InfoSymmetryFinite ShapesPatternsReflectionsRotationsTranslationsGlidesClassifyingInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingThe Mathematics of SymmetryBeth Kirby and Carl LeeUniversity of KentuckyMA 111Fall 2009Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingInfoSymmetryFinite ShapesPatternsReflectionsRotationsTranslationsGlidesClassifyingSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingCourse InformationText: Peter Tannenbaum, Excursions in Modern Mathematics,second custom edition for the University of Kentucky, Pearson.Course Website:http://www.ms.uky.edu/∼lee/ma111fa09/ma111fa09.htmlSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying11.0 Introduction to SymmetrySymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingAre These Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Assume this extends forever to the left and rightSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Assume this extends forever to the left and rightSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Assume this extends forever in all directionsSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Assume this extends forever in all directionsSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Assume this extends forever in all directionsSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Assume this extends forever in all directionsSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Assume this extends forever in all directionsSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs the Image of the Sun Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingIs This Symmetrical?Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying11.6 Symmetry of Finite ShapesSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesLet’s look at the symmetries of some finite shapes — shapesthat do not extend forever in any direction, but are confined toa bounded region of the plane.Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesThis shape has 1 line or axis of reflectional symmetry. It hassymmetry of type D1.Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesThis shape has 3 axes of reflectional symmetry.Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesIt has a 120 degree angle of rotational symmetry. (Rotatecounterclockwise for positive angles.)Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesBy performing this rotation again we have a 240 degree angleof rotational symmetry.Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesIf we perform the basic 120 degree rotation 3 times, we bringthe shape back to its starting position. We say that this shapehas 3-fold rotational symmetry. With 3 reflections and 3-foldrotational symmetry, this shape has symmetry type D3.Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesSymmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesThis shape has 2 axes of reflectional symmetry.Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesIt has a 180 degree angle of rotational symmetry.Symmetry UKInfo Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides ClassifyingSymmetries of Finite ShapesIf we perform the basic 180 degree rotation 2 times, we bringthe shape back to its starting position. We say that this shapehas 2-fold rotational symmetry. With 2 reflections and 2-foldrotational symmetry, this shape has symmetry type D2.Symmetry UKInfo Symmetry