Uniform tilings of the hyperbolic plane



Uniform tilings of the hyperbolic plane
by Basudeb Datta and Subhojoy Gupta

Abstract. A uniform 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 uniform tiling with a given vertex type, and pose some open questions.

https://arxiv.org/abs/1806.11393

h/t Henry Segerman for sharing with me.

Comments

  1. wendy kriegerJuly 4, 2018 at 8:22 PM

    John Conway, Marek Ctrnack and I worked out most of the rules for this. I even tried for a simple motation for it all, such as 3% 3/ 9/ 2/ , but it's froghteningly complex.

    ReplyDelete
  2. Roice NelsonJuly 4, 2018 at 9:35 PM

    wendy krieger, are the rules you worked out available somewhere for study?

    ReplyDelete
  3. wendy kriegerJuly 4, 2018 at 11:58 PM

    I wrote something up about them on some private list, but see this figure:

    plus.google.com - Decorating the Orbifold The orbifold is a description of a 2D symmetry devis...

    ReplyDelete
  4. wendy kriegerJuly 5, 2018 at 1:02 AM

    Have a look here and pages to the 'right' (arrow).
    superliminal.com - 883333-1.gif

    ReplyDelete
  5. Melinda GreenJuly 12, 2018 at 2:53 AM

    Nice to see a Tyler link and know it's not completely forgotten.

    ReplyDelete
  6. wendy kriegerJuly 12, 2018 at 5:12 AM

    We have not forgotten it at all. Brilliant peice, if i should say so.

    ReplyDelete

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