Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying The Mathematics of Symmetry Beth Kirby and Carl Lee University of Kentucky MA 111 Fall 2009 Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Course Information Text 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 html Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11 0 Introduction to Symmetry Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Are These Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Assume this extends forever to the left and right Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Assume this extends forever to the left and right Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Assume this extends forever in all directions Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Assume this extends forever in all directions Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Assume this extends forever in all directions Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Assume this extends forever in all directions Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Assume this extends forever in all directions Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is the Image of the Sun Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Is This Symmetrical Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying 11 6 Symmetry of Finite Shapes Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Let s look at the symmetries of some finite shapes shapes that do not extend forever in any direction but are confined to a bounded region of the plane Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes This shape has 1 line or axis of reflectional symmetry It has symmetry of type D1 Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes This shape has 3 axes of reflectional symmetry Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes It has a 120 degree angle of rotational symmetry Rotate counterclockwise for positive angles Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes By performing this rotation again we have a 240 degree angle of rotational symmetry Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes If we perform the basic 120 degree rotation 3 times we bring the shape back to its starting position We say that this shape has 3 fold rotational symmetry With 3 reflections and 3 fold rotational symmetry this shape has symmetry type D3 Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes This shape has 2 axes of reflectional symmetry Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes It has a 180 degree angle of rotational symmetry Symmetry UK Info Symmetry Finite Shapes Patterns Reflections Rotations Translations Glides Classifying Symmetries of Finite Shapes If we perform the basic 180 degree rotation 2 times we bring the shape back to its starting position We say that this shape has 2 fold rotational symmetry With 2 reflections and 2 fold rotational symmetry this shape has symmetry type D2 Symmetry UK
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