Editing Symmetry (section)

==In mathematics==

===In geometry===
{{main|Symmetry (geometry)}}
[[File:The armoured triskelion on the flag of the Isle of Man.svg|thumb|upright=0.8|The [[triskelion]] has 3-fold rotational symmetry.]]

A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.<ref>E. H. Lockwood, R. H. Macmillan, ''[[Geometric symmetry (book)|Geometric Symmetry]]'', London: Cambridge Press,
1978</ref> This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:

* An object has [[reflectional symmetry]] (line or mirror symmetry) if there is a line (or in [[three-dimensional space|3D]] a plane) going through it which divides it into two pieces that are mirror images of each other.<ref>{{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}}</ref>
*An object has [[rotational symmetry]] if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.<ref>{{cite book | author=Singer, David A. | year=1998 | title=Geometry: Plane and Fancy | url=https://archive.org/details/geometryplanefan0000sing | url-access=registration | publisher=Springer Science & Business Media}}</ref>
*An object has [[translational symmetry]] if it can be [[translation (geometry)|translated]] (moving every point of the object by the same distance) without changing its overall shape.<ref>Stenger, Victor J. (2000) and Mahou Shiro (2007). ''Timeless Reality''. Prometheus Books. Especially chapter 12. Nontechnical.</ref>
*An object has [[helical symmetry]] if it can be simultaneously translated and rotated in three-dimensional space along a line known as a [[screw axis]].<ref>Bottema, O, and B. Roth, ''Theoretical Kinematics,'' Dover Publications (September 1990)</ref>
*An object has [[scale symmetry]] if it does not change shape when it is expanded or contracted.<ref>Tian Yu Cao ''Conceptual Foundations of Quantum Field Theory'' Cambridge University Press p.154-155</ref> [[Fractals]] also exhibit a form of scale symmetry, where smaller portions of the fractal are [[similarity (geometry)|similar]] in shape to larger portions.<ref name="Gouyet">{{cite book | last = Gouyet | first = Jean-François | title = Physics and fractal structures | publisher = Masson Springer | location = Paris/New York | year = 1996 | isbn = 978-0-387-94153-0 }}</ref>
*Other symmetries include [[glide reflection]] symmetry (a reflection followed by a translation) and [[improper rotation|rotoreflection]] symmetry (a combination of a rotation and a reflection<ref>{{Cite web|url=https://encyclopedia2.thefreedictionary.com/rotoreflection+axis|title=Rotoreflection Axis|website=TheFreeDictionary.com|access-date=2019-11-12}}</ref>).

===In logic===

A [[binary relation|dyadic relation]] ''R'' = ''S'' × ''S'' is symmetric if for all elements ''a'', ''b'' in ''S'', whenever it is true that ''Rab'', it is also true that ''Rba''.<ref>Josiah Royce, Ignas K. Skrupskelis (2005) ''The Basic Writings of Josiah Royce: Logic, loyalty, and community (Google eBook)'' Fordham Univ Press, p. 790</ref> Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul.

In propositional logic, symmetric binary [[logical connective]]s include ''[[logical conjunction|and]]'' (∧, or &), ''[[logical disjunction|or]]'' (∨, or |) and ''[[if and only if]]'' (↔), while the connective ''if'' (→) is not symmetric.<ref>{{Cite web|url=https://cs.uwaterloo.ca/~a23gao/cs245_f19/slides/lec02_prop_syntax_nosol.pdf|title=Propositional Logic: Introduction and Syntax|last=Gao|first=Alice|date=2019|website=University of Waterloo — School of Computer Science|access-date=2019-11-12}}</ref> Other symmetric logical connectives include ''[[logical nand|nand]]'' (not-and, or ⊼), ''[[xor]]'' (not-biconditional, or ⊻), and ''[[logical nor|nor]]'' (not-or, or ⊽).

===Other areas of mathematics===
{{main|Symmetry in mathematics}}
Generalizing from geometrical symmetry in the previous section, one can say that a [[mathematical object]] is ''symmetric'' with respect to a given [[Operation (mathematics)|mathematical operation]], if, when applied to the object, this operation preserves some property of the object.<ref>Christopher G. Morris
 (1992) ''Academic Press Dictionary of Science and Technology'' Gulf Professional Publishing</ref> The set of operations that preserve a given property of the object form a [[group (mathematics)|group]].

In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include [[even and odd functions]] in [[calculus]], [[symmetric group]]s in [[abstract algebra]], [[symmetric matrix|symmetric matrices]] in [[linear algebra]], and [[Galois group]]s in [[Galois theory]]. In [[statistics]], symmetry also manifests as [[symmetric probability distribution]]s, and as [[skewness]]—the asymmetry of distributions.<ref>{{cite journal | author = Petitjean, M. | title = Chirality and Symmetry Measures: A Transdisciplinary Review | journal = Entropy | year = 2003 | volume = 5 | issue = 3 | pages=271–312 (see section 2.9) | doi = 10.3390/e5030271| bibcode = 2003Entrp...5..271P | doi-access = free }}</ref>