## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 45, Number 3 (2015), 309-339.

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

#### 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 $pi/4$ 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 $I$ .

#### Article information

**Source**

Hiroshima Math. J., Volume 45, Number 3 (2015), 309-339.

**Dates**

First available in Project Euclid: 24 November 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1448323768

**Digital Object Identifier**

doi:10.32917/hmj/1448323768

**Mathematical Reviews number (MathSciNet)**

MR3429167

**Zentralblatt MATH identifier**

1331.52028

**Subjects**

Primary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]

Secondary: 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

**Keywords**

Anisohedral tile continuous deformation Archimedean dual isohedral tiling spherical monohedral deltoidal tiling spherical monohedral rhombus-faced tiling

#### Citation

Sakano, Yudai; Akama, Yohji. Anisohedral spherical triangles and classification of spherical tilings by congruent kites, darts and rhombi. Hiroshima Math. J. 45 (2015), no. 3, 309--339. doi:10.32917/hmj/1448323768. https://projecteuclid.org/euclid.hmj/1448323768