Semi-regular Tilings of the Hyperbolic Plane

  • Basudeb Datta
  • Subhojoy GuptaEmail author


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.


Semi-regular tilings Hyperbolic tilings Archimedean tilings Semi-equivelar maps Vertex-transitive tilings 

Mathematics Subject Classification

Primary 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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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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