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A pattern is symmetric if there is at least one symmetry rotation translation reflection glide reflection that leaves the pattern unchanged Rotation To rotate an object means to turn it around Every rotation has a center and an angle Translation To translate an object means to move it without rotating or reflecting it Every translation has a direction and a distance Reflection To reflect an object means to produce its mirror image Every reflection has a mirror line A reflection of an R is a backwards R Glide Reflection A glide reflection combines a reflection with a translation along the direction of the mirror line Glide reflections are the only type of symmetry that involve more than one step Symmetries create patterns that help us organize our world conceptually Symmetric patterns occur in nature and are invented by artists craftspeople musicians choreographers and mathematicians In mathematics the idea of symmetry gives us a precise way to think about this subject We will talk about plane symmetries those that take place on a flat plane but the ideas generalize to spatial symmetries too Plane symmetry involves moving all points around the plane so that their positions relative to each other remain the same although their absolute positions may change Symmetries preserve distances angles sizes and shapes 1For example rotation by 90 degrees about a fixed point is an example of a plane symmetry 2Another basic type of symmetry is a reflection The reflection of a figure in the plane about a line moves its reflected image to where it would appear if you viewed it using a mirror placed on the line Another way to make a reflection is to fold a piece of paper and trace the figure onto the other side of the fold 3A third
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