Filling spherical surfaces by mixed triangle and square tiles.

Abstract

We present four classes of highly symmetric defect patterns on a spherical surface tiled with triangular and square tiles. These patterns accommodate both stretched triangular and square lattices and are analyzed in terms of their symmetries. Both spherical and corresponding polyhedron views are considered, with emphasis on three-dimensional point-group symmetries. In addition to the original patterns, alternative defect configurations are explored, including those generated by the kaleidoscopic operation, originally suggested by Caspar and Klug for triangular tiling, as well as the cut-and-rotate operation applied through a great circle on the sphere. While these alternatives can lower the space-group symmetries, they provide a broader understanding of the system's possible configurations. For a fixed square surface area fraction, we also examine a scenario that identifies the likely ground state among the four primary classes and their alternatives.