Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
Symmetry in mathematics
In formal terms, a mathematical object is symmetric with respect to a given operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).
The symmetric function of a two-dimensional figure is a line such that, for each perpendicular constructed, if the perpendicular intersects the figure at a distance 'd' from the axis along the perpendicular, then there exists another intersection of the shape and the perpendicular, at the same distance 'd' from the axis, in the opposite direction along the perpendicular.
Another way to think about the symmetric function is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror images.
Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry.
Symmetric geometrical shapes
For an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between 2/3 and 1 for any convex shape.
For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space.
In certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry; in such contexts in modern physics the term parity or P-symmetry is used for both.
Advanced types of reflection symmetry
For more general types of reflection there are correspondingly more general types of reflection symmetry. For example:
- with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc.)
- with respect to circle inversion.
Mirrored symmetry is also found in the design of ancient structures, including Stonehenge.
- Johnson, Anthony (2008). Solving Stonehenge: The New Key to an Ancient Enigma. Thames & Hudson.
- Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson.
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