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.
"Classification of spherical tilings by congruent quadrangles over pseudo-double wheels (I) -- a special tiling by congruent concave quadrangles." Hiroshima Math. J. 43 (3) 285 - 304, November 2013. https://doi.org/10.32917/hmj/1389102577