Tiling Euclidean Polygons Mapped From Their Hyperbolic Equivalent

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Tiling Euclidean Polygons Mapped From Their Hyperbolic Equivalent

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2017-06

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Abstract

The concept of repeating artistic patterns, for instance spirals, waves, snail shells, tilings etc., have been in existence for centuries now. It was during 1900's that a noted Dutch graphic artist M.C. Escher worked extensively in this world of art which was inspired by mathematics. Escher painstakingly hand-drew such perceptive repeating patterns (which were mostly Euclidean in nature) and his famous hyperbolic patterns: Circle Limit I, II, III and IV which were based on regular tessellations. This research work concentrates on leveraging hyperbolic and Euclidean geometry in art, drawing inspiration from Escher's work. Various Euclidean, non-Euclidean and spherical repeating patterns are special forms of tessellations. At the core of these patterns lies an idea, proposed by Dr. Dunham, that a small congruent sub-pattern, called a motif, which when reflected and rotated will generate the entire pattern. This work focuses on transforming the central polygon in a hyperbolic pattern to obtain its Euclidean counterpart. This counterpart will further tile a planar region to generate a Euclidean tiling. There are various interesting applications that allow the user to draw such repeating patterns programmatically, however none of them show the reverse mapping from a hyperbolic to a Euclidean pattern. We enhance an existing Java application by creating a bridge that connects hyperbolic patterns to their Euclidean equivalents and facilitates the user to work with tilings. The results are expected to show a transformation from hyperbolic to Euclidean patterns followed by tiling of the Euclidean pattern on a planar region.

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University of Minnesota M.S. thesis.June 2017. Major: Computer Science. Advisor: Douglas Dunham. 1 computer file (PDF); vi, 76 pages.

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Prakash, Manoj Naik. (2017). Tiling Euclidean Polygons Mapped From Their Hyperbolic Equivalent. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/190582.

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