Title

Edge-to-Edge Tilings with Vertex, Edge and Tile Transivity Properties

Author

Mark Tomenes

Date of Award

2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Ma. Louise Antonette N. De Las Peñas, PhD

Abstract

Edge-to-edge tilings with vertex, edge and tile transitivity properties An (a, b, c) tiling T is an a-isogonal, b-isotoxal and c-isohedral tiling, that is, T forms, respectively, a orbits of vertices, b orbits of edges and c orbits of tiles under its symmetry group. This study gives a method of constructing an (a, b, c) tiling of the Euclidean (E 2 ), hyperbolic (H2 ) or spherical (S 2 ) plane. The method allows for a complete enumeration of the (a, 2, c) tilings of E 2 and S 2 as well as a listing of (a, 3, c) tilings of E 2 derived from tilings by regular polygons, which will be presented. Moreover, classes of (1 , b, c) tilings of E 2 , H2 and S 2 will be shown, particularly those derived from [p n ] tilings.

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