Construction of (a, b, c) tilings of the Euclidean plane, hyperbolic plane, and the sphere
DOI:
https://doi.org/10.55016/ojs/cdm.v20i2.75933Abstract
An $(a,b,c)$ tiling forms under its symmetry group $a$ orbits of vertices; $b$ orbits of edges; and $c$ orbits of tiles. This paper discusses a method to arrive at an $(a,b,c)$ tiling of the Euclidean plane ($\mathbb{E}^2$), hyperbolic plane ($\mathbb{H}^2$) or 2-dimensional sphere ($\mathbb{S}^2$). An application of the method facilitates the complete enumeration of the $(a,2,c)$ tiling of $\mathbb{E}^2$ and $\mathbb{S}^2$ as well as a listing of $(a,3,c)$ tilings of $\mathbb{E}^2$.
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