Editing Reflection symmetry

{{Short description|Invariance under a mathematical reflection}}
{{Use mdy dates|date=November 2024}}{{refimprove|date=October 2015}}
{{distinguish|Point reflection}}
{{Redirect|Mirror symmetry||}}
[[Image:Symmetry.png|thumb|250px|right|Figures with the axes of [[symmetry]] drawn in. The figure with no axes is [[asymmetry|asymmetric]].]]

In [[mathematics]], '''reflection symmetry''', '''line symmetry''', '''mirror symmetry''', or '''mirror-image symmetry''' is [[symmetry]] with respect to a [[Reflection (mathematics)|reflection]]. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.

In [[Two-dimensional space|2-dimensional space]], there is a line/axis of symmetry, in [[Three-dimensional space|3-dimensional space]], there is a [[plane (mathematics)|plane]] of symmetry. An object or figure which is indistinguishable from its transformed image is called [[mirror image|mirror symmetric]]. 

==Symmetric function== 
[[File:Empirical Rule.PNG|thumb|A [[normal distribution]] bell curve is an example of a symmetric function|left]]

In formal terms, a [[mathematical object]] is symmetric with respect to a given [[mathematical operation|operation]] such as reflection, [[Rotational symmetry|rotation]], or [[Translational symmetry|translation]], if, when applied to the object, this operation preserves some property of the object.<ref name=Stewart32>{{cite book | title=What Shape is a Snowflake? Magical Numbers in Nature | publisher=Weidenfeld & Nicolson | author=Stewart, Ian | year=2001 | page=32}}</ref> The set of operations that preserve a given property of the object form a [[group (algebra)|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 image]]s.<ref name=Stewart32/> 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 a [[cone]] and [[sphere]] have infinitely many planes of symmetry.

==Symmetric geometrical shapes==
{| class="wikitable floatright" align=right
|+ 2D shapes w/reflective symmetry
|[[File:Isosceles_trapezoid.svg|100px]]
|[[File:GeometricKite.svg|100px]]
|-
!colspan=2|[[isosceles trapezoid]] and [[kite (geometry)|kite]]
|-
|[[File:Hexagon p2 symmetry.png|100px]]
|[[File:Hexagon d3 symmetry.png|100px]]
|-
!colspan=2|[[Hexagon]]s
|-
|[[File:Octagon p2 symmetry.png|100px]]
|[[File:Octagon d2 symmetry.png|100px]]
|-
!colspan=2|[[octagon]]s
|}

[[Triangle]]s with reflection symmetry are [[isosceles]]. [[Quadrilateral]]s with reflection symmetry are [[kite (geometry)|kite]]s, (concave) deltoids, [[rhombi]],<ref>{{cite book |last1=Gullberg |first1=Jan |author-link=Jan Gullberg |title=Mathematics: From the Birth of Numbers |url=https://archive.org/details/mathematicsfromb1997gull |url-access=registration |date=1997 |publisher=W. W. Norton |isbn=0-393-04002-X|pages=[https://archive.org/details/mathematicsfromb1997gull/page/394 394–395]}}</ref> and [[isosceles trapezoid]]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 (geometry)|axiality]] 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 any [[Convex polygon|convex 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.<ref>{{Cite journal |last=O’Brien |first=David |last2=McShane |first2=Pauric |last3=Thornton |first3=Sean |title=The Group of Symmetries of the Cube |url=https://maths.nuigalway.ie/~rquinlan/groups/exhibition/mcshane_obrien_thornton.pdf |journal=NUI Galway}}</ref>

==Advanced types of reflection symmetry==
For more general types of [[reflection (mathematics)|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 [[inversive geometry|circle inversion]].

