Symmetry | Research & Encyclopedia Articles

Symmetry is a visual characteristic in the shape and design of an object. Take a starfish for an example of an object with symmetry. If the arms of the starfish are all exactly the same length and width, then the starfish could be folded in half and the two sides would exactly match each other. The line on which the starfish was folded is called a line of symmetry. Any object that can be folded in half and each side matches the other is said to have line symmetry.

When an object has line symmetry, one half of the object is a reflection of the other half. Just imagine that the line of symmetry is a mirror. The actual object and its reflection in the "mirror" would exactly match each other. A human face, if vertically bisected into two halves (left and right), would reveal its bilateral symmetry: that is, the left half ideally matches the right half.

It is possible for an object to have more than one line of symmetry. Imagine a square. A vertical line of symmetry could be drawn down the middle of the square and the left and right sides would be symmetrical. Also, a horizontal line could be drawn across the middle of the square and the top and bottom would be symmetrical. As shown in the figure below, a star or a starfish has five lines of symmetry.

Alternately, the ability of an object to match its original figure with less than a full turn about its center is called rotational or point symmetry. For example, if a starfish embedded in the sand on a beach is picked up, rotated less than 360°, and then set back down exactly into its original imprint in the sand, the starfish would be said to have rotational symmetry.

Many natural and manmade items have reflection and/or rotation symmetry. Some examples are pottery, weaving, and quilting designs; architectural features such as windows, doors, roofs, hinges, tile floors, brick walls, railings, fences, bridge supports, or arches; many kinds of flowers and leaves; honeycomb; the cross section of an apple or a grapefruit; snowflakes; an open umbrella; letters of the alphabet; kaleidoscope designs; a pinwheel, windmill, or ferris wheel; some national flags (but not the U.S. flag); a ladder; a baseball diamond; or a stop sign.

Mathematicians use symmetries to reduce the complexity of a mathematical problem. In other words, one can apply what is known about just one of multiple symmetric pieces to other symmetric pieces and thereby learn more about the whole object.

Transformations.

Schattschneider, D. Visions of symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher. New York: W. H. Freeman and Company, 1990.

Search for Math: Symmetry. The Math Forum at Swarthmore College. <http://www.forum.swarthmore.edu />.

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