Editing Symmetry (section)

==In the arts==
{{Further|Mathematics and art}}

There exists a list of journals and newsletters known to deal, at least in part, with symmetry and the arts.<ref>{{cite journal | last1 = Bouissou, C. | last2 = Petitjean, M. | title = Asymmetric Exchanges | journal = Journal of Interdisciplinary Methodologies and Issues in Science | year = 2018 | volume = 4 | pages = 1–18 | url = https://hal.archives-ouvertes.fr/hal-01782438v2/document | doi = 10.18713/JIMIS-230718-4-1 | doi-access = free}} (see  appendix 1)</ref>

===In architecture===
{{Further|Mathematics and architecture}}
[[File:Taj Mahal, Agra views from around (85).JPG|thumb|Seen from the side, the [[Taj Mahal]] has bilateral symmetry; from the top (in plan), it has fourfold symmetry.]]

Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic [[cathedral]]s and [[The White House]], through the layout of the individual [[floor plan]]s, and down to the design of individual building elements such as [[mosaic|tile mosaics]]. [[Islam]]ic buildings such as the [[Taj Mahal]] and the [[Lotfollah mosque]] make elaborate use of symmetry both in their structure and in their ornamentation.<ref>[http://members.tripod.com/vismath/kim/ Williams: Symmetry in Architecture]. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.</ref><ref>[http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml Aslaksen: Mathematics in Art and Architecture]. Math.nus.edu.sg. Retrieved on 2013-04-16.</ref> Moorish buildings like the [[Alhambra]] are ornamented with complex patterns made using translational and reflection symmetries as well as rotations.<ref>{{cite book |author=Derry, Gregory N. |title=What Science Is and How It Works |url=https://books.google.com/books?id=Dk-xS6nABrYC&pg=PA269 |year=2002 |publisher=Princeton University Press |isbn=978-1-4008-2311-6 |pages=269–}}</ref>

It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures";<ref name=Dunlap>{{cite news |last1=Dunlap |first1=David W. |title=Behind the Scenes: Edgar Martins Speaks |url=http://lens.blogs.nytimes.com/2009/07/31/behind-10/?_r=0 |newspaper=New York Times |access-date=11 November 2014 |date=31 July 2009 | quote=“My starting point for this construction was a simple statement which I once read (and which does not necessarily reflect my personal views): ‘Only a bad architect relies on symmetry; instead of symmetrical layout of blocks, masses and structures, Modernist architecture relies on wings and balance of masses.’}}</ref> [[Modernist architecture]], starting with [[International style (architecture)|International style]], relies instead on "wings and balance of masses".<ref name=Dunlap/>

===In pottery and metal vessels===
[[File:Makingpottery.jpg|thumb|right|Clay pots thrown on a [[pottery wheel]] acquire rotational symmetry.]]

Since the earliest uses of [[pottery wheel]]s to help shape clay vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives.

Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient [[Chinese people|Chinese]], for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.<ref>[http://www.chinavoc.com/arts/handicraft/bronze.htm The Art of Chinese Bronzes] {{Webarchive|url=https://web.archive.org/web/20031211185121/http://chinavoc.com/arts/handicraft/bronze.htm |date=2003-12-11 }}. Chinavoc (2007-11-19). Retrieved on 2013-04-16.</ref>

===In carpets and rugs===
[[File:Farsh1.jpg|thumb|upright=1.5|Persian rug with rectangular symmetry]]

A long tradition of the use of symmetry in [[carpet]] and rug patterns spans a variety of cultures. American [[Navajo people|Navajo]] Indians used bold diagonals and rectangular motifs. Many [[Oriental rugs]] have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of a [[rectangle]]—that is, [[Motif (visual arts)|motifs]] that are reflected across both the horizontal and vertical axes (see {{slink|Klein four-group|Geometry}}).<ref>[https://web.archive.org/web/20010203155200/http://marlamallett.com/default.htm Marla Mallett Textiles & Tribal Oriental Rugs]. The Metropolitan Museum of Art, New York.</ref><ref>[http://navajocentral.org/rugs.htm Dilucchio: Navajo Rugs]. Navajocentral.org (2003-10-26). Retrieved on 2013-04-16.</ref>

