Tilings Generated by Non-Parallel Projection Schemes

This thesis defines and investigates rational and irrational 2:1 X-projection schemes and non-parallel projection schemes with strips at rational gradients. Both irrational 2:1 X-projection schemes and non-parallel projection schemes with strips at rational gradients are shown to produce tilings with infinitely many prototiles, with the tilings produced by the second of these schemes nonetheless shown to display a property similar to repetitivity. Rational 2:1 X-projection schemes are shown to produce tilings with a finite number of prototiles, with a subset of these filings shown to be repetitive. The points in the fundamental domain of our lattice L that correspond to translates of these filings are also investigated, with these points shown to be either dense in a finite number of lines or dense in the fundamental domain. This also leads to a proof of repetitivity in all rational 2:1 X-projection tangs and aperiodicity in a subset of these tilings. The tiling spaces of such filings are also investigated. In addition, the proportions in which the prototiles in a rational 2:1 X-projection tiling appear are also looked at, and a possible explanation of the values observed is provided.