Hiroshima Mathematical Journal

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

Yudai Sakano and Yohji Akama

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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

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


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