===In quilts===
[[File:kitchen kaleid.svg|thumb|upright=0.65|Kitchen [[kaleidoscope]] [[quilt]] block]]

As [[quilt]]s are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry.<ref>[http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm Quate: Exploring Geometry Through Quilts] {{Webarchive|url=https://web.archive.org/web/20031231031119/http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm |date=2003-12-31 }}. Its.guilford.k12.nc.us. Retrieved on 2013-04-16.</ref>

===In other arts and crafts===
{{Further|Islamic geometric patterns}}

Symmetries appear in the design of objects of all kinds. Examples include [[beadwork]], [[furniture]], [[sand painting]]s, [[knot]]work, [[masks]], and [[musical instruments]]. Symmetries are central to the art of [[M.C. Escher]] and the many applications of [[tessellation]] in art and craft forms such as [[wallpaper]], ceramic tilework such as in [[Islamic geometric patterns|Islamic geometric decoration]], [[batik]], [[ikat]], carpet-making, and many kinds of [[textile]] and [[embroidery]] patterns.<ref>{{cite book |last1=Cucker |first1=Felipe |author1-link=Felipe Cucker|title=Manifold Mirrors: The Crossing Paths of the Arts and Mathematics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-72876-8 |pages=77–78, 83, 89, 103}}</ref>

Symmetry is also used in designing logos.<ref>{{cite web|title=How to Design a Perfect Logo with Grid and Symmetry|url=https://www.designmantic.com/how-to/how-to-design-a-perfect-logo}}</ref> By creating a logo on a grid and using the theory of symmetry, designers can organize their work, create a symmetric or asymmetrical design, determine the space between letters, determine how much negative space is required in the design, and how to accentuate parts of the logo to make it stand out.

===In music===
[[File:Major and minor triads, circles, dozenal.png|thumb|[[Major and minor]] triads on the white piano keys are symmetrical to the D.]]

Symmetry is not restricted to the visual arts. Its role in the history of [[music]] touches many aspects of the creation and perception of music.

====Musical form====

Symmetry has been used as a [[musical form|formal]] constraint by many composers, such as the [[arch form|arch (swell) form]] (ABCBA) used by [[Steve Reich]], [[Béla Bartók]], and [[James Tenney]]. In classical music, [[Johann Sebastian Bach]] used the symmetry concepts of permutation and invariance.<ref>see ("Fugue No. 21," [http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf pdf] {{Webarchive|url=https://web.archive.org/web/20050913002304/http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf |date=2005-09-13 }} or [http://jan.ucc.nau.edu/~tas3/wtc/ii21.html Shockwave] {{Webarchive|url=https://web.archive.org/web/20051026015256/http://jan.ucc.nau.edu/~tas3/wtc/ii21.html |date=2005-10-26 }})</ref>

====Pitch structures====
Symmetry is also an important consideration in the formation of [[scale (music)|scale]]s and [[chord (music)|chords]], traditional or [[tonality|tonal]] music being made up of non-symmetrical groups of [[pitch (music)|pitches]], such as the [[diatonic scale]] or the [[major chord]]. [[Symmetrical scale]]s or chords, such as the [[whole tone scale]], [[augmented chord]], or diminished [[seventh chord]] (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are [[ambiguous]] as to the [[Key (music)|key]] or tonal center, and have a less specific [[diatonic functionality]]. However, composers such as [[Alban Berg]], [[Béla Bartók]], and [[George Perle]] have used axes of symmetry and/or [[interval cycle]]s in an analogous way to [[musical key|keys]] or non-[[tonality|tonal]] tonal [[Tonic (music)|center]]s.<ref name=Perle1992>{{Cite journal |title=Symmetry, the twelve-tone scale, and tonality |first=George |last=Perle |author-link=George Perle |journal=Contemporary Music Review |volume=6 |issue=2 |year=1992 |pages=81–96 |doi=10.1080/07494469200640151}}</ref> George Perle explains that "C–E, D–F♯, [and] Eb–G, are different instances of the same [[interval (music)|interval]] … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:"<ref name=Perle1992/>

