Construction operations to create new aperiodic tilings: local isomorphism classes and simplified matching rules

This thesis studies several constructions to produce aperiodic tilings with particular properties. The first chapter of this thesis gives a constructive method, exchanging edge to edge matching rules for a small atlas of permitted patches, that can decrease the number of prototiles needed to tile a space. We present a single prototile that can only tile R3 aperiodically, and a pair of square prototiles that can only tile R2 aperiodically. The thesis then details a construction that superimposes two unit square tilings to create new aperiodic tilings. We show with this method that tiling spaces can be constructed with any desired number of local isomorphism classes, up to (and including) an infinite value. Hyperbolic variants are also detailed. The final chapters of the thesis apply the concept of Toeplitz arrays to this construction, allowing it to be iterated. This gives a general method to produce new aperiodic tilings, from a set of unit square tilings. Infinite iterations of the construction are then studied. We show that infinite superimpositions of periodic tilings are describable as substitution tilings, and also that most Robinson tilings can be constructed by infinite superimpositions of given periodic tilings. Possible applications of the thesis are then briefly considered.