# Reflection symmetry

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Reflection symmetry

Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection.

In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric (see mirror image). Also see pattern.

The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it 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 image. 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, for the same reason.

If the letter T is reflected along a vertical axis, it appears the same. Note that this is sometimes called horizontal symmetry, and sometimes vertical symmetry. One can better use an unambiguous formulation, e.g. "T has a vertical symmetry axis."

The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kites and the isosceles trapezoids.

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.

Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane.

In certain contexts there is rotational symmetry anyway. Then mirror-image symmetry is equivalent with inversion symmetry; in such contexts in modern physics the term P-symmetry is used for both (P stands for parity).

For more general types of reflection there are corresponding more general types of reflection symmetry. Examples:
*with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc).
*with respect to circle inversion.

ee also

*Rotational symmetry
*Translational symmetry

References

*cite book |title=Symmetry |last=Weyl |first=Hermann |authorlink=Hermann Weyl |coauthors= |date=1982 |origdate=1952 |publisher=Princeton University Press |location=Princeton |isbn=0-691-02374-3 |pages= |url= |ref=Weyl 1982

* [http://republika.pl/fraktal/mapping.html Mapping with symmetry - source in Delphi]
* [http://www.mathsisfun.com/geometry/symmetry-reflection.html Reflection Symmetry Examples] from Math Is Fun

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