{|
|-
|D
| 
|D♯
| 
|'''E'''
| 
|F
| 
|F♯
| 
|G
| 
|G♯
|-
|D
| 
|C♯
| 
|'''C'''
| 
|B
| 
|A♯
| 
|A
| 
|G♯
|}

Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0).<ref name=Perle1992/>

{|
|rowspan=3|+
|2
|
|3
|
|'''4'''
|
|5
|
|6
|
|7
|
|8
|-
|2
|
|1
| 
|'''0'''
|
|11
|
|10
|
|9
|
|8
|-
|4
|
|4
|
|4
|
|4
|
|4
|
|4
|
|4
|}

Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are [[enharmonic]] with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal [[chord progression|progressions]] in the works of [[Romantic music|Romantic]] composers such as [[Gustav Mahler]] and [[Richard Wagner]] form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, [[Alexander Scriabin]], [[Edgard Varèse]], and the Vienna school. At the same time, these progressions signal the end of tonality.<ref name=Perle1992/><ref name="Perle1990">{{cite book |author-link=George Perle |author=Perle, George |year=1990 |title=The Listening Composer |url=https://archive.org/details/listeningcompose00perl |url-access=limited |page=[https://archive.org/details/listeningcompose00perl/page/n31 21] |publisher=University of California Press |isbn=978-0-520-06991-6}}</ref>

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's [[String Quartet (Berg)|''Quartet'', Op. 3]] (1910).<ref name="Perle1990"/>

====Equivalency====

[[Tone row]]s or [[pitch class]] [[Set theory (music)|sets]] which are [[Invariant (music)|invariant]] under [[Permutation (music)|retrograde]] are horizontally symmetrical, under [[Melodic inversion|inversion]] vertically. See also [[Asymmetric rhythm]].

===In aesthetics===
{{Main|Symmetry (physical attractiveness)}}

The relationship of symmetry to [[aesthetics]] is complex. Humans find [[bilateral symmetry]] in faces physically attractive;<ref name="Grammer1994">{{cite journal |last1=Grammer |first1=K. |last2=Thornhill |first2=R. |date=1994 |title=Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness |journal=Journal of Comparative Psychology |location=Washington, D.C. |volume=108 |issue=3 |pages=233–42 |doi=10.1037/0735-7036.108.3.233|pmid=7924253 |s2cid=1205083 }}</ref> it indicates health and genetic fitness.<ref>{{cite book |last1=Rhodes |first1=Gillian |last2=Zebrowitz |first2=Leslie A. |author2-link=Leslie Zebrowitz |title=Facial Attractiveness: Evolutionary, Cognitive, and Social Perspectives |publisher=[[Ablex]] |year=2002 |isbn=1-56750-636-4}}</ref><ref name="Jones2001">Jones, B. C., Little, A. C., Tiddeman, B. P., Burt, D. M., & Perrett, D. I. (2001). Facial symmetry and judgements of apparent health Support for a “‘ good genes ’” explanation of the attractiveness – symmetry relationship, 22, 417–429.</ref> Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. Rudolf Arnheim suggested that people prefer shapes that have some symmetry, and enough complexity to make them interesting.<ref>{{cite book |last=Arnheim |first=Rudolf |title=Visual Thinking |url=https://archive.org/details/visualthinking00rudo |url-access=registration |publisher=University of California Press |year=1969}}</ref>

===In literature===
Symmetry can be found in various forms in [[literature]], a simple example being the [[palindrome]] where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, such as the rise and fall pattern of ''[[Beowulf]]''.<ref>{{cite web | url=http://trace.tennessee.edu/cgi/viewcontent.cgi?article=1925&context=utk_gradthes | title=Symmetrical Aesthetics of Beowulf | publisher=University of Tennessee, Knoxville | year=2009 |author1=Jenny Lea Bowman}}</ref>