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Divisible tilings in the hyperbolic plane. (English) Zbl 0962.05017

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.

MSC:

05B45 Combinatorial aspects of tessellation and tiling problems
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20H15 Other geometric groups, including crystallographic groups
51F15 Reflection groups, reflection geometries
51M10 Hyperbolic and elliptic geometries (general) and generalizations
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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