Semi-regular Tilings of the Hyperbolic Plane
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A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
KeywordsSemi-regular tilings Hyperbolic tilings Archimedean tilings Semi-equivelar maps Vertex-transitive tilings
Mathematics Subject ClassificationPrimary 52C20 Secondary 52A45 51M20
The first author is supported by SERB, DST (Grant No. MTR/2017/000410). The second author acknowledges the SERB, DST (Grant No. MT/2017/000706) and the Infosys Foundation for their support. The authors are also supported by the UGC Centre for Advanced Studies. The authors thank the referee for several suggestions that significantly improved this article. Several figures in this article were made using the L2Primitives and Tess packages for Mathematica, available online at the Wolfram Library Archive.
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