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Full-Text Articles in Mathematics
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines …
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Mathematical Sciences Technical Reports (MSTR)
We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.
Tilings Which Split A Mirror, Jim Belk
Tilings Which Split A Mirror, Jim Belk
Mathematical Sciences Technical Reports (MSTR)
We consider the mirror of a reflection which consists of its subset of fixed points. We investigate a number of conditions on the tiling that guarantee that the surface splits at a mirror.
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
We have all seen many of the beautiful patterns obtained by tiling the hyperbolic plane H by repeated reflection in the sides of a "kaleidoscopic" polygon. Though there are such patterns on the sphere and the euclidean plane, these positively curved and fiat geometries lack the richness we see in the hyperbolic plane. Many of these patterns have been popularized by the beautiful art of M.C. Escher. For a list of references and a more complete discussion on the construction of artistic tilings see [6].