Inmathematics,reflection symmetry,line symmetry,mirror symmetry, ormirror-image symmetry issymmetry with respect to areflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.

In2-dimensional space, there is a line/axis of symmetry, in3-dimensional space, there is aplane of symmetry. An object or figure which is indistinguishable from its transformed image is calledmirror symmetric.

In formal terms, amathematical object is symmetric with respect to a givenoperation such as reflection,rotation, ortranslation, if, when applied to the object, this operation preserves some property of the object.^[1] The set of operations that preserve a given property of the object form agroup. 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 eachperpendicular 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'smirror images.^[1] 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, while acone andsphere have infinitely many planes of symmetry.

Triangles with reflection symmetry areisosceles.Quadrilaterals with reflection symmetry arekites, (concave) deltoids,rhombi,^[2] andisosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges. For an arbitrary shape, theaxiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between two-thirds and 1 for anyconvex shape.

In 3D, the cube in which the plane can configure in all of the three axes that can reflect the cube has 9 planes of reflective symmetry.^[3]

For more general types ofreflection there are correspondingly more general types of reflection symmetry. For example:

• with respect to a non-isometricaffine involution (anoblique reflection in a line, plane, etc.)
• with respect tocircle inversion.

Animals that are bilaterally symmetric have reflection symmetry around thesagittal plane, which divides the body vertically into left and right halves, with one of each sense organ and limb pair on either side. Most animals are bilaterally symmetric, likely because this supportsforward movement andstreamlining.^[4][5][6]

Mirror symmetry is often used inarchitecture, as in the facade ofSanta Maria Novella,Florence.^[7] It is also found in the design of ancient structures such asStonehenge.^[8] Symmetry was a core element in some styles of architecture, such asPalladianism.^[9]

• Patterns in nature
• Point reflection symmetry
• Coxeter group theory aboutReflection groups inEuclidean space
• Rotational symmetry (different type of symmetry)
• Chirality

• Stewart, Ian (2001).What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson.

• Weyl, Hermann (1982) [1952].Symmetry. Princeton: Princeton University Press.ISBN 0-691-02374-3.