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

TL;DR

This paper investigates the properties of quotient maps derived from 2- and 3-uniform tilings of the plane when projected onto the torus, focusing on bounds of vertex orbits in these quotient maps.

Contribution

It establishes bounds on the number of vertex orbits for quotient maps of 2- and 3-uniform plane tilings on the torus, extending understanding of their symmetry properties.

Findings

Bounds on vertex orbits for 2-uniform tilings on the torus.

Bounds on vertex orbits for 3-uniform tilings on the torus.

Analysis of the relationship between plane tilings and their torus quotients.

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.