Joint work with Yang Liu. Motivated by applications in modeling glass structures in architecture, we consider the geometry and computation of mesh surfaces with planar hexagonal faces from the point of view of discrete differential geometry. It is shown that the mesh structure is naturally related to conjugate curve networks on its underlying smooth surface; furthermore, the shape of each hexagonal face is in the limit related to the Dupin indicatrix. These results are used in combination with an optimization method to compute a mesh surface with planar hexagonal faces to approximate a given smooth surface.