## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 43, Number 3 (2013), 285-304.

### Classification of spherical tilings by congruent quadrangles over pseudo-double wheels (I) -- a special tiling by congruent concave quadrangles

#### Abstract

Every simple quadrangulation of the sphere is
generated by a graph called a pseudo-double wheel with two local
expansions (Brinkmann et al. "Generation of simple quadrangulations of
the sphere.'' Discrete Math., Vol. 305, No. 1-3, pp. 33--54, 2005). So,
toward a classification of the spherical tilings by congruent quadrangles,
we propose to classify those with the tiles being convex and the graphs
being pseudo-double wheels. In this paper, we verify that a certain
series of assignments of edge-lengths to pseudo-double wheels does not
admit a tiling by congruent convex quadrangles. Actually, we prove the
series admits only one tiling by twelve congruent *concave*
quadrangles such that the symmetry of the tiling has only three
perpendicular 2-fold rotation axes, and the tiling seems to be new.

#### Article information

**Source**

Hiroshima Math. J., Volume 43, Number 3 (2013), 285-304.

**Dates**

First available in Project Euclid: 7 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1389102577

**Digital Object Identifier**

doi:10.32917/hmj/1389102577

**Mathematical Reviews number (MathSciNet)**

MR3161319

**Zentralblatt MATH identifier**

1295.52024

**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] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

**Keywords**

Monohedral tiling spherical quadrangle pseudo-double wheel

#### Citation

Akama, Yohji. Classification of spherical tilings by congruent quadrangles over pseudo-double wheels (I) -- a special tiling by congruent concave quadrangles. Hiroshima Math. J. 43 (2013), no. 3, 285--304. doi:10.32917/hmj/1389102577. https://projecteuclid.org/euclid.hmj/1389102577