2015/11/01 by Yudai Sakano, Yohji Akama · 1 citation
Engineering · Mathematics · #Structural Analysis and Optimization #Advanced Materials and Mechanics #Rhombus #Mathematics #Substitution tiling #Combinatorics #Hexagonal tiling #Projection (relational algebra) #Penrose tiling #Series (stratigraphy) #Spherical trigonometry #Tessellation (computer graphics) #Geometry #Quasicrystal #Grid #Algorithm
paper · doi:10.32917/hmj/1448323768
openalex publication_date 2015/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
We classify all spherical monohedral (kite/dart/rhombus)-faced tilings, as follows: The set of spherical monohedral rhombus-faced tilings consists of (1) the central projection of the rhombic dodecahedron, (2) the central projection of the rhombic triacontahedron, (3) a series of non-isohedral tilings, and (4) a series of tilings which are topologically trapezohedra (here a trapezohedron is the dual of an antiprism.). The set of spherical tilings by congruent kites consists of (1) the central projection T of the tetragonal icosikaitetrahedron, (2) the central projection of the tetragonal hexacontahedron, (3) a non-isohedral tiling obtained from T by gliding a hemisphere of T with pi/4 radian, and (4) a continuously deformable series of tilings which are topologically trapezohedra. The set of spherical tilings by congruent darts is a continuously deformable series of tilings which are topologically trapezohedra. In the above explanation, unless otherwise stated, the tilings we have enumerated are isohedral and admit no continuous deformation. We prove that if a spherical (kite/dart/rhombus) admits an edge-to-edge spherical monohedral tiling, then it also does a spherical isohedral tiling. We also prove that the set of anisohedral, spherical triangles (i.e., spherical triangles admitting spherical monohedral triangular tilings but not any spherical isohedral triangular tilings) consists of a certain, infinite series of isosceles triangles I , and an infinite series of right scalene triangles which are the bisections of I .