Tilings and Amalgamations

Wingate, William Jonathan G (1989). Tilings and Amalgamations. MPhil thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000fc18


This thesis investigates tilings of the Euclidean Plane and their amalgamations. For any tiling an amalgamation is a tiling produced by joining together tiles of the original tiling. This definition can be extended to cover any suitable adjacency structure, such as a graph.

The first chapter of the thesis reviews some of the basic concepts and results in the theory of tilings. Chapter 2 introduces amalgamations, both of tilings and of infinite graphs. Chapter 3 discusses tesseral arithmetic^ and shows how the theory of amalgamations can be used to produce addressing systems of the plane.

The second part of the thesis concentrates on classifying and enumerating amalgamators. In chapter 4, we list the possible types of amalgamation of each of the eleven Laves nets. In chapter 5 an algorithm to enumerate a particular class of amalgamations is developed, and the results of running this on a computer are presented. Chapter 6 contains some theoretical results about tiling hierarchies^ sequences of tilings produced by successive amalgamations.

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