Quotient maps of 2,3-uniform tilings of the plane on the torus

Dipendu Maity; Debashis Bhowmik; Marbarisha M. Kharkongor

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arXiv · 2101.04373

Quotient maps of 2,3-uniform tilings of the plane on the torus

Dipendu Maity·Debashis Bhowmik·Marbarisha M. Kharkongor

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Abstract

A 2-uniform tiling is an edge-to-edge tiling by regular polygons having $2$ distinct transitivity classes of vertices. There are 20 distinct 2-uniform tilings (these are of $14$ different types) on the plane, and since the plane is the universal cover of the torus, it is natural to explore maps on the torus that correspond to the 2-uniform tilings. In this article, we discuss that if a map is the quotient of a plane's $2$-uniform lattice then what would be the bounds of the number of vertex orbits. A 3-uniform tiling is an edge-to-edge tiling by regular polygons having $3$ distinct transitivity classes of vertices. There are $61$ distinct $3$-uniform tilings on the plane. In this article, we discuss that if a map is the quotient of a plane's $3$-uniform lattice then what would be the bounds of the number of vertex orbits.

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Dipendu Maity, Debashis Bhowmik, Marbarisha M. Kharkongor. 2021-01-12. Quotient maps of 2,3-uniform tilings of the plane on the torus. https://arxiv.org/abs/2101.04373

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