Divisible tilings in the hyperbolic plane (Q1583605)

An edge-to-edge tiling \({\mathcal T}\) of the hyperbolic plane by polygons is called kaleidoscopic if it is invariant under the reflexions in the lines that contain the edges of \({\mathcal T}\). Such a tiling is generated from a single tile \(P\), the mastertile, by repeated reflections in its sides. In a kaleidoscopic tiling \({\mathcal T}\), if the reflection lines are unions of edges of \({\mathcal T}\), then \({\mathcal T}\) is called geodesic; in this case each interior angle of a tile must be of the form \(\pi/k\) for some integer \(k\). A geodesic kaleidoscopic tiling \({\mathcal T}_1\) subdivides another geodesic kaleidoscopic tiling \({\mathcal T}_2\) if each tile of \({\mathcal T}_2\) is a union of finitely many tiles of \({\mathcal T}_1\). If \({\mathcal T}_1\) subdivides \({\mathcal T}_2\), then the pair \((P_1,P_2)\) which consists of a mastertile \(P_1\) of \({\mathcal T}_1\) and \(P_2\) of \({\mathcal T}_2\) with \(P_1\subset P_2\) is called a divisible tiling pair. The authors enumerate all divisible tiling pairs \((P_1, P_2)\) of the hyperbolic plane in which \(P_2\) is a quadrangle and \(P_1\) a triangle (in this case the triangle tilings for \(P_1\) are kaleidoscopically subdividing the quadrilateral tilings for \(P_2\)). There are a finite number of 1-, 2-, and 3-parameter families as well as a finite number of exceptional cases. The authors also enumerate the less complex cases of triangle tilings subdividing triangle tilings, and quadrilateral tilings subdividing quadrilateral tilings.