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Full-Text Articles in Physical Sciences and Mathematics
Hyperbolic Billiard Paths, Rebecca Lehman, Chad White
Hyperbolic Billiard Paths, Rebecca Lehman, Chad White
Mathematical Sciences Technical Reports (MSTR)
A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by wordlength and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard spectrum for the (2, 3, 7)-tiling.
Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil
Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil
Mathematical Sciences Technical Reports (MSTR)
In contribution to the classification of all tilings of low-genus surfaces, the kaleidoscopic and non-kaleidoscopic tilings by quadrilaterals are given up to genus 12. As part of their classification, the algebraic structure of the conformal tiling groups and the geometric structure of the tiles are specified. In addition, several infinite classes of tilings and tiling groups are presented.