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The role of the corona in determining global properties of homogeneous tilings in the Euclidean and hyperbolic planes
Abstract
The corona of a tile T in a tiling ${\cal F}$ is the set of all tiles in ${\cal F}$ which meet T. The corona of any other configuration: an edge, vertex, or simply connected patch of tiles in ${\cal F}$ is defined analogously. The first part of this thesis establishes that for a homogeneous, edge-to-edge tiling ${\cal F}$ of E$\sp2$ or H$\sp2$ by tiles topologically equivalent to triangles or quadrilaterals, congruence of the coronas of all tiles in ${\cal F}$ is sufficient for ${\cal F}$ to be isohedral. (A tiling is isohedral if its group of symmetries acts transitively on its tiles.) A tiling of E$\sp2$ is called balanced if the relative frequencies of the numbers of vertices and edges to tiles can be computed. The major portion of the thesis extends this definition to H$\sp2$ and establishes a technique using recurrence relations for computing these relative frequencies for homogeneous tilings by topological triangles or quadrilaterals. These frequencies are irrational numbers which depend on the rate at which the tiling grows by the accretion of successive coronas to an initial configuration. Every homogeneous tiling of H$\sp2$ by topological triangles is balanced, with the exception of those tilings of homogeneous type (4.2m.2p), m $\not=$ p. In E$\sp2$, every isohedral tiling is balanced. Examples are given of both balanced and non-balanced isohedral tilings of homogeneous type (4$\sp3$.6) in H$\sp2$. The results are independent of the initial configuration. It is conjectured that the growth rate and so the balance of an isohedral tiling of H$\sp2$ depends on the configuration of the coronas.
Subject Area
Mathematics
Recommended Citation
Moran, Judith Flagg, "The role of the corona in determining global properties of homogeneous tilings in the Euclidean and hyperbolic planes" (1990). Doctoral Dissertations Available from Proquest. AAI9101647.
https://scholarworks.umass.edu/dissertations/AAI9101647