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

TL;DR

This paper classifies spherical tilings by congruent quadrangles over pseudo-double wheel graphs, discovering a unique tiling with twelve congruent concave quadrangles exhibiting specific symmetry, advancing understanding of spherical quadrangulations.

Contribution

It introduces a classification of spherical tilings by congruent quadrangles over pseudo-double wheels, identifying a novel tiling with concave quadrangles and specific symmetry properties.

Findings

A series of edge-length assignments does not produce convex tilings.

A unique tiling with twelve congruent concave quadrangles is found.

The tiling has three perpendicular 2-fold rotation axes.

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 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 new.