiled with a mesh of planar convex polygons that have only valence-3 vertices.