On Heesch's problem and other tiling problems

In this dissertation we present some new results on several open tiling problems, including the long-standing Heesch's Problem. We give a survey of Heesch's Problem and discuss its connections with other famous and long-standing open tiling problems, including the Tiling Problem and the Einstein Problem.; In a number of settings, including $E2$, $H2$, and $E3$, we examine monotiles that are “combinatorially imbalanced.” To study these imbalances we introduce a combinatorial analysis which, in certain circumstances, may be applied to give an upper bound on the Heesch number of the monotile in question. Using this analysis, we provide evidence in several settings, including $E2$, that the combinatorial imbalances do not provide a uniform upper bound on Heesch numbers, thus allowing the possibility of arbitrarily high Heesch numbers.; Generalizing Ammann's Heesch number 3 example, we display an infinite family of monotiles with Heesch number 3, a finite family of monotiles with Heesch number 4, and an infinite family of monotiles with Heesch number 5. These are some of the more interesting results of this thesis since previously the highest known Heesch number was 3 and corresponded to the single example by Ammann.; We also prove several sharp results on Heesch's Problem and the Tiling Problem in the hyperbolic plane. We show that the combinatorial imbalance techniques which proved successful in giving high Heesch numbers in the Euclidean plane do not give high Heesch numbers in the hyperbolic plane. We also solve the Tiling Problem for a class of hyperbolic polygons whose edges are all marked with either a bump or a nick. Further, we show that Heesch's Problem does not have interesting solutions for this class of tiles.; Finally, we give a surprising counterexample to a conjecture by Friedman by exhibiting a monotile based on Voderberg's tile which has surround number 2. We provide figures describing our result. Moreover, we exhibit monotiles with the property that 2 copies can surround N copies for any positive integer N. We also provide some three-dimensional results concerning surround numbers.