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.