## Course Description

Repeating symmetry patterns and tilings are present all around us, from the brickwork on campus, to designs on tapestries and wallpaper, to paintings like those of M.C. Escher, to crystals including snowflakes and quartz. In this class, students will explore symmetry patterns, learn to identify and classify two-dimensional patterns, and use software to create their own tiling designs. Students will relate tiling patterns to their folded up counterparts, called orbifolds, and use mathematical ideas of curvature and cone points to determine which patterns are possible and which patterns can never be achieved. In addition to analyzing repeating patterns of tiles, students will examine non-periodic patterns, such as Penrose’s kite and dart tilings, and use mathematical ideas of self-similarity and limits to understand why these patterns can never exactly repeat. Course assignments will include readings, mathematical investigations, design projects such as virtual and physical kaleidoscopes, quizzes, and a final project. The final project will allow students to delve more deeply into a theoretical topic (e.g. hyperbolic tilings or crystallographic groups) or an application (e.g. quasicrystals or basket weave patterns). There are no prerequisites.

## Course Notes

Table of Contents

Part1_TypesOfSymmetry

Part2_IsometriesOfThePlane

Part3_ReflectionsOfReflections

Part4_ClassificationOfIsometries

Part5_CombiningIsometries

Part6_OrderInversesAndCommutativity

Part7_Kaleidescopes

Part8_Rosettes

Part9_FiniteFigures

Part10_FriezePatterns

Part11_ClassifyingFriezePatterns

Part12_PaperCuttingAndMusicalFriezePatterns

Part13_WallpaperPatterns

Part14_OrbifoldNotationAndTheMagicTheorem

Part15_DrawingWallpaperPatterns

Part16_Topology

Part17_ClassificationOfSurfaces

Part18_EulerCharacteristic

Part19_EulerCharcteristicAndTopology

Part20_Curvature

Part21_GaussBonnetTheorem

Part22_Orbifolds

Part23_OrbifoldsAndTheMagicTheorem

Part24_SphericalSymmetryPatterns

Part25_FriezePatternNotation

Part26_HyperbolicGeometry

Part27_HyperbolicSymmetryPatterns

Part28_Tilings

Part29_PentagonalTiles

Part30_PenroseTiles

Final Project Ideas

Wall tiling at the Alhambra, in Granada, Spain, illustrating wallpaper group 333.

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