Tiling in Hyperbolic Geometry and the Area of
Tiling in Hyperbolic Geometry and the Area of Triangles Breana Figueroa Department of Mathematics and Computer Science, Longwood University, Farmville, VA 23909 Introduction The goal of this project was to examine the properties of hyperbolic geometry using visual representations in the form of computer models and paper model approximations. Using information related to tiling the hyperbolic plane, it is then possible to construct polygonal approximations of the hyperbolic plane to help better visual and understand some of its more startling features, including why triangles are “skinny” and the need to have more than 360 degrees of angles around each vertex in the construction. Tiling with Equilateral Triangles and Other Models Ideal Triangles Euclidean Plane Definitions • Points: All points may be defined as such that . • Lines: Lines may be defined as: Hyperbolic Plane Since we are limited by the application software, we are not able to perfectly construct an ideal triangle— a triangle with three ideal vertexes (or vertexes with angles of zero degrees). However, this representation above is similar to how an ideal triangle would appear using the Poincaré Disk Model. The area of a triangle in hyperbolic geometry is proportional to the defect, and as such, an ideal triangle has the largest possible area in hyperbolic geometry. Future Work • Angle Measure: The Euclidean angle measure on the angle between tangent lines to any two rays. • Distance: Given two points A and B in a hyperbolic plane, we define distances as: • Poincaré Disk Model: A two-dimensional representation of the hyperbolic plane. • Defect: The difference of the interior angle sum of a hyperbolic polygon from the Euclidean interior angle sum of the same regular convex polygon. Comparisons In Euclidean geometry, the most equilateral triangles that can fit around a single vertex is six. In hyperbolic geometry, we are able to fit more since triangles are not restricted to have an interior angle sum of 180 degrees. Using the formulas listed below for finding the interior angle and side length of an equilateral hyperbolic triangle, we are able to create the hyperbolic tilings using an online application as depicted above. I will continue research in this area in the spring and summer of 2018 by creating educational labs intended to guide students through computer based investigations of hyperbolic geometry. The labs will be designed to have students investigate theorems of Euclidean geometry in the setting of hyperbolic geometry. Acknowledgements Cormier Honors College Dr. Thomas Wears – Faculty Mentor
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