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Mathematical Symmetry

Mathematical symmetry plays an important and pervasive role in nature and in human design. There are four types of mathematical symmetry the plane:
- Translational symmetry - An object, such as a tessellation, possesses translational symmetry if it can be translated (moved) by some distance and remain unchanged. A tessellation or pattern with translational symmetry is repeating, like a wallpaper or fabric pattern. There are seventeen distinct symmetry groups in which all repeating patterns in the plane fall, sometimes referred to as the Wallpaper Groups.
- Rotational symmetry - An object possesses rotational symmetry if it can be rotated through some angle and remain unchanged. Examples of objects with rotational symmetry include automobile wheels, flowers, and kaleidoscope patterns. Repeating patterns in the plane can only possess 2-, 3-, 4-, and 6-fold rotationally symmetry. (If a pattern possesses
*n*-fold rotational symmetry about a point, it can be rotated by 1/*n* of a full revolution about that point and remain unchanged.
- Glide reflection symmetry - An object possesses glide reflection symmetry if it can be translated by some distance and mirrored about a line and remain unchanged.
- Mirror symmetry - This is a special case of glide reflection symmetry, where the glide distance is zero.

Less strictly defined, self similarity can also be considered as a type of symmetry. This is the characteristic feature of fractal objects. If one can zoom in on a portion of an object and see something that is similar to (looks like) a reduced version of another portion of the object, then that object possesses self similarity. This type of symmetry is widespread in nature. Examples include the system of arteries in the human body and the coastline of a continent.

Click here to browse puzzles that teach about symmetry.

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