TheInfoList

250px, Figures with the axes of
asymmetric Asymmetric may refer to: *Asymmetry in geometry, chemistry, and physics Computing *Asymmetric cryptography, in public-key cryptography *Asymmetric digital subscriber line, Internet connectivity *Asymmetric multiprocessing, in computer architecture ...
. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. In conclusion, a line of symmetry splits the shape in half and those halves should be identical.

# Symmetric function In formal terms, a
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs. ...
is symmetric with respect to a given
operation Operation or Operations may refer to: Science and technology * Surgical operation or surgery, in medicine * Operation (mathematics), a calculation from zero or more input values (called operands) to an output value ** Arity, number of arguments or ...
such as reflection,
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center and perpendicular to ...
or
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''translating' ...
, 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 A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
. 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 In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
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 image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances su ...
s. 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

Triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, d ...
s with reflection symmetry are
isosceles In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version th ...
.
Quadrilateral A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle), tetragon (in analogy to pentagon and hexagon), and 4-gon (in ...
s with reflection symmetry are
kite . This sparless, ram-air inflated kite, has a complex bridle formed of many strings attached to the face of the wing. A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift an ...
s, (concave) deltoids,
rhombi In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
, and
isosceles trapezoid In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined ... s. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges. 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.

# Mathematical equivalents

For each line or plane of reflection, the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient s ...
is isomorphic with ''Cs'' (see
point groups in three dimensions In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that ...
), one of the three types of order two (
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inputs ...
s), hence algebraically ''C2''. The
fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of ...
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 Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the re ...
or P-symmetry is used for both.

# Advanced types of reflection symmetry

For more general types of
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
there are correspondingly more general types of reflection symmetry. For example: * with respect to a non-isometric
affine involutionIn Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space R''n''. Such involutions are easy to characterize and they can be described geometrically. Linear involutions To give a li ...
(an
oblique reflection In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will still stay so under affine transformations. C ...
in a line, plane, etc.) * with respect to
circle inversion In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry be ...
.

# In nature

Animals that are bilaterally symmetric have reflection symmetry in the sagittal 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 supports forward movement and streamlining.

# In architecture Mirror symmetry is often used in
architecture upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted), Plan of the second floor (attic storey) of the Hôtel de Brionne in Paris – 1734. Architecture (Latin ''architect ...
, as in the facade of
Santa Maria Novella Santa Maria Novella is a church in Florence, Italy, situated opposite, and lending its name to, the city's main railway station. Chronologically, it is the first great basilica in Florence, and is the city's principal Dominican church. The church ... ,
Florence Florence ( ; it, Firenze ) is a city in Central Italy and the capital city of the Tuscany region. It is the most populated city in Tuscany, with 383,084 inhabitants in 2013, and over 1,520,000 in its metropolitan area. Florence was a centre o ... . It is also found in the design of ancient structures such as
Stonehenge Stonehenge is a prehistoric monument on Salisbury Plain in Wiltshire, England, west of Amesbury. It consists of an outer ring of vertical Sarsen standing stones, each around high, wide, and weighing around 25 tons, topped by connecting hor ...
.Johnson, Anthony (2008). ''Solving Stonehenge: The New Key to an Ancient Enigma''. Thames & Hudson. Symmetry was a core element in some styles of architecture, such as
Palladianism '', in an English translation published in London, 1736. Palladian architecture is a European architectural style derived from and inspired by the designs of the Venetian architect Andrea Palladio (1508–1580). What is recognised as Palladian ar ...
.

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Point reflection In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it ...
symmetry

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