Anisohedral spherical triangles and classification of spherical tilings by congruent kites, darts and rhombi

Yudai Sakano, Yohji Akama

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

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 Π/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.

Original languageEnglish
Pages (from-to)309-339
Number of pages31
JournalHiroshima Mathematical Journal
Volume45
Issue number3
DOIs
Publication statusPublished - 2015 Nov

Keywords

  • Anisohedral tile
  • Archimedean dual
  • Continuous deformation
  • Glide
  • Isohedral tiling
  • Spherical monohedral deltoidal tiling
  • Spherical monohedral rhombus-faced tiling

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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