==In nature==
[[File:Maja crispata (Maia verrucosa) - Museo Civico di Storia Naturale Giacomo Doria - Genoa, Italy - DSC03222 Cropped.JPG|thumb|Many animals, such as this [[spider crab]] ''[[Maja crispata]]'', are bilaterally symmetric.|left]]
{{main|Bilateral symmetry}}

[[Bilateria|Animals that are bilaterally symmetric]] have reflection symmetry around 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 [[Animal locomotion|forward movement]] and [[Streamlines, streaklines, and pathlines|streamlining]].<ref>{{cite web |last=Valentine |first=James W. |title=Bilateria |url=http://www.accessscience.com/abstract.aspx?id=802620&referURL=http%3a%2f%2fwww.accessscience.com%2fcontent.aspx%3fid%3d802620 |url-status=dead |archive-url=https://web.archive.org/web/20071117021029/http://www.accessscience.com/abstract.aspx?id=802620&referURL=http%3a%2f%2fwww.accessscience.com%2fcontent.aspx%3fid%3d802620 |archive-date=November 17, 2007 |access-date=29 May 2013 |publisher=AccessScience}}</ref><ref name=Finnerty>{{cite journal | url=http://faculty.weber.edu/rmeyers/PDFs/Finnerty%20-%20symmetry%20evol.pdf | title=Did internal transport, rather than directed locomotion, favor the evolution of bilateral symmetry in animals? | author=Finnerty, John R. | journal=BioEssays | year=2005 | volume=27 | issue=11 | pages=1174–1180 | doi=10.1002/bies.20299 | pmid=16237677}}</ref><ref name=Berkeley>{{cite web | url=http://evolution.berkeley.edu/evolibrary/article/arthropods_04 | title=Bilateral (left/right) symmetry | publisher=Berkeley | access-date=14 June 2014}}</ref>
==In architecture==

[[File:Santa Maria Novella.jpg|thumb|Mirror symmetry is often used in [[architecture]], as in the facade of [[Santa Maria Novella]], [[Florence]], 1470.]]

{{main|Mathematics and architecture}}

Mirror symmetry is often used in [[architecture]], as in the facade of [[Santa Maria Novella]], [[Florence]].<ref name="Tavernor1998">{{cite book |last=Tavernor|first=Robert |title=On Alberti and the Art of Building |url=https://books.google.com/books?id=hOs2zXz7M7wC&pg=PA103 |year=1998 |publisher=Yale University Press |isbn=978-0-300-07615-8 |pages=102–106 |quote=More accurate surveys indicate that the facade lacks a precise symmetry, but there can be little doubt that Alberti intended the composition of number and geometry to be regarded as perfect. The facade fits within a square of 60 Florentine braccia}}</ref> It is also found in the design of ancient structures such as [[Stonehenge]].<ref name="Johnson, Anthony 2008">Johnson, Anthony (2008). ''Solving Stonehenge: The New Key to an Ancient Enigma''. Thames & Hudson.</ref> Symmetry was a core element in some styles of architecture, such as [[Palladianism]].<ref>{{cite web |last1=Waters |first1=Suzanne |title=Palladianism |url=https://www.architecture.com/Explore/ArchitecturalStyles/Palladianism.aspx |publisher=Royal Institution of British Architects |access-date=29 October 2015}}</ref>

==See also==
* [[Patterns in nature]]
* [[Point reflection]] symmetry
* [[Coxeter group|Coxeter group theory]] about [[Reflection group]]s in [[Euclidean space]]
* [[Rotational symmetry]] (different type of symmetry)
* [[Chirality]]

== References ==
{{reflist}}

==Bibliography==

===General===

* {{cite book | title=What Shape is a Snowflake? Magical Numbers in Nature | publisher=Weidenfeld & Nicolson | author=Stewart, Ian | year=2001}}

===Advanced===

* {{cite book |title=Symmetry |last=Weyl |first=Hermann |author-link=Hermann Weyl |year=1982 |orig-year=1952 |publisher=Princeton University Press |location=Princeton |isbn=0-691-02374-3 |ref=Weyl 1982}}
{{commons category|Reflection symmetry}}
[[Category:Elementary geometry]]
[[Category:Euclidean symmetries]]