2012/09/29 by Yohji Akama, Akama, Yohji
Computer Science · Engineering · Mathematics · #Advanced Graph Theory Research #Cellular Automata and Applications #graph theory and CDMA systems #math.CO #math.MG #msc:05B45 #msc:05C10 #msc:51M20 #msc:52C20
paper · pdf · doi:10.48550/arxiv.1210.0152
17 pages, 6 figures
arxiv created 2012/10/05 · arxiv updated 2012/10/08
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.