A class of maps on the torus and their vertex orbits
Marbarisha M. Kharkongor, Dipendu Maity

TL;DR
This paper investigates the symmetry properties of tilings on the plane, focusing on maps derived from vertex-homogeneous lattices on the torus and establishing bounds on vertex orbits.
Contribution
It provides new bounds on the number of vertex orbits for maps that are quotients of plane's $k$-vertex-homogeneous lattices with $k \, \ge \, 4$.
Findings
Derived sharp bounds for vertex orbits in such maps.
Analyzed symmetry classes in plane tilings.
Extended understanding of vertex-homogeneity in toroidal maps.
Abstract
A tiling (edge-to-edge) of the plane is a family of tiles that cover the plane without gaps or overlaps. Vertex figure of a vertex in a tiling to be the union of all edges incident to that vertex. A tiling is -vertex-homogeneous if any two vertices with congruent vertex figures are symmetric with each other and the vertices form precisely transitivity classes with respect to the group of all symmetries of the tiling. In this article, we discuss that if a map is the quotient of a plane's -vertex-homogeneous lattice () then what would be the sharp bounds of the number of vertex orbits.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsQuasicrystal Structures and Properties · graph theory and CDMA systems · Cellular Automata and Applications
