### 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.

Original language | English |
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Pages (from-to) | 285-304 |

Number of pages | 20 |

Journal | Hiroshima Mathematical Journal |

Volume | 43 |

Issue number | 3 |

Publication status | Published - 2013 Nov 1 |

### Keywords

- Monohedral tiling
- Pseudo-double wheel
- Spherical quadrangle

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology