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{{short description|Relationship between mathematics and art}}
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[[File:Dürer Melancholia I.jpg|thumb|250px|Mathematics in art: [[Albrecht Dürer]]'s copper plate engraving ''[[Melencolia I]]'', 1514. Mathematical references include a compass for [[geometry]], a [[magic square]] and a [[Dürer's solid|truncated rhombohedron]], while measurement is indicated by the scales and [[hourglass]].<ref name=Ziegler/>]]
[[File:Paolo uccello, studio di vaso in prospettiva 02.jpg|thumb|250px|[[Wire-frame model|Wireframe drawing]]<ref name="ColomboDel Bimbo2005"/> of a vase as a [[solid of revolution]]<ref name="ColomboDel Bimbo2005">{{cite journal |last1=Colombo |first1=C. |last2=Del Bimbo |first2=A. |last3=Pernici |first3=F. |title=Metric 3D reconstruction and texture acquisition of surfaces of revolution from a single uncalibrated view |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=27 |issue=1 |year=2005 |pages=99–114 |doi=10.1109/TPAMI.2005.14|pmid=15628272 |citeseerx=10.1.1.58.8477 |s2cid=13387519 }}</ref> by [[Paolo Uccello]]. 15th century]]
{{Math topics TOC}}

'''Mathematics and art''' are related in a variety of ways. [[Mathematics]] has itself been described as an [[art]] [[mathematical beauty|motivated by beauty]]. Mathematics can be discerned in arts such as [[Music and mathematics|music]], [[dance]], [[painting]], [[Mathematics and architecture|architecture]], [[sculpture]], and [[Mathematics and fiber arts|textiles]]. This article focuses, however, on mathematics in the visual arts.

Mathematics and art have a long historical relationship. [[List of mathematical artists|Artists have used mathematics]] since the 4th century BC when the Greek [[sculpture|sculptor]] [[Polykleitos]] wrote [[Polykleitos#Canon of Polykleitos|his ''Canon'']], prescribing proportions [[Polykleitos#Conjectured reconstruction|conjectured to have been based]] on the ratio 1:{{radic|2}} for the ideal male nude. Persistent popular claims have been made for the use of the [[golden ratio]] in ancient art and architecture, without reliable evidence. In the Italian [[Renaissance]], [[Luca Pacioli]] wrote the influential treatise ''[[De divina proportione]]'' (1509), illustrated with woodcuts by [[Leonardo da Vinci]], on the use of the golden ratio in art. Another Italian painter, [[Piero della Francesca]], developed [[Euclid]]'s ideas on [[Perspective (visual)|perspective]] in treatises such as ''De Prospectiva Pingendi'', and in his paintings. The engraver [[Albrecht Dürer]] made many references to mathematics in his work ''[[Melencolia I]]''. In modern times, the [[graphic artist]] [[M. C. Escher]] made intensive use of [[tessellation]] and [[hyperbolic geometry]], with the help of the mathematician [[H. S. M. Coxeter]], while the [[De Stijl]] movement led by [[Theo van Doesburg]] and [[Piet Mondrian]] explicitly embraced geometrical forms. Mathematics has inspired textile arts such as [[quilting]], [[knitting]], [[cross-stitch]], [[crochet]], [[embroidery]], [[weaving]], [[Turkish carpet|Turkish]] and other [[carpet]]-making, as well as [[kilim]]. In [[Islamic art]], symmetries are evident in forms as varied as Persian [[girih]] and Moroccan [[zellige]] tilework, [[Mughal architecture|Mughal]] [[jali]] pierced stone screens, and widespread [[muqarnas]] vaulting.

Mathematics has directly influenced art with conceptual tools such as [[linear perspective]], the analysis of [[symmetry]], and mathematical objects such as [[polyhedron|polyhedra]] and the [[Möbius strip]]. [[Magnus Wenninger]] creates colourful [[stellation|stellated polyhedra]], originally as models for teaching. Mathematical concepts such as [[recursion]] and logical paradox can be seen in paintings by [[René Magritte]] and in engravings by M. C. Escher. [[Computer art]] often makes use of [[fractal]]s including the [[Mandelbrot set]], and sometimes explores other mathematical objects such as [[cellular automata]]. Controversially, the artist [[David Hockney]] has argued that artists from the Renaissance onwards made use of the [[camera lucida]] to draw precise representations of scenes; the architect Philip Steadman similarly argued that [[Vermeer]] used the [[camera obscura]] in his distinctively observed paintings.

Other relationships include the algorithmic analysis of artworks by [[X-ray fluorescence spectroscopy]], the finding that traditional [[batik]]s from different regions of [[Java]] have distinct [[fractal dimension]]s, and stimuli to mathematics research, especially [[Filippo Brunelleschi]]'s theory of perspective, which eventually led to [[Girard Desargues]]'s [[projective geometry]]. A persistent view, based ultimately on the [[Pythagoreanism|Pythagorean]] notion of harmony in music, holds that everything was arranged by Number, that God is the geometer of the world, and that therefore the world's [[sacred geometry|geometry is sacred]].

==Origins: from ancient Greece to the Renaissance==
{{further|Artistic canons of body proportions}}
{{Anchor|Polykleitos}}

===Polykleitos's ''Canon'' and ''symmetria''===

[[File:Doryphoros MAN Napoli Inv6011-2.jpg|thumb|upright=0.6|Roman copy in marble of ''[[Doryphoros]]'', originally a bronze by [[Polykleitos]]]]
{{further|Polykleitos}}

[[Polykleitos]] the elder (c. 450–420 BC) was a [[Greeks|Greek]] [[sculptor]] from the school of [[Ancient Argos|Argos]], and a contemporary of [[Phidias]]. His works and statues consisted mainly of bronze and were of athletes. According to the philosopher and mathematician [[Xenocrates]], Polykleitos is ranked as one of the most important sculptors of [[classical antiquity]] for his work on the ''[[Doryphorus]]'' and the statue of [[Hera]] in the [[Heraion of Argos]].<ref name=Stewart>{{cite journal |last=Stewart |first=Andrew |title=Polykleitos of Argos," One Hundred Greek Sculptors: Their Careers and Extant Works |journal=[[Journal of Hellenic Studies]] |date=November 1978 |volume=98 |pages=122–131 |doi=10.2307/630196 |jstor=630196|s2cid=162410725 }}</ref> While his sculptures may not be as famous as those of Phidias, they are much admired. In [[Polykleitos#Canon of Polykleitos|his ''Canon'']], a treatise he wrote designed to document the "perfect" [[body proportions]] of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body.<ref name=Stewart/>

The ''Canon'' itself has been lost but it is conjectured that Polykleitos used a sequence of proportions where each length is that of the diagonal of a square drawn on its predecessor, 1:{{radic|2}} (about 1:1.4142).<ref name=Tobin>{{cite journal |last=Tobin |first=Richard |title=The ''Canon'' of Polykleitos |journal=[[American Journal of Archaeology]] |volume=79 |issue=4 |date=October 1975 |pages=307–321 |doi=10.2307/503064|jstor=503064 |s2cid=191362470 }}</ref>

The influence of the ''Canon'' of Polykleitos is immense in [[Classical Greek]], [[Ancient Rome|Roman]], and [[Renaissance]] sculpture, with many sculptors following Polykleitos's prescription. While none of Polykleitos's original works survive, Roman copies demonstrate his ideal of physical perfection and mathematical precision. Some scholars argue that [[Pythagoreanism|Pythagorean]] thought influenced the ''Canon'' of Polykleitos.<ref name=Raven>{{cite journal |last=Raven |first=J. E. |title=Polyclitus and Pythagoreanism |journal=[[Classical Quarterly]] |date=1951 |volume=1 |issue=3–4 |pages=147– |doi=10.1017/s0009838800004122|s2cid=170092094 }}</ref> The ''Canon'' applies the basic mathematical concepts of Greek geometry, such as the ratio, proportion, and ''symmetria'' (Greek for "harmonious proportions") and turns it into a system capable of describing the human form through a series of continuous [[geometric series|geometric progressions]].<ref name=Tobin/>

===Perspective and proportion===

{{main|Perspective (graphical)}}

[[File:Brunelleschi's perspective experiment.jpg|thumb|[[Filippo Brunelleschi|Brunelleschi]]'s experiment with [[linear perspective]]]]

In classical times, rather than making distant figures smaller with [[linear perspective]], painters sized objects and figures according to their thematic importance. In the Middle Ages, some artists used [[reverse perspective]] for special emphasis. The Muslim mathematician [[Alhazen]] (Ibn al-Haytham) described a theory of optics in his ''[[Book of Optics]]'' in 1021, but never applied it to art.<ref name=StAndrews/> The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand [[nature]] and the [[arts]]. Two major motives drove artists in the late Middle Ages and the Renaissance towards mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas. Second, philosophers and artists alike were convinced that mathematics was the true essence of the physical world and that the entire universe, including the arts, could be explained in geometric terms.<ref name=Emmer>{{cite book |editor=Emmer, Michelle |title=The Visual Mind II |url=https://archive.org/details/visualmindiileon00mich |url-access=registration |publisher=MIT Press |date=2005 |isbn=978-0-262-05048-7}}</ref>

The rudiments of perspective arrived with [[Giotto di Bondone|Giotto]] (1266/7 – 1337), who attempted to draw in perspective using an algebraic method to determine the placement of distant lines. In 1415, the Italian [[architect]] [[Filippo Brunelleschi]] and his friend [[Leon Battista Alberti]] demonstrated the geometrical method of applying perspective in Florence, using [[similar triangles]] as formulated by Euclid, to find the apparent height of distant objects.<ref>{{cite book |last=Vasari |first=Giorgio |author-link=Giorgio Vasari |date=1550 |title=Lives of the Artists |publisher=Torrentino |page=Chapter on Brunelleschi|title-link=Lives of the Artists }}</ref><ref>{{cite book |last1=Alberti |first1=Leon Battista |author-link=Leon Battista Alberti |last2=Spencer |first2=John R. |title=On Painting |date=1956 |publisher=Yale University Press |orig-year=1435 |url=http://www.noteaccess.com/Texts/Alberti/}}</ref> Brunelleschi's own perspective paintings are lost, but [[Masaccio]]'s painting of the Holy Trinity shows his principles at work.<ref name=StAndrews/><ref>{{cite book |last=Field |first=J. V. |author-link=Judith V. Field |title=The Invention of Infinity: Mathematics and Art in the Renaissance |date=1997 |publisher=Oxford University Press |isbn=978-0-19-852394-9}}</ref><ref>{{cite web |last=Witcombe |first=Christopher L. C. E. |title=Art History Resources |url=http://arthistoryresources.net/renaissance-art-theory-2012/perspective.html |access-date=5 September 2015}}</ref>

[[File:San Romano Battle (Paolo Uccello, London) 01.jpg|thumb|left|upright=1.2<!--width for very low image-->|[[Paolo Uccello]] made innovative use of [[linear perspective|perspective]] in ''[[The Battle of San Romano]]'' (c.&nbsp;1435–1460).]]

The Italian painter [[Paolo Uccello]] (1397–1475) was fascinated by perspective, as shown in his paintings of ''[[The Battle of San Romano]]'' (c. 1435–1460): broken lances lie conveniently along perspective lines.<ref name="Hart">{{cite web |last=Hart |first=George W. |author-link=George W. Hart |title=Polyhedra in Art |url=http://www.georgehart.com/virtual-polyhedra/art.html |access-date=24 June 2015}}</ref><ref name="CunninghamReich2014">{{cite book |last1=Cunningham |first1=Lawrence |last2=Reich |first2=John |last3=Fichner-Rathus |first3=Lois |title=Culture and Values: A Survey of the Western Humanities |url=https://books.google.com/books?id=0t0bCgAAQBAJ&pg=PA375 |date=1 January 2014 |publisher=Cengage Learning |isbn=978-1-285-44932-6 |page=375 |quote=which illustrate Uccello's fascination with perspective. The jousting combatants engage on a battlefield littered with broken lances that have fallen in a near-grid pattern and are aimed toward a vanishing point somewhere in the distance.}}</ref>

The painter [[Piero della Francesca]] (c. 1415–1492) exemplified this new shift in Italian Renaissance thinking. He was an expert [[mathematician]] and [[geometer]], writing books on [[solid geometry]] and [[Perspective (graphical)|perspective]], including ''[[De prospectiva pingendi]] (On Perspective for Painting)'', ''Trattato d'Abaco (Abacus Treatise)'', and ''[[De quinque corporibus regularibus]] (On the Five Regular Solids)''.<ref>{{cite book |last=della Francesca |first=Piero |author-link=Piero della Francesca |title=De prospectiva pingendi |editor=G. Nicco Fasola |location=Florence |year=1942 |orig-year=c. 1474}}</ref><ref>{{cite book |last=della Francesca |first=Piero |author-link=Piero della Francesca |title=Trattato d'Abaco |editor=G. Arrighi |location=Pisa |year=1970 |orig-year=Fifteenth century}}</ref><ref>{{cite book |last=della Francesca |first=Piero |author-link=Piero della Francesca |title=L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli |editor=G. Mancini |year=1916}}</ref> The historian [[Vasari]] in his ''[[Lives of the Painters]]'' calls Piero the "greatest geometer of his time, or perhaps of any time."<ref>{{cite book |last=Vasari |first=Giorgio |author-link=Giorgio Vasari |title=Le Opere, volume 2 |editor=G. Milanesi |date=1878 |page=490}}</ref> Piero's interest in perspective can be seen in his paintings including the [[Polyptych of Perugia]],<ref>{{cite book |last=Zuffi |first=Stefano  |title=Piero della Francesca |url=https://archive.org/details/pierodellafrance00zuff |url-access=limited |publisher=L'Unità – Mondadori Arte |year=1991 |page=[https://archive.org/details/pierodellafrance00zuff/page/n52 53]}}</ref> the ''San Agostino altarpiece'' and ''[[The Flagellation of Christ]]''. His work on geometry influenced later mathematicians and artists including [[Luca Pacioli]] in his ''De divina proportione'' and [[Leonardo da Vinci]]. Piero studied classical mathematics and the works of [[Archimedes]].<ref>{{cite book |last=Heath |first=T. L. |title=The Thirteen Books of Euclid's Elements |url=https://archive.org/details/bub_gb_lxkPAAAAIAAJ |publisher=Cambridge University Press |year=1908 |page=[https://archive.org/details/bub_gb_lxkPAAAAIAAJ/page/n101 97]}}</ref> He was taught commercial arithmetic in "abacus schools"; his writings are formatted like abacus school textbooks,<ref>{{cite book |last=Grendler |first=P. |contribution=What Piero Learned in School: Fifteenth-Century Vernacular Education |title=Piero della Francesca and His Legacy |editor=M.A. Lavin |publisher=University Press of New England |year=1995 |pages=161–176}}</ref> perhaps including Leonardo Pisano ([[Fibonacci]])'s 1202 ''[[Liber Abaci]]''. [[Linear perspective]] was just being introduced into the artistic world. Alberti explained in his 1435 ''[[De pictura]]'': "light rays travel in straight lines from points in the observed scene to the eye, forming a kind of [[pyramid]] with the eye as vertex." A painting constructed with linear perspective is a [[Cross section (geometry)|cross-section]] of that pyramid.<ref>{{cite book |last1=Alberti |first1=Leon Battista |author-link=Leon Battista Alberti |title=On Painting |editor= Kemp, Martin |last2=Grayson |first2=Cecil (trans.) |publisher=Penguin Classics |year=1991}}</ref>

In ''De Prospectiva Pingendi'', Piero transforms his empirical observations of the way aspects of a figure change with point of view into mathematical proofs. His treatise starts in the vein of Euclid: he defines the point as "the tiniest thing that is possible for the eye to comprehend".{{efn|In Piero's Italian: "una cosa tanto picholina quanto e possible ad ochio comprendere".}}<ref name=Emmer/> He uses [[deductive logic]] to lead the reader to the perspective representation of a three-dimensional body.<ref>{{cite web |last=Peterson |first=Mark |url=http://www.mtholyoke.edu/courses/rschwart/mac/Italian/geometry.shtml |title=The Geometry of Piero della Francesca |quote=In Book I, after some elementary constructions to introduce the idea of the apparent size of an object being actually its angle subtended at the eye, and referring to Euclid's Elements Books I and VI, and Euclid's Optics, he turns, in Proposition 13, to the representation of a square lying flat on the ground in front of the viewer. What should the artist actually draw? After this, objects are constructed in the square (tilings, for example, to represent a tiled floor), and corresponding objects are constructed in perspective; in Book II prisms are erected over these planar objects, to represent houses, columns, etc.; but the basis of the method is the original square, from which everything else follows. |access-date=2009-04-19 |archive-date=2016-07-01 |archive-url=https://web.archive.org/web/20160701204113/https://www.mtholyoke.edu/courses/rschwart/mac/Italian/geometry.shtml |url-status=dead }}</ref>

The artist [[David Hockney]] [[Hockney-Falco thesis|argued]] in his book ''[[Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters]]'' that artists started using a [[camera lucida]] from the 1420s, resulting in a sudden change in precision and realism, and that this practice was continued by major artists including [[Jean-Auguste-Dominique Ingres|Ingres]], [[Jan van Eyck|Van Eyck]], and [[Michelangelo Merisi|Caravaggio]].<ref>{{cite book |last=Hockney |first=David |title=Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters |year=2006 |publisher=Thames and Hudson |isbn=978-0-500-28638-8}}</ref> Critics disagree on whether Hockney was correct.<ref>{{cite news |last1=Van Riper |first1=Frank |title=Hockney's 'Lucid' Bomb At the Art Establishment |url=https://www.washingtonpost.com/wp-srv/photo/essays/vanRiper/030220.htm |newspaper=The Washington Post |access-date=4 September 2015}}</ref><ref>{{cite news |last=Marr |first=Andrew |author-link=Andrew Marr |title=What the eye didn't see |url=https://www.theguardian.com/theobserver/2001/oct/07/featuresreview.review1 |newspaper=The Guardian |access-date=4 September 2015 |date=7 October 2001}}</ref> Similarly, the architect Philip Steadman argued controversially<ref>{{cite web |last1=Janson |first1=Jonathan |title=An Interview with Philip Steadman |url=http://www.essentialvermeer.com/interviews_newsletter/steadman_interview.html#.VeqxKJdUWHg |publisher=Essential Vermeer |access-date=5 September 2015 |date=25 April 2003}}</ref> that [[Vermeer]] had used a different device, the [[camera obscura]], to help him create his distinctively observed paintings.<ref>{{cite book |last=Steadman |first=Philip |title=Vermeer's Camera: Uncovering the Truth Behind the Masterpieces |date=2002 |publisher=Oxford |isbn=978-0-19-280302-3 |url-access=registration |url=https://archive.org/details/vermeerscameraun0000stea }}</ref>

In 1509, [[Luca Pacioli]] (c. 1447–1517) published ''[[De divina proportione]]'' on [[mathematical]] and [[artistic]] [[Proportionality (mathematics)|proportion]], including in the human face. [[Leonardo da Vinci]] (1452–1519) illustrated the text with woodcuts of regular solids while he studied under Pacioli in the 1490s. Leonardo's drawings are probably the first illustrations of skeletonic solids.<ref>{{cite web |url=http://www.georgehart.com/virtual-polyhedra/pacioli.html |title=Luca Pacioli's Polyhedra |last=Hart |first=George |author-link=George W. Hart |access-date=13 August 2009}}</ref> These, such as the [[rhombicuboctahedron]], were among the first to be drawn to demonstrate perspective by being overlaid on top of each other. The work discusses perspective in the works of [[Piero della Francesca]], [[Melozzo da Forlì]], and [[Marco Palmezzano]].<ref>{{cite news |last=Morris |first=Roderick Conway |title=Palmezzano's Renaissance:From shadows, painter emerges |url=https://www.nytimes.com/2006/01/27/arts/27iht-conway.html?pagewanted=all&_r=0 |newspaper=New York Times |access-date=22 July 2015 |date=27 January 2006}}</ref> Leonardo studied Pacioli's ''Summa'', from which he copied tables of proportions.<ref>{{cite web |url=http://www.dartmouth.edu/~matc/math5.geometry/unit14/unit14.html |last=Calter |first=Paul |title=Geometry and Art Unit 1 |publisher=[[Dartmouth College]] |access-date=13 August 2009 |archive-date=21 August 2009 |archive-url=https://web.archive.org/web/20090821025325/http://www.dartmouth.edu/~matc/math5.geometry/unit14/unit14.html |url-status=dead }}</ref> In ''[[Mona Lisa]]'' and ''[[The Last Supper (Leonardo)|The Last Supper]]'', Leonardo's work incorporated linear perspective with a [[vanishing point]] to provide apparent depth.<ref>{{cite book |last=Brizio |first=Anna Maria |title=Leonardo the Artist |url=https://archive.org/details/leonardoartist00briz |url-access=registration |publisher=McGraw-Hill |year=1980|isbn=9780070079311 }}</ref>  ''The Last Supper'' is constructed in a tight ratio of 12:6:4:3, as is [[Raphael]]'s ''[[The School of Athens]]'', which includes Pythagoras with a tablet of ideal ratios, sacred to the Pythagoreans.<ref name="Ladwein2006">{{cite book |last=Ladwein |first=Michael |title=Leonardo Da Vinci, the Last Supper: A Cosmic Drama and an Act of Redemption |url=https://books.google.com/books?id=wMFem_x1M04C&pg=PA62 |year=2006 |publisher=Temple Lodge Publishing |isbn=978-1-902636-75-7|pages=61–62}}</ref><ref>{{cite book |last=Turner |first=Richard A. |title=Inventing Leonardo |url=https://archive.org/details/inventingleonard00turn |url-access=registration |publisher=Alfred A. Knopf |year=1992|isbn=9780679415510 }}</ref> In ''[[Vitruvian Man]]'', Leonardo expressed the ideas of the Roman architect [[Vitruvius]], innovatively showing the male figure twice, and centring him in both a circle and a square.<ref>{{cite web |last=Wolchover |first=Natalie |title=Did Leonardo da Vinci copy his famous 'Vitruvian Man'? |url=https://www.nbcnews.com/id/wbna46204318 |publisher=NBC News |access-date=27 October 2015 |date=31 January 2012}}</ref>

As early as the 15th century, [[curvilinear perspective]] found its way into paintings by artists interested in image distortions. [[Jan van Eyck]]'s 1434 ''[[Arnolfini Portrait]]'' contains a convex mirror with reflections of the people in the scene,<ref>{{cite journal |last1=Criminisi |first1=A. |last2=Kempz |first2=M. |last3=Kang |first3=S. B. |url=http://research.microsoft.com/pubs/72436/Criminisi_ReflectionsOfReality_2003.pdf |title=Reflections of Reality in Jan van Eyck and Robert Campin |journal=Historical Methods |volume=37 |issue=3 |pages=109–121 |year=2004 |doi=10.3200/hmts.37.3.109-122|s2cid=14289312 }}</ref> while [[Parmigianino]]'s ''Self-portrait in a Convex Mirror'', c. 1523–1524, shows the artist's largely undistorted face at the centre, with a strongly curved background and artist's hand around the edge.<ref name=Cucker299>{{cite book |last=Cucker |first=Felipe|author-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=299–300<!--Parmigianino-->, 306–307<!--van Eyck-->}}</ref>

Three-dimensional space can be represented convincingly in art, as in [[technical drawing]], by means other than perspective. [[Oblique projection]]s, including cavalier perspective (used by French military artists to depict fortifications in the 18th century), were used continuously and ubiquitously by Chinese artists from the first or second centuries until the 18th century. The Chinese acquired the technique from India, which acquired it from Ancient Rome. Oblique projection is seen in Japanese art, such as in the [[Ukiyo-e]] paintings of [[Torii Kiyonaga]] (1752–1815).<ref name=Cucker269>{{cite book |last=Cucker |first=Felipe |author-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=269–278}}</ref>

<gallery mode="nolines" heights="180px" widths="180px">
File:Pacioli De Divina Proportione Head Equilateral Triangle 1509.jpg|Woodcut from [[Luca Pacioli]]'s 1509 ''[[De divina proportione]]'' <!--NOT golden ratio--> with an [[equilateral triangle]] on a human face
File:Camera Lucida in use drawing small figurine.jpg|[[Camera lucida]] in use. ''[[Scientific American]]'', 1879
File:Camera obscura2.jpg|Illustration of an artist using a [[camera obscura]]. 17th century
File:Da Vinci Vitruve Luc Viatour.jpg|Proportion: [[Leonardo da Vinci|Leonardo]]'s ''[[Vitruvian Man]]'', c. 1490
File:Masaccio, trinità.jpg|[[Brunelleschi]]'s theory of [[perspective (graphical)|perspective]]: [[Masaccio]]'s [[Holy Trinity (Masaccio)|''Trinità'']], c. 1426–1428, in the [[Basilica of Santa Maria Novella]]
File:Della Pittura Alberti perspective pillars on grid.jpg|Diagram from [[Leon Battista Alberti]]'s 1435 ''[[Della Pittura]]'', with pillars in perspective on a grid
File:Piero - The Flagellation.jpg|[[Linear perspective]] in [[Piero della Francesca]]'s ''[[Flagellation of Christ (Piero della Francesca)|Flagellation of Christ]]'', c. 1455–1460
File:The Arnolfini Portrait, détail (2).jpg|[[Curvilinear perspective]]: [[convex mirror]] in [[Jan van Eyck]]'s ''[[Arnolfini Portrait]]'', 1434
File:Parmigianino Selfportrait.jpg|[[Parmigianino]], ''[[Self-portrait in a Convex Mirror]]'', c. 1523–1524
File:Pythagoras with tablet of ratios.jpg|Pythagoras with tablet of ratios, in [[Raphael]]'s ''[[The School of Athens]]'', 1509
<!--File:Jiajing Emperor on his state barge.jpg|[[Oblique projection]]: the Emperor [[Jiajing]] on his state barge. [[Ming dynasty]] scroll painting, China, c. 1538-->
File:Xu Yang - Entrance and yard of a yamen.jpg|[[Oblique projection]]: ''Entrance and yard of a yamen''. Detail of scroll about [[Suzhou]] by Xu Yang, ordered by the [[Qianlong Emperor]]. 18th century
File:3 Brettspiele.jpg|[[Oblique projection]]: women playing [[Shogi]], [[Go (game)|Go]] and [[Sugoroku#Ban-sugoroku|Ban-sugoroku]] board games. Painting by [[Torii Kiyonaga]], Japan, c. 1780
</gallery>

===Golden ratio===

{{further|List of works designed with the golden ratio}}

The [[golden ratio]] (roughly equal to 1.618) was known to [[Euclid]].<ref>{{cite web |last=Joyce |first=David E. |author-link=David E. Joyce (mathematician) |title=Euclid's Elements, Book II, Proposition 11 |url=http://aleph0.clarku.edu/~djoyce/elements/bookII/propII11.html |publisher=Clark University |access-date=24 September 2015 |date=1996}}</ref> The golden ratio has persistently been claimed<ref>{{cite journal |author1=Seghers, M. J. |author2=Longacre, J. J. |author3=Destefano, G. A. |title=The Golden Proportion and Beauty |journal=Plastic and Reconstructive Surgery |volume=34 |issue=4 |pages=382–386 |date=1964 |doi=10.1097/00006534-196410000-00007|s2cid=70643014 }}</ref><ref>{{cite book |author=Mainzer, Klaus |title=Symmetries of Nature: A Handbook for Philosophy of Nature and Science |publisher=Walter de Gruyter |date=1996 |page=118}}</ref><ref>{{cite web |url=http://mathsforeurope.digibel.be/amphi.htm |title=Mathematical properties in ancient theatres and amphitheatres |access-date=29 January 2014 |archive-url=https://web.archive.org/web/20170715212252/http://mathsforeurope.digibel.be/amphi.htm |archive-date=15 July 2017 |url-status=dead }}</ref><ref>{{cite web |url=http://www.the-colosseum.net/architecture/ellipsis.htm |title=Architecture: Ellipse? |publisher=The-Colosseum.net |access-date=29 January 2014 |archive-date=11 December 2013 |archive-url=https://web.archive.org/web/20131211055626/http://www.the-colosseum.net/architecture/ellipsis.htm |url-status=dead }}</ref> in modern times to have been used in art and [[Mathematics and architecture|architecture]] by the ancients in Egypt, Greece and elsewhere, without reliable evidence.<ref name=Markowsky>{{cite journal |author=Markowsky, George |url=http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf |title=Misconceptions about the Golden Ratio |journal=The College Mathematics Journal |volume=23 |issue=1 |pages=2–19 |date=January 1992 |doi=10.2307/2686193 |jstor=2686193 |access-date=2015-06-26 |archive-url=https://web.archive.org/web/20080408200850/http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf |archive-date=2008-04-08 |url-status=dead }}</ref> The claim may derive from confusion with "golden mean", which to the Ancient Greeks meant "avoidance of excess in either direction", not a ratio.<ref name=Markowsky/> [[Pyramidology|Pyramidologists]] since the 19th century have argued on dubious mathematical grounds for the golden ratio in pyramid design.{{efn|The ratio of the slant height to half the base length is 1.619, less than 1% from the golden ratio, suggesting the use of the [[Kepler triangle]] (face angle 51°49').<ref name=Markowsky/><ref>{{cite book |first=Socrates G. |last=Taseos |title=Back in Time 3104 B.C. to the Great Pyramid |publisher=SOC Publishers |date=1990}}</ref> However, other ratios are within measurement error of the same shape, and historical evidence suggests that simple integer ratios are more likely to have been used.<ref>{{cite journal |last=Bartlett |first=Christopher |date=May 2014 |doi=10.1007/s00004-014-0193-9 |issue = 2 |journal = Nexus Network Journal |pages = 299–311 |title = The Design of The Great Pyramid of Khufu |volume = 16| s2cid = 122021107 |doi-access = free }}</ref><ref>{{cite book |last=Herz-Fischler |first=Roger |isbn=0-88920-324-5 |mr=1788996 |publisher=[[Wilfrid Laurier University Press]] | location = Waterloo, Ontario |title=The Shape of the Great Pyramid |year=2000}}</ref>}}<!--end efn--> The [[Parthenon]], a 5th-century BC temple in Athens, has been claimed to use the golden ratio in its [[façade]] and floor plan,<ref>{{cite book |author=Huntley, H.E. |title=The Divine Proportion |url=https://archive.org/details/divineproportion0000hunt |url-access=registration |publisher=Dover |year=1970}}</ref><ref>{{cite book |author=Hemenway, Priya |year=2005 |title=Divine Proportion: Phi In Art, Nature, and Science |publisher=Sterling |page=96}}</ref><ref>{{cite web |last1=Usvat |first1=Liliana |title=Mathematics of the Parthenon |url=http://www.mathematicsmagazine.com/Articles/Mathematics_ofTheParthenon.php#.VYqGl0Z0dIQ |publisher=Mathematics Magazine |access-date=24 June 2015}}</ref> but these claims too are disproved by measurement.<ref name=Markowsky/> The [[Great Mosque of Kairouan]] in Tunisia has similarly been claimed to use the golden ratio in its design,<ref>{{cite journal |last1=Boussora |first1=Kenza |last2=Mazouz |first2=Said |title=The Use of the Golden Section in the Great Mosque of Kairouan |journal=Nexus Network Journal |volume=6 |issue=1 |date=Spring 2004 |pages=7–16 |doi=10.1007/s00004-004-0002-y |quote=The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organisation. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period. |doi-access=free }}</ref> but the ratio does not appear in the original parts of the mosque.<ref>{{cite journal |last1=Brinkworth |first1=Peter |last2=Scott |first2=Paul |title=The Place of Mathematics |journal=Australian Mathematics Teacher |volume=57 |issue=3 |year=2001 |page=2}}</ref> The historian of architecture [[Frederik Macody Lund]] argued in 1919 that the [[Cathedral of Chartres]] (12th century), [[Notre-Dame of Laon]] (1157–1205) and [[Notre Dame de Paris]] (1160) are designed according to the golden ratio,<ref>{{cite book |last=Chanfón Olmos |first=Carlos |title=Curso sobre Proporción. Procedimientos reguladors en construcción |publisher=Convenio de intercambio Unam–Uady. México – Mérica |date=1991}}</ref> drawing regulator lines to make his case. Other scholars argue that until Pacioli's work in 1509, the golden ratio was unknown to artists and architects.<ref>{{cite book |last=Livio |first=Mario |author-link=Mario Livio |year=2002 |title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number |url=https://archive.org/details/goldenratio00mari |url-access=registration |bibcode=2002grsp.book.....L |isbn=9780767908160 }}</ref> For example, the height and width of the front of Notre-Dame of Laon have the ratio 8/5 or 1.6, not 1.618. Such [[Fibonacci ratio]]s quickly become hard to distinguish from the golden ratio.<ref>{{cite journal |last=Smith |first=Norman A. F. |title=Cathedral Studies: Engineering or History |url=http://pubs-newcomen.com/tfiles/73ap095.pdf |journal=[[Transactions of the Newcomen Society]] |date=2001 |volume=73 |pages=95–137 |doi=10.1179/tns.2001.005 |s2cid=110300481 |url-status=usurped |archive-url=https://www.webcitation.org/6dh0wipJN?url=http://pubs-newcomen.com/tfiles/73ap095.pdf |archive-date=2015-12-11 }}</ref> After Pacioli, the golden ratio is more definitely discernible in artworks including Leonardo's ''[[Mona Lisa]]''.<ref>{{cite news|last1=McVeigh|first1=Karen|title=Why golden ratio pleases the eye: US academic says he knows art secret|url=https://www.theguardian.com/artanddesign/2009/dec/28/golden-ratio-us-academic|newspaper=The Guardian|access-date=27 October 2015|date=28 December 2009}}</ref>

Another ratio, the only other morphic number,<ref>{{cite journal |last1=Aarts |first1=J. |last2=Fokkink |first2=R. |last3=Kruijtzer |first3=G. |title=Morphic numbers |journal=Nieuw Arch. Wiskd. |series=5 |volume=2 |issue=1 |year=2001 |pages=56–58 |url=http://www.nieuwarchief.nl/serie5/pdf/naw5-2001-02-1-056.pdf}}</ref> was named the [[plastic_ratio|plastic number]]{{efn|'[[Plastic arts|Plastic]]' named the ability to take on a chosen three-dimensional shape.}} in 1928 by the Dutch architect [[Hans van der Laan]] (originally named ''le nombre radiant'' in French).<ref name=Padovan>{{cite journal |url=http://www.nexusjournal.com/conferences/N2002-Padovan.html |title=Dom Hans van Der Laan and the Plastic Number |last=Padovan |first=Richard |author-link=Richard Padovan |journal=Nexus IV: Architecture and Mathematics |editor1=Williams, Kim|editor2=Francisco Rodrigues, Jose |pages=181–193 |year=2002}}</ref> Its value is the solution of the [[cubic function|cubic equation]]

:<math>x^3=x+1\,</math>,

an irrational number which is approximately 1.325. According to the architect [[Richard Padovan]], this has characteristic ratios {{sfrac|3|4}} and {{sfrac|1|7}}, which govern the limits of human perception in relating one physical size to another. Van der Laan used these ratios when designing the 1967 [[St. Benedictusberg Abbey]] church in the Netherlands.<ref name=Padovan/>

<gallery mode="nolines" heights="180px" widths="180px">
File:Mathematical Pyramid.svg|Base:hypotenuse(b:a) ratios for the [[Great Pyramid of Giza|Pyramid of Khufu]] could be: 1:φ ([[Kepler triangle]]), 3:5 ([[Special right triangles|3-4-5 Triangle]]), or 1:4/π
File:Laon Cathedral's regulator lines.jpg|Supposed ratios: [[Notre-Dame of Laon]]
File:Mona Lisa Golden Ratio.jpg|Golden rectangles superimposed on the [[Mona Lisa]]
File:Interieur bovenkerk, zicht op de middenbeuk met koorbanken voor de monniken - Mamelis - 20536587 - RCE.jpg|The 1967 [[St. Benedictusberg Abbey]] church by [[Hans van der Laan]] has [[plastic ratio]] proportions.
</gallery>

===Planar symmetries===
{{further|Planar symmetry|Wallpaper group|Islamic geometric patterns|Kilim}}
[[File:Carpet with Double Medallion.jpg|thumb|upright|left|Powerful presence:<ref name=Salingaros/> carpet with double medallion. Central [[Anatolia]] (Konya – Karapınar), circa 1600. [[Alâeddin Mosque]]]]

[[Planar symmetry|Planar symmetries]] have for millennia been exploited in artworks such as [[carpet]]s, lattices, textiles and tilings.<ref name=Cucker89/><ref name=Lerner/><ref name=Ellison/><ref name=Castera/>

Many traditional rugs, whether pile carpets or flatweave [[kilim]]s, are divided into a central field and a framing border; both can have symmetries, though in handwoven carpets these are often slightly broken by small details, variations of pattern and shifts in colour introduced by the weaver.<ref name=Cucker89>{{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=89–102}}</ref> In kilims from [[Anatolia]], the [[motif (textile arts)|motifs]] used are themselves usually symmetrical. The general layout, too, is usually present, with arrangements such as stripes, stripes alternating with rows of motifs, and packed arrays of roughly hexagonal motifs. The field is commonly laid out as a wallpaper with a [[wallpaper group]] such as pmm, while the border may be laid out as a frieze of [[frieze group]] pm11, pmm2 or pma2. Turkish and Central Asian kilims often have three or more borders in different frieze groups. Weavers certainly had the intention of symmetry, without explicit knowledge of its mathematics.<ref name=Cucker89/>
The mathematician and architectural theorist [[Nikos Salingaros]] suggests that the "powerful presence"<ref name=Salingaros/> (aesthetic effect) of a "great carpet"<ref name=Salingaros/> such as the best Konya two-medallion carpets of the 17th century is created by mathematical techniques related to the theories of the architect [[Christopher Alexander]]. These techniques include making opposites couple; opposing colour values; differentiating areas geometrically, whether by using complementary shapes or balancing the directionality of sharp angles; providing small-scale complexity (from the knot level upwards) and both small- and large-scale symmetry; repeating elements at a hierarchy of different scales (with a ratio of about 2.7 from each level to the next). Salingaros argues that "all successful carpets satisfy at least nine of the above ten rules", and suggests that it might be possible to create a metric from these rules.<ref name=Salingaros>{{cite journal |last=Salingaros |first=Nikos |title=The 'life' of a carpet: an application of the Alexander rules |url=http://zeta.math.utsa.edu/~yxk833/life.carpet.html |journal=8th International Conference on Oriental Carpets |date=November 1996 |access-date=2015-10-02 |archive-date=2016-03-05 |archive-url=https://web.archive.org/web/20160305001146/http://zeta.math.utsa.edu/~yxk833/life.carpet.html |url-status=dead }} Reprinted in {{cite book |editor1-last=Eiland |editor1-first=M. |editor2-last=Pinner |editor2-first=M. |title=Oriental Carpet and Textile Studies V |date=1998 |publisher=Conference on Oriental Carpets |location=Danville, CA}}</ref>

Elaborate lattices are found in Indian [[Jali]] work, carved in marble to adorn tombs and palaces.<ref name=Lerner>{{cite book |last1=Lerner |first1=Martin |title=The flame and the lotus: Indian and Southeast Asian art from the Kronos collections |date=1984 |publisher=Metropolitan Museum of Art |edition=Exhibition Catalogue |url=http://libmma.contentdm.oclc.org/cdm/compoundobject/collection/p15324coll10/id/105494}}</ref> Chinese lattices, always with some symmetry, exist in 14 of the 17 wallpaper groups; they often have mirror, double mirror, or rotational symmetry. Some have a central medallion, and some have a border in a frieze group.<ref name=Cucker103>{{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=103–106}}</ref> Many Chinese lattices have been analysed mathematically by Daniel S. Dye; he identifies [[Sichuan]] as the centre of the craft.<ref>{{cite book |last1=Dye |first1=Daniel S. |title=Chinese Lattice Designs |url=https://archive.org/details/chineselatticede00dyed |url-access=registration |date=1974 |publisher=Dover |pages=[https://archive.org/details/chineselatticede00dyed/page/30 30–39]|isbn=9780486230962 }}</ref>

[[File:Girih tiles.svg|thumb|upright|[[Girih tiles]]]]

Symmetries are prominent in [[textile arts]] including [[quilting]],<ref name=Ellison>{{cite book |last1=Ellison |first1=Elaine |last2=Venters |first2=Diana |publisher=Key Curriculum |title=Mathematical Quilts: No Sewing Required |year=1999}}</ref> [[knitting]],<ref>{{cite journal |last1=b<!--sic, l/c-->elcastro |first1=s<!--sic, l/c-->arah-m<!--sic, l/c-->arie |title=Adventures in Mathematical Knitting |url=http://www.americanscientist.org/issues/feature/adventures-in-mathematical-knitting/1 |journal=American Scientist |volume=101 |issue=2 |page=124 |date=2013 |doi=10.1511/2013.101.124 |access-date=2015-06-24 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304032908/http://www.americanscientist.org/issues/feature/adventures-in-mathematical-knitting/1 |url-status=dead }}</ref> [[cross-stitch]], [[crochet]],<ref>{{cite book |author=Taimina, Daina |title=Crocheting Adventures with Hyperbolic Planes|title-link= Crocheting Adventures with Hyperbolic Planes |year=2009 |publisher=A K Peters |isbn=978-1-56881-452-0}}</ref> [[embroidery]]<ref>Snook, Barbara. ''Florentine Embroidery''. Scribner, Second edition 1967.</ref><ref>Williams, Elsa S. ''Bargello: Florentine Canvas Work''. Van Nostrand Reinhold, 1967.</ref> and [[weaving]],<ref>{{cite journal |doi=10.2307/2690105 |author1=Grünbaum, Branko |author-link1=Branko Grünbaum |author2=Shephard, Geoffrey C. |title=Satins and Twills: An Introduction to the Geometry of Fabrics |date=May 1980 |journal=[[Mathematics Magazine]] |volume=53 |issue=3 |pages=139–161 |jstor=2690105 |hdl=10338.dmlcz/104026 |hdl-access=free }}</ref> where they may be purely decorative or may be marks of status.<ref name=Gamwell423>{{cite book |last1=Gamwell |first1=Lynn|author1-link= Lynn Gamwell |title=Mathematics and Art: A Cultural History |date=2015 |publisher=Princeton University Press |page=423 |isbn=978-0-691-16528-8}}</ref> [[Rotational symmetry]] is found in circular structures such as [[dome]]s; these are sometimes elaborately decorated with symmetric patterns inside and out, as at the 1619 [[Sheikh Lotfollah Mosque]] in [[Isfahan]].<ref>{{cite book |last1=Baker |first1=Patricia L. |last2=Smith |first2=Hilary |title=Iran |url=https://books.google.com/books?id=a40CkMNqU8AC&pg=PA108 |edition=3 |year=2009 |publisher=Bradt Travel Guides |isbn=978-1-84162-289-7 |page=107}}</ref> Items of embroidery and [[lace]] work such as tablecloths and table mats, made using bobbins or by [[tatting]], can have a wide variety of reflectional and rotational symmetries which are being explored mathematically.<ref>{{cite journal |last1=Irvine |first1=Veronika |last2=Ruskey |first2=Frank |title=Developing a Mathematical Model for Bobbin Lace |journal=[[Journal of Mathematics and the Arts]] |date=2014 |volume=8 |issue=3–4 |pages=95–110 |doi=10.1080/17513472.2014.982938 |arxiv=1406.1532|bibcode=2014arXiv1406.1532I |s2cid=119168759 }}</ref>

[[Islamic art]] [[Islamic geometric patterns|exploits symmetries]] in many of its artforms, notably in [[girih tiles|girih]] tilings. These are formed using a set of five tile shapes, namely a regular decagon, an elongated hexagon, a bow tie, a rhombus, and a regular pentagon. All the sides of these tiles have the same length; and all their angles are multiples of 36° (π/5 [[radian]]s), offering fivefold and tenfold symmetries. The tiles are decorated with [[strapwork]] lines (girih), generally more visible than the tile boundaries. In 2007, the physicists [[Peter Lu]] and [[Paul Steinhardt]]  argued that girih resembled [[quasicrystalline]] [[Penrose tiling]]s.<ref>{{cite journal |last1=Lu |first1=Peter J. |last2=Steinhardt |first2=Paul J. |year=2007 |title=Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture |journal=[[Science (journal)|Science]] |volume=315 |pages=1106–1110 |doi=10.1126/science.1135491 |pmid=17322056 |issue=5815 |bibcode=2007Sci...315.1106L|s2cid=10374218 }}</ref> Elaborate geometric [[zellige]] tilework is a distinctive element in [[Morocco|Moroccan]] architecture.<ref name=Castera>{{cite book |last1=Castera |first1=Jean Marc |last2=Peuriot |first2=Francoise |title=Arabesques. Decorative Art in Morocco |date=1999 |publisher=Art Creation Realisation |isbn=978-2-86770-124-5}}</ref> [[Muqarnas]] vaults are three-dimensional but were designed in two dimensions with drawings of geometrical cells.<ref>{{cite web |last1=van den Hoeven |first1=Saskia |last2=van der Veen |first2=Maartje |title=Muqarnas-Mathematics in Islamic Arts |url=http://www.wiskuu.nl/muqarnas/Muqarnas_english8.pdf |access-date=15 January 2016 |archive-url=https://web.archive.org/web/20130927070005/http://www.wiskuu.nl/muqarnas/Muqarnas_english8.pdf |archive-date=27 September 2013}}</ref><!--also at http://www.jphogendijk.nl/projects/muqarnas2010.pdf-->

<gallery mode="nolines" heights="180px" widths="180px">
File:Hotamis Kilim.jpg|Hotamis kilim (detail), central [[Anatolia]], early 19th century
File:Ming flower brocade (cropped)2.jpg|Detail of a [[Ming Dynasty]] brocade, using a [[Hexagonal tiling|chamfered hexagonal lattice]] pattern
File:Salim Chishti Tomb-2.jpg|''[[Jali|Jaali]]'' marble lattice at tomb of [[Salim Chishti]], [[Fatehpur Sikri]], [[India]]
<!--need a Chinese Lattice as in Dye or Cucker-->
File:Florentine Bargello Pattern.png|[[Symmetry|Symmetries]]: Florentine [[Bargello (needlework)|Bargello pattern]] tapestry work
File:Isfahan Lotfollah mosque ceiling symmetric.jpg|Ceiling of the [[Sheikh Lotfollah Mosque]], [[Isfahan]], 1619
File:Frivolité.jpg|[[Rotational symmetry]] in [[lace]]: [[tatting]] work
File:Darb-i Imam shrine spandrel.JPG|[[Girih tiles]]: patterns at large and small scales on a [[spandrel]] from the [[Darb-i Imam]] shrine, Isfahan, 1453
File:Fes Medersa Bou Inania Mosaique2.jpg|[[Tessellation]]s: [[zellige]] mosaic tiles at [[Bou Inania Madrasa]], [[Fes]], [[Morocco]]
File:Mezquita Shah, Isfahán, Irán, 2016-09-20, DD 64 (detail).jpg|The complex geometry and tilings of the [[muqarnas]] vaulting in the Sheikh Lotfollah Mosque, Isfahan
File:Topkapi Scroll p294 muqarnas.JPG|Architect's plan of a muqarnas quarter vault. [[Topkapı Scroll]]
File:Tupa-inca-tunic.png|[[Topa Inca Yupanqui|Tupa Inca]] tunic from [[Peru]], 1450 –1540, an [[Andean textiles|Andean textile]] denoting high rank<ref name=Gamwell423/>
</gallery>

===Polyhedra===

The [[Platonic solid]]s and other [[polyhedron|polyhedra]] are a recurring theme in Western art. They are found, for instance, in a marble mosaic featuring the [[small stellated dodecahedron]], attributed to Paolo Uccello, in the floor of the [[San Marco di Venezia|San Marco Basilica]] in Venice;<ref name="Hart"/> in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations for [[Luca Pacioli]]'s 1509 book ''The Divine Proportion'';<ref name="Hart"/> as a glass [[rhombicuboctahedron]] in Jacopo de Barbari's portrait of Pacioli, painted in 1495;<ref name="Hart"/> in the truncated polyhedron (and various other mathematical objects) in [[Albrecht Dürer]]'s engraving [[Melencolia I]];<ref name="Hart"/> and in [[Salvador Dalí]]'s painting ''The Last Supper'' in which Christ and his disciples are pictured inside a giant [[dodecahedron]].<ref>{{cite journal |last=Markowsky |first=George |title=Book review: ''The Golden Ratio'' |journal=Notices of the American Mathematical Society |volume=52 |issue=3 |pages=344–347 |date=March 2005 |url=https://www.ams.org/notices/200503/rev-markowsky.pdf}}</ref>

[[Albrecht Dürer]] (1471–1528) was a [[German people|German]] Renaissance [[printmaker]] who made important contributions to polyhedral literature in his 1525 book, ''Underweysung der Messung (Education on Measurement)'', meant to teach the subjects of [[linear perspective]], [[geometry]] in [[architecture]], [[Platonic solids]], and [[regular polygons]]. Dürer was likely influenced by the works of [[Luca Pacioli]] and [[Piero della Francesca]] during his trips to [[Italy]].<ref>{{cite book |last=Panofsky |first=E. |title=The Life and Art of Albrecht Durer |publisher=Princeton |year=1955}}</ref> While the examples of perspective in ''Underweysung der Messung'' are underdeveloped and contain inaccuracies, there is a detailed discussion of polyhedra. Dürer is also the first to introduce in text the idea of [[Net (polyhedron)|polyhedral nets]], polyhedra unfolded to lie flat for printing.<ref>{{cite web |last=Hart |first=George W. |url=http://www.georgehart.com/virtual-polyhedra/durer.html |title=Dürer's Polyhedra |access-date=13 August 2009}}</ref> Dürer published another influential book on [[human proportions]] called ''Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion)'' in 1528.<ref>{{cite book |last=Dürer |first=Albrecht |author-link=Albrecht Dürer |title=Hierinn sind begriffen vier Bucher von menschlicher Proportion|url=https://archive.org/details/hierinnsindbegri00dure |access-date=24 June 2015 |location=Nuremberg |date=1528}}</ref>

Dürer's well-known engraving ''[[Melencolia I]]'' depicts a frustrated thinker sitting by a [[truncated triangular trapezohedron]] and a [[magic square]].<ref name=Ziegler/> These two objects, and the engraving as a whole, have been the subject of more modern interpretation than the contents of almost any other print,<ref name=Ziegler/><ref>{{cite journal |author=Schreiber, P. |title=A New Hypothesis on Durer's Enigmatic Polyhedron in His Copper Engraving 'Melencolia I' |journal=[[Historia Mathematica]] |volume=26 |issue=4 |pages=369–377 |date=1999 |doi=10.1006/hmat.1999.2245|doi-access=free }}</ref><ref>{{cite book |first=Campbell |last=Dodgson |title=Albrecht Dürer |location= London |publisher=Medici Society |year=1926 |page=94}}</ref> including a two-volume book by Peter-Klaus Schuster,<ref name="Schuster">{{cite book |last=Schuster |first=Peter-Klaus |title=Melencolia I: Dürers Denkbild |pages=17–83 |location=Berlin |publisher=Gebr. Mann Verlag |year=1991}}</ref> and an influential discussion in [[Erwin Panofsky]]'s monograph of Dürer.<ref name=Ziegler>{{cite news |last=Ziegler |first=Günter M. |title=Dürer's polyhedron: 5 theories that explain Melencolia's crazy cube |url=https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/dec/03/durers-polyhedron-5-theories-that-explain-melencolias-crazy-cube |newspaper=[[The Guardian]] |access-date=27 October 2015 |date=3 December 2014}}</ref><ref name="Panofsky">{{cite book |last1=Panofsky |first1=Erwin |author1-link=Erwin Panofsky |last2=Klibansky |first2=Raymond |author2-link=Raymond Klibansky |last3=Saxl |first3=Fritz |author3-link=Fritz Saxl |title=Saturn and Melancholy |publisher=Basic Books |year=1964 |url=http://quod.lib.umich.edu/cgi/t/text/text-idx?c=genpub;cc=genpub;q1=Klibansky;op2=and;op3=and;rgn=works;rgn1=author;rgn2=author;rgn3=author;view=toc;idno=0431529.0001.001}}</ref>

[[Salvador Dalí]]'s 1954 painting ''[[Corpus Hypercubus]]'' uniquely depicts the cross of Christ as an unfolded three-dimensional net for a [[hypercube]], also known as a [[tesseract]]: the unfolding of a tesseract into these eight cubes is analogous to unfolding the sides of a cube into a cross shape of six squares, here representing the divine perspective with a four-dimensional regular polyhedron.<ref name="Rucker">{{cite book |last=Rucker |first=Rudy |title=The Fourth Dimension: Toward a Geometry of Higher Reality |publisher=Courier Corporation |date=2014 |url=https://books.google.com/books?id=0xReBAAAQBAJ |isbn=978-0-486-79819-6 }}</ref><ref name=CorpusHypercubus>{{cite web |title=Crucifixion (Corpus Hypercubus) |url=http://www.metmuseum.org/collection/the-collection-online/search/488880 |publisher=Metropolitan Museum of Art |access-date=5 September 2015}}</ref> The painting shows the figure of Christ in front of the tessaract; he would normally be shown fixed with nails to the cross, but there are no nails in the painting. Instead, there are four small cubes in front of his body, at the corners of the frontmost of the eight tessaract cubes. The mathematician [[Thomas Banchoff]] states that Dalí was trying to go beyond the three-dimensional world, while the poet and art critic [[Kelly Grovier]] says that "The painting seems to have cracked the link between the spirituality of Christ's salvation and the materiality of geometric and physical forces. It appears to bridge the divide that many feel separates science from religion."<ref name="Macdonald 2016">{{cite web |last1=Macdonald |first1=Fiona |title=The painter who entered the fourth dimension |url=https://www.bbc.com/culture/article/20160511-the-painter-who-entered-the-fourth-dimension |publisher=[[BBC]] |access-date=8 February 2022 |date=11 May 2016}}</ref>

<gallery mode="nolines" heights="180px" widths="180px">
File:Leonardo polyhedra.png|The first printed illustration of a [[rhombicuboctahedron]], by [[Leonardo da Vinci]], published in ''[[De Divina Proportione]]'', 1509
File:Icosahedron-spinoza.jpg|[[Icosahedron]] as a part of the monument to [[Baruch Spinoza]], [[Amsterdam]]
</gallery>

===Fractal dimensions===

[[File:Batik pedalaman - parang klithik.JPG|thumb|[[Batik]]s from [[Surakarta]], Java, like this ''parang klithik'' sword pattern, have a [[fractal dimension]] between 1.2 and 1.5.]]

Traditional Indonesian wax-resist [[batik]] designs on cloth combine [[representation (arts)|representational]] motifs (such as floral and vegetal elements) with abstract and somewhat chaotic elements, including imprecision in applying the wax resist, and random variation introduced by cracking of the wax. Batik designs have a [[fractal dimension]] between 1 and 2, varying in different regional styles. For example, the batik of [[Cirebon]] has a fractal dimension of 1.1; the batiks of [[Yogyakarta]] and [[Surakarta]] (Solo) in Central [[Java]] have a fractal dimension of 1.2 to 1.5; and the batiks of [[Rembang Regency|Lasem]] on the north coast of Java and of [[Tasikmalaya]] in West Java have a fractal dimension between 1.5 and 1.7.<ref>{{cite journal |last1=Lukman |first1=Muhamad |last2=Hariadi |first2=Yun |last3=Destiarmand |first3=Achmad Haldani |title=Batik Fractal: Traditional Art to Modern Complexity |journal=Proceeding Generative Art X, Milan, Italy |date=2007 }}</ref>

The [[drip painting]] works of the modern artist [[Jackson Pollock]] are similarly distinctive in their fractal dimension. His 1948 ''Number 14'' has a coastline-like dimension of 1.45, while his later paintings had successively higher fractal dimensions and accordingly more elaborate patterns. One of his last works, ''Blue Poles'', took six months to create, and has the fractal dimension of 1.72.<ref>{{cite news |last=Ouellette |first=Jennifer |title=Pollock's Fractals |url=http://discovermagazine.com/2001/nov/featpollock |access-date=26 September 2016 |agency=Discover Magazine |date=November 2001}}</ref>

==A complex relationship==

The astronomer [[Galileo Galilei]] in his ''[[Il Saggiatore]]'' wrote that "[The universe] is written in [[patterns in nature|the language of mathematics]], and its characters are triangles, circles, and other geometric figures."<ref>{{cite book |last=Galilei |first=Galileo |author-link=Galileo Galilei |title=The Assayer |year=1623}}, as translated in {{cite book |last=Drake |first=Stillman |author-link=Stillman Drake |year=1957 |title=Discoveries and Opinions of Galileo |publisher=Doubleday |pages=[https://archive.org/details/discoveriesopini00gali_0/page/237 237–238] |isbn=978-0-385-09239-5 |url=https://archive.org/details/discoveriesopini00gali_0/page/237 }}</ref> Artists who strive and seek to study nature must first, in Galileo's view, fully understand mathematics. Mathematicians, conversely, have sought to interpret and analyse art through the lens of geometry and rationality. The mathematician [[Felipe Cucker]] suggests that mathematics, and especially geometry, is a source of rules for "rule-driven artistic creation", though not the only one.<ref>{{cite book |last=Cucker |first=Felipe |author-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 |page=381}}</ref> Some of the many strands of the resulting complex relationship<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 |page=10}}</ref> are described below.

[[File:Ghhardy@72.jpg|thumb|upright|The mathematician [[G. H. Hardy]] defined a set of criteria for [[mathematical beauty]].]]

===Mathematics as an art===

{{main|Mathematical beauty}}

The mathematician Jerry P. King describes mathematics as an art, stating that "the keys to mathematics are beauty and elegance and not dullness and technicality", and that beauty is the motivating force for mathematical research.<ref>{{cite book |last=King |first=Jerry P. |title=The Art of Mathematics |date=1992 |publisher=Fawcett Columbine |isbn=978-0-449-90835-8 |pages=8–9}}</ref> King cites the mathematician [[G. H. Hardy]]'s 1940 essay ''[[A Mathematician's Apology]]''. In it, Hardy discusses why he finds two theorems of [[classical antiquity|classical times]] as first rate, namely [[Euclid]]'s proof there are infinitely many [[prime number]]s, and the proof that the square root of 2 is [[irrational number|irrational]]. King evaluates this last against Hardy's criteria for [[mathematical beauty|mathematical elegance]]: "''seriousness, depth, generality, unexpectedness, inevitability'', and ''economy''" (King's italics), and describes the proof as "aesthetically pleasing".<ref>{{cite book |last=King |first=Jerry P. |title=The Art of Mathematics |date=1992 |publisher=Fawcett Columbine |isbn=978-0-449-90835-8 |pages=135–139}}</ref> The Hungarian mathematician [[Paul Erdős]] agreed that mathematics possessed beauty but considered the reasons beyond explanation: "Why are numbers beautiful? It's like asking why is [[Symphony No. 9 (Beethoven)|Beethoven's Ninth Symphony]] beautiful. If you don't see why, someone can't tell you. I ''know'' numbers are beautiful."<ref>{{cite book |last=Devlin |first=Keith |title=The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip |publisher=Basic Books |year=2000 |page=140 |chapter=Do Mathematicians Have Different Brains? |chapter-url=https://books.google.com/books?id=AJdmfYEaLG4C |isbn=978-0-465-01619-8}}</ref>

===Mathematical tools for art===

{{further|List of mathematical artists|fractal art|computer art}}

Mathematics can be discerned in many of the arts, such as [[music]], [[dance]],<ref>{{cite web |last=Wasilewska |first=Katarzyna |title=Mathematics in the World of Dance |url=http://archive.bridgesmathart.org/2012/bridges2012-453.pdf |publisher=Bridges |access-date=1 September 2015 |date=2012}}</ref> [[painting]], [[architecture]], and [[sculpture]]. Each of these is richly associated with mathematics.<ref name=Malkevitch>{{cite web |last=Malkevitch |first=Joseph |title=Mathematics and Art |url=https://www.ams.org/samplings/feature-column/fcarc-art1 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref> Among the connections to the visual arts, mathematics can provide tools for artists, such as the rules of [[linear perspective]] as described by [[Brook Taylor]] and [[Johann Lambert]], or the methods of [[descriptive geometry]], now applied in software modelling of solids, dating back to Albrecht Dürer and [[Gaspard Monge]].<ref>{{cite web |last=Malkevitch |first=Joseph |title=Mathematics and Art. 2. Mathematical tools for artists |url=https://www.ams.org/samplings/feature-column/fcarc-art2 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref> Artists from Luca Pacioli in the [[Middle Ages]] and Leonardo da Vinci and Albrecht Dürer in the [[Renaissance]] have made use of and developed mathematical ideas in the pursuit of their artistic work.<ref name=Malkevitch/><ref>{{cite web |title=Math and Art: The Good, the Bad, and the Pretty |url=http://www.maa.org/meetings/calendar-events/math-and-art-the-good-the-bad-and-the-pretty |publisher=Mathematical Association of America |access-date=2 September 2015 |archive-date=9 September 2015 |archive-url=https://web.archive.org/web/20150909085252/http://www.maa.org/meetings/calendar-events/math-and-art-the-good-the-bad-and-the-pretty |url-status=dead }}</ref> The use of perspective began, despite some embryonic usages in the architecture of Ancient Greece, with Italian painters such as [[Giotto]] in the 13th century; rules such as the [[vanishing point]] were first formulated by [[Brunelleschi]] in about 1413,<ref name=StAndrews>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |title=Mathematics and art – perspective |url=http://www-history.mcs.st-and.ac.uk/HistTopics/Art.html |publisher=University of St Andrews |access-date=1 September 2015 |date=January 2003}}</ref> his theory influencing Leonardo and Dürer. [[Isaac Newton]]'s work on the [[visible spectrum|optical spectrum]] influenced [[Johann Wolfgang Goethe|Goethe]]'s ''[[Theory of Colours]]'' and in turn artists such as [[Philipp Otto Runge]], [[J. M. W. Turner]],<ref>{{cite web |last=Cohen |first=Louise |title=How to spin the colour wheel, by Turner, Malevich and more |url=http://www.tate.org.uk/context-comment/articles/how-to-spin-the-colour-wheel |publisher=Tate Gallery |access-date=4 September 2015 |date=1 July 2014}}</ref> the [[Pre-Raphaelites]] and [[Wassily Kandinsky]].<ref>{{cite book |last=Kemp |first=Martin |title=The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat |date=1992 |publisher=Yale University Press |isbn=978-968-867-185-6}}</ref><ref name="Gage1999">{{cite book |last=Gage |first=John |title=Color and Culture: Practice and Meaning from Antiquity to Abstraction |url=https://books.google.com/books?id=oq_GtjmoTNgC&pg=PA207 |year=1999 |publisher=University of California Press |isbn=978-0-520-22225-0 |page=207}}</ref> Artists may also choose to analyse the [[symmetry]] of a scene.<ref>{{cite web |last=Malkevitch |first=Joseph |title=Mathematics and Art. 3. Symmetry |url=https://www.ams.org/samplings/feature-column/fcarc-art3 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref> Tools may be applied by mathematicians who are exploring art, or artists inspired by mathematics, such as [[M. C. Escher]] (inspired by [[H. S. M. Coxeter]]) and the architect [[Frank Gehry]], who more tenuously argued that [[computer aided design]] enabled him to express himself in a wholly new way.<ref>{{cite web |last=Malkevitch |first=Joseph |title=Mathematics and Art. 4. Mathematical artists and artist mathematicians |url=https://www.ams.org/samplings/feature-column/fcarc-art4 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref>

[[File:Octopod by syntopia.jpg|thumb|''Octopod'' by Mikael Hvidtfeldt Christensen. [[Algorithmic art]] produced with the software ''Structure Synth'']]

The artist Richard <!--no, not that Richard Wright-->Wright argues that mathematical objects that can be constructed can be seen either "as processes to simulate phenomena" or as works of "[[computer art]]". He considers the nature of mathematical thought, observing that [[fractals]] were known to mathematicians for a century before they were recognised as such. Wright concludes by stating that it is appropriate to subject mathematical objects to any methods used to "come to terms with cultural artifacts like art, the tension between objectivity and subjectivity, their metaphorical meanings and the character of representational systems." He gives as instances an image from the [[Mandelbrot set]], an image generated by a [[cellular automaton]] algorithm, and a [[rendering (computer graphics)|computer-rendered]] image, and discusses, with reference to the [[Turing test]], whether [[algorithm]]ic products can be art.<ref>{{cite journal |last=Wright |first=Richard |title=Some Issues in the Development of Computer Art as a Mathematical Art Form |journal=[[Leonardo (journal)|Leonardo]] |date=1988 |volume=1 |issue=Electronic Art, supplemental issue |pages=103–110 |doi=10.2307/1557919 |jstor=1557919}}</ref> Sasho Kalajdzievski's ''Math and Art: An Introduction to Visual Mathematics'' takes a similar approach, looking at suitably visual mathematics topics such as tilings, fractals and hyperbolic geometry.<ref>{{cite book |last=Kalajdzievski |first=Sasho |title=Math and Art: An Introduction to Visual Mathematics |date=2008 |publisher=Chapman and Hall |isbn=978-1-58488-913-7}}</ref>

Some of the first works of computer art were created by [[Desmond Paul Henry]]'s "Drawing Machine 1", an [[analogue computer|analogue machine]] based on a [[bombsight]] computer and exhibited in 1962.<ref name=Beddard>{{cite web |last=Beddard |first=Honor |title=Computer art at the V&A |url=http://www.vam.ac.uk/content/journals/research-journal/issue-02/computer-art-at-the-v-and-a/ |publisher=Victoria and Albert Museum |access-date=22 September 2015|date=2011-05-26 }}</ref><ref>{{cite news |title=Computer Does Drawings: Thousands of lines in each |agency=The Guardian |date=17 September 1962}} in Beddard, 2015.</ref> The machine was capable of creating complex, abstract, asymmetrical, curvilinear, but repetitive line drawings.<ref name=Beddard/><ref>{{cite book |last=O'Hanrahan |first=Elaine |year=2005 |title=Drawing Machines: The machine produced drawings of Dr. D. P. Henry in relation to conceptual and technological developments in machine-generated art (UK 1960–1968). Unpublished MPhil. Thesis. |publisher=John Moores University, Liverpool}} in Beddard, 2015.</ref> More recently, [[Hamid Naderi Yeganeh]] has created shapes suggestive of real world objects such as fish and birds, using formulae that are successively varied to draw families of curves or angled lines.<ref>{{cite news |title=Catch of the day: mathematician nets weird, complex fish |url=https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/feb/24/catch-of-the-day-mathematician-nets-weird-complex-fish |date=24 February 2015 |first=Alex |last=Bellos  |newspaper=The Guardian |access-date=25 September 2015}}</ref><ref name=Yeganeh2>{{cite web|url=https://www.ams.org/mathimagery/displayimage.php?album=40&pid=684#top_display_media|title="A Bird in Flight (2016)," by Hamid Naderi Yeganeh|publisher=[[American Mathematical Society]]|date=March 23, 2016|access-date=April 6, 2017|archive-date=April 2, 2019|archive-url=https://web.archive.org/web/20190402131123/http://www.ams.org/mathimagery/displayimage.php?album=40&pid=684#top_display_media|url-status=dead}}</ref><ref>{{cite news |title=Next da Vinci? Math genius using formulas to create fantastical works of art |url= http://www.cnn.com/2015/09/17/arts/math-art/ |date=September 18, 2015 |first=Stephy |last=Chung |work=[[CNN]]}}</ref> Artists such as Mikael Hvidtfeldt Christensen create works of [[generative art|generative]] or [[algorithmic art]] by writing scripts for a software system such as ''Structure Synth'': the artist effectively directs the system to apply a desired combination of mathematical operations to a chosen set of data.<ref>{{cite web |last=Levin |first=Golan |title=Generative Artists |url=http://cmuems.com/2013/a/resources/artists-generative/ |publisher=CMUEMS |access-date=27 October 2015 |date=2013}} This includes a link to [http://blog.hvidtfeldts.net/index.php/generative-art-links/ Hvidtfeldts Syntopia].</ref><ref>{{cite web |last=Verostko |first=Roman |author-link=Roman Verostko |title=The Algorists |url=http://www.verostko.com/algorist.html |access-date=27 October 2015}}</ref>

<gallery mode=nolines widths=180px heights=180px>
File:Bathsheba Grossman geometric art.jpg|Mathematical sculpture by [[Bathsheba Grossman]], 2007
File:Hartmut Skerbisch.jpg|[[Fractal]] sculpture: ''3D Fraktal 03/H/dd'' by [[:de:Hartmut Skerbisch|Hartmut Skerbisch]], 2003
File:FWF Samuel Monnier détail.jpg|[[Fibonacci word]]: detail of artwork by Samuel Monnier, 2009
File:Wiki.picture by drawing machine 1.jpg|[[Computer art]] image produced by [[Desmond Paul Henry]]'s "Drawing Machine 1", exhibited 1962
File:A Bird in Flight by Hamid Naderi Yeganeh 2016.jpg|''[[A Bird in Flight]]'', by [[Hamid Naderi Yeganeh]], 2016, constructed with a family of mathematical curves.
</gallery>

===From mathematics to art===

[[File:Les Demoiselles d'Avignon.jpg|thumb|upright|[[Proto-Cubism]]: [[Pablo Picasso]]'s 1907 painting ''[[Les Demoiselles d'Avignon]]'' uses a [[fourth dimension in art|fourth dimension]] projection to show a figure both full face and in profile.<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=315–317}}</ref>]]

{{further|Proto-Cubism|tessellation|M. C. Escher|Mathematics of paper folding|Mathematics and fiber arts}}

The mathematician and [[theoretical physicist]] [[Henri Poincaré]]'s ''[[Science and Hypothesis]]'' was widely read by the [[Cubism|Cubists]], including [[Pablo Picasso]] and [[Jean Metzinger]].<ref name="Miller"/><ref>{{cite book |last=Miller |first=Arthur I. |title=Insights of Genius: Imagery and Creativity in Science and Art |publisher=Springer |year=2012 |isbn=978-1-4612-2388-7}}</ref> Being thoroughly familiar with [[Bernhard Riemann]]'s work on non-Euclidean geometry, Poincaré was more than aware that [[Euclidean geometry]] is just one of many possible geometric configurations, rather than as an absolute objective truth. The possible existence of a [[Fourth dimension in art|fourth dimension]] inspired artists to question classical [[Perspective (graphical)#Renaissance&nbsp;: Mathematical basis|Renaissance perspective]]: [[non-Euclidean geometry]] became a valid alternative.<ref name=Henderson>{{cite book |last=Henderson |first=Linda Dalrymple |author-link=Linda Dalrymple Henderson |title=The Fourth Dimension and Non-Euclidean geometry in Modern Art |publisher=Princeton University Press |year=1983}}</ref><ref>{{cite book |last1=Antliff |first1=Mark |author2=Leighten, Patricia Dee |title=Cubism and Culture |publisher=Thames & Hudson |year=2001 |url=http://eres.lndproxy.org/edoc/AH317Antliff-09.pdf |archive-url=https://web.archive.org/web/20200726154229/http://eres.lndproxy.org/edoc/AH317Antliff-09.pdf |url-status=dead |archive-date=26 July 2020 }}</ref><ref>{{cite book |last=Everdell |first=William R. |author-link=William Everdell |title=The First Moderns: Profiles in the Origins of Twentieth-Century Thought |year=1997 |publisher=University of Chicago Press |isbn=978-0-226-22480-0 |page=[https://archive.org/details/firstmodernsprof00ever/page/312 312] |url=https://archive.org/details/firstmodernsprof00ever/page/312 }}</ref> The concept that painting could be expressed mathematically, in colour and form, contributed to Cubism, the art movement that led to [[abstract art]].<ref>{{cite book |author=Green, Christopher |title=Cubism and its Enemies, Modern Movements and Reaction in French Art, 1916–1928 |publisher=Yale University Press |year=1987 |pages=13–47}}</ref> Metzinger, in 1910, wrote that: "[Picasso] lays out a free, mobile perspective, from which that ingenious mathematician [[Maurice Princet]] has deduced a whole geometry".<ref>{{cite journal |last=Metzinger |first=Jean |author-link=Jean Metzinger |date=October–November 1910 |title=Note sur la peinture |journal=Pan |page=60}} in {{cite book |last=Miller |title=Einstein, Picasso |year=2001 |url=https://archive.org/details/einsteinpicassos00mill |url-access=registration |publisher=Basic Books |page=[https://archive.org/details/einsteinpicassos00mill/page/167 167]|isbn=9780465018598 }}</ref> Later, Metzinger wrote in his memoirs:

<blockquote>Maurice Princet joined us often&nbsp;... it was as an artist that he conceptualized mathematics, as an aesthetician that he invoked ''n''-dimensional continuums. He loved to get the artists interested in the [[Schlegel diagram|new views on space]] that had been opened up by [[Victor Schlegel|Schlegel]] and some others. He succeeded at that.<ref>{{cite book |last=Metzinger |first=Jean |author-link=Jean Metzinger |title=Le cubisme était né |year=1972 |publisher=Éditions Présence |pages=43–44}} in {{cite book |last=Ferry |first=Luc |author-link=Luc Ferry |others=Robert De Loaiza, trans. |title=Homo Aestheticus: The Invention of Taste in the Democratic Age |year=1993 |publisher=University of Chicago Press |isbn=978-0-226-24459-4 |page=[https://archive.org/details/homoaestheticusi0000ferr/page/215 215] |url=https://archive.org/details/homoaestheticusi0000ferr/page/215 }}</ref></blockquote>

The impulse to make teaching or research models of mathematical forms naturally creates objects that have symmetries and surprising or pleasing shapes. Some of these have inspired artists such as the [[Dadaism|Dadaists]] [[Man Ray]],<ref>{{cite web |title=Man Ray–Human Equations A Journey from Mathematics to Shakespeare. February 7 – May 10, 2015 |date=7 February 2015 |url=http://www.phillipscollection.org/events/2015-02-07-exhibition-man-ray-human-equations |publisher=Phillips Collection|access-date=5 September 2015}}</ref> [[Marcel Duchamp]]<ref>{{cite journal |last1=Adcock |first1=Craig |title=Duchamp's Eroticism: A Mathematical Analysis |journal=Iowa Research Online |date=1987 |volume=16 |issue=1 |pages=149–167 |url=http://ir.uiowa.edu/cgi/viewcontent.cgi?article=1208&context=dadasur|archive-url=https://web.archive.org/web/20150907181449/http://ir.uiowa.edu/cgi/viewcontent.cgi?article=1208&context=dadasur|url-status=dead|archive-date=September 7, 2015}}</ref> and [[Max Ernst]],<ref name="Elder2013">{{cite book |last=Elder |first=R. Bruce |title=DADA, Surrealism, and the Cinematic Effect |url=https://books.google.com/books?id=mhXaAgAAQBAJ&pg=PA602 |year=2013 |publisher=Wilfrid Laurier University Press |isbn=978-1-55458-641-7 |page=602}}</ref><ref>{{cite book |author=Tubbs, Robert |title=Mathematics in Twentieth-Century Literature and Art: Content, Form, Meaning |url=https://books.google.com/books?id=h1vBAwAAQBAJ&pg=PA118 |year=2014 |publisher=JHU Press |page=118 |isbn=978-1-4214-1402-7}}</ref> and following Man Ray, [[Hiroshi Sugimoto]].<ref>{{cite web|title=Hiroshi Sugimoto Conceptual Forms and Mathematical Models February 7 – May 10, 2015 |date=7 February 2015 |url=http://www.phillipscollection.org/events/2015-02-07-exhibition-hiroshi-sugimoto |publisher=Phillips Collection |access-date=5 September 2015}}</ref>

[[File:Objet mathematique by Man Ray.jpg|thumb|upright|left|[[Enneper surface]]s as [[Dadaism]]: [[Man Ray]]'s 1934 ''Objet mathematique'']]

Man Ray photographed some of the mathematical models in the [[Institut Henri Poincaré]] in Paris, including ''Objet mathematique'' (Mathematical object). He noted that this represented [[Enneper surface]]s with constant [[negative curvature]], derived from the [[pseudo-sphere]]. This mathematical foundation was important to him, as it allowed him to deny that the object was "abstract", instead claiming that it was as real as the urinal that Duchamp made into a work of art. Man Ray admitted that the object's [Enneper surface] formula "meant nothing to me, but the forms themselves were as varied and authentic as any in nature." He used his photographs of the mathematical models as figures in his series he did on [[Shakespeare]]'s plays, such as his 1934 painting ''Antony and Cleopatra''.<ref>{{cite book |last=Tubbs |first=Robert |title=Mathematics in 20th-Century Literature and Art |date=2014 |publisher=Johns Hopkins |isbn=978-1-4214-1380-8 |pages=8–10}}</ref> The art reporter Jonathan Keats, writing in ''ForbesLife'', argues that Man Ray photographed "the elliptic paraboloids and conic points in the same sensual light as his pictures of [[Kiki de Montparnasse]]", and "ingeniously repurposes the cool calculations of mathematics to reveal the topology of desire".<ref>{{cite web |last=Keats |first=Jonathon |title=See How Man Ray Made Elliptic Paraboloids Erotic At This Phillips Collection Photography Exhibit |url=https://www.forbes.com/sites/jonathonkeats/2015/02/13/see-how-man-ray-made-elliptic-paraboloids-sexy-at-this-phillips-collection-photography-exhibit/ |work=Forbes |access-date=10 September 2015 |date=13 February 2015}}</ref> Twentieth century sculptors such as [[Henry Moore]], [[Barbara Hepworth]] and [[Naum Gabo]] took inspiration from mathematical models.<ref>{{cite book |last1=Gamwell |first1=Lynn |title=Mathematics and Art: A Cultural History |date=2015 |publisher=Princeton University Press |pages=311–312 |isbn=978-0-691-16528-8}}</ref> Moore wrote of his 1938 ''Stringed Mother and Child'': "Undoubtedly the source of my stringed figures was the [[Science Museum, London|Science Museum]]&nbsp;... I was fascinated by the mathematical models I saw there&nbsp;... It wasn't the scientific study of these models but the ability to look through the strings as with a bird cage and to see one form within another which excited me."<ref>{{cite book |editor-last=Hedgecoe |editor-first=John |contribution=Henry Moore: Text on His Sculpture |title=Henry Spencer Moore |date=1968 |publisher=Simon and Schuster |page=105}}</ref>

[[File:Theo van Doesburg 122.jpg|thumb|upright=0.4|[[Theo van Doesburg]]'s ''Six Moments in the Development of Plane to Space'', 1926 or 1929]]

The artists [[Theo van Doesburg]] and [[Piet Mondrian]] founded the [[De Stijl]] movement, which they wanted to "establish a visual vocabulary {{sic|comprised |hide=y|of}} elementary geometrical forms comprehensible by all and adaptable to any discipline".<ref name=tate>{{cite web |url=http://www.tate.org.uk/learn/online-resources/glossary/d/de-stijl |title=De Stijl |access-date=11 September 2015 |work=Tate Glossary |publisher=The Tate}}</ref><ref>{{cite book |last=Curl |first=James Stevens |title=A Dictionary of Architecture and Landscape Architecture |url=https://archive.org/details/dictionaryofarch00curl_0 |url-access=registration |year=2006 |edition=Second |publisher=Oxford University Press |isbn=978-0-19-860678-9}}</ref> Many of their artworks visibly consist of ruled squares and triangles, sometimes also with circles. De Stijl artists worked in painting, furniture, interior design and architecture.<ref name=tate/> After the breakup of De Stijl, Van Doesburg founded the [[Avant-garde]] [[Art Concret]] movement, describing his 1929–1930 ''[https://commons.wikimedia.org/wiki/File:Theo_van_Doesburg_218.jpg Arithmetic Composition]'', a series of four black squares on the diagonal of a squared background, as "a structure that can be controlled, a ''definite'' surface without chance elements or individual caprice", yet "not lacking in spirit, not lacking the universal and not&nbsp;... empty as there is ''everything'' which fits the internal rhythm". The art critic Gladys Fabre observes that two progressions are at work in the painting, namely the growing black squares and the alternating backgrounds.<ref>{{cite book |last=Tubbs |first=Robert |title=Mathematics in Twentieth-Century Literature and Art: Content, Form, Meaning  |year=2014 |publisher=JHU Press |pages=44–47 |isbn=978-1-4214-1402-7}}</ref>

The mathematics of [[tessellation]], polyhedra, shaping of space, and self-reference provided the graphic artist [[M. C. Escher]] (1898—1972) with a lifetime's worth of materials for his woodcuts.<ref>{{cite web |url=http://www.nga.gov/collection/gallery/ggescher/ggescher-main1.html |title=Tour: M.C. Escher – Life and Work |publisher=NGA |access-date=13 August 2009 |url-status=dead |archive-url=https://web.archive.org/web/20090803124602/http://www.nga.gov/collection/gallery/ggescher/ggescher-main1.html |archive-date=3 August 2009 }}</ref><ref>{{cite web |url=http://www.mathacademy.com/pr/minitext/escher/ |title=MC Escher |publisher=Mathacademy.com |date=1 November 2007 |access-date=13 August 2009}}</ref> In the ''Alhambra Sketch'', Escher showed that art can be created with polygons or regular shapes such as triangles, squares, and hexagons. Escher used irregular polygons when tiling the plane and often used reflections, [[glide reflection]]s, and [[Translation (geometry)|translations]] to obtain further patterns. Many of his works contain impossible constructions, made using geometrical objects which set up a contradiction between perspective projection and three dimensions, but are pleasant to the human sight. Escher's ''[[Ascending and Descending]]'' is based on the "[[Penrose stairs|impossible staircase]]" created by the medical scientist [[Lionel Penrose]] and his son the mathematician [[Roger Penrose]].<ref>{{cite journal |last1=Penrose |first1=L.S. |last2=Penrose |first2=R. |title=Impossible objects: A special type of visual illusion |journal=[[British Journal of Psychology]] |year=1958 |volume=49 |issue=1 |pages=31–33 |doi=10.1111/j.2044-8295.1958.tb00634.x |pmid=13536303}}</ref><ref>{{cite book |last1=Kirousis |first1=Lefteris M. |last2=Papadimitriou |first2=Christos H. |title=26th Annual Symposium on Foundations of Computer Science (SFCS 1985) |chapter=The complexity of recognizing polyhedral scenes |author2-link=Christos Papadimitriou |doi=10.1109/sfcs.1985.59 |pages=175–185 |year=1985|isbn=978-0-8186-0644-1 |citeseerx=10.1.1.100.4844 }}</ref><ref>{{cite book |last=Cooper |first=Martin |title=Inequality, Polarization and Poverty |contribution=Tractability of Drawing Interpretation |doi=10.1007/978-1-84800-229-6_9 |isbn=978-1-84800-229-6 |pages=[https://archive.org/details/linedrawinginter00coop_873/page/n218 217]–230 |publisher=Springer-Verlag |url=https://archive.org/details/linedrawinginter00coop_873 |url-access=limited |year=2008}}</ref>

Some of Escher's many tessellation drawings were inspired by conversations with the mathematician [[H. S. M. Coxeter]] on [[hyperbolic geometry]].<ref>{{cite book |author=Roberts, Siobhan |author-link=Siobhan Roberts|title=King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry |page=Chapter 11 |contribution='Coxetering' with M.C. Escher |publisher=Walker |year=2006}}</ref> Escher was especially interested in five specific polyhedra, which appear many times in his work. The [[Platonic solids]]—tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons—are especially prominent in ''Order and Chaos'' and ''Four Regular Solids''.<ref>{{cite book |author=Escher, M.C. |title=The World of MC Escher |publisher=Random House |date=1988}}</ref> These stellated figures often reside within another figure which further distorts the viewing angle and conformation of the polyhedrons and provides a multifaceted perspective artwork.<ref>{{cite book |author1=Escher, M.C. |author2=Vermeulen, M.W. |author3=Ford, K. |title=Escher on Escher: Exploring the Infinite |year=1989 |publisher=HN Abrams}}</ref>

The visual intricacy of mathematical structures such as tessellations and polyhedra have inspired a variety of mathematical artworks. [[Stewart Coffin]] makes polyhedral puzzles in rare and beautiful woods; [[George W. Hart]] works on the theory of [[polyhedra]] and sculpts objects inspired by them; [[Magnus Wenninger]] makes "especially beautiful" models of [[List of Wenninger polyhedron models|complex stellated polyhedra]].<ref>{{cite web |last1=Malkevitch |first1=Joseph |title=Mathematics and Art. 5. Polyhedra, tilings, and dissections |url=https://www.ams.org/samplings/feature-column/fcarc-art5 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref>

The distorted perspectives of [[anamorphosis]] have been explored in art since the sixteenth century, when [[Hans Holbein the Younger]] incorporated a severely distorted skull in his 1533 painting ''[[The Ambassadors (Holbein)|The Ambassadors]]''. Many artists since then, including Escher, have make use of anamorphic tricks.<ref>{{cite book |author=Marcolli, Matilde |author-link=Matilde Marcolli |title=The notion of Space in Mathematics through the lens of Modern Art |publisher=Century Books |date=July 2016 |url=http://www.its.caltech.edu/~matilde/SpaceMathArticle.pdf |pages=23–26}}</ref>

The mathematics of [[topology]] has inspired several artists in modern times. The sculptor [[John Robinson (sculptor)|John Robinson]] (1935–2007) created works such as ''Gordian Knot'' and ''Bands of Friendship'', displaying [[knot theory]] in polished bronze.<ref name=Emmer/> Other works by Robinson explore the topology of [[torus]]es. ''Genesis'' is based on [[Borromean rings]] – a set of three circles, no two of which link but in which the whole structure cannot be taken apart without breaking.<ref>{{cite web |url=http://www.bradshawfoundation.com/jr/ |title=John Robinson |date=2007 |publisher=Bradshaw Foundation |access-date=13 August 2009}}</ref> The sculptor [[Helaman Ferguson]] creates complex [[Surface (topology)|surface]]s and other [[topological space|topological objects]].<ref>{{cite web |url=http://www.helasculpt.com/index.html |title=Helaman Ferguson web site |publisher=Helasculpt.com |access-date=13 August 2009 |url-status=dead |archive-url=https://web.archive.org/web/20090411034819/http://www.helasculpt.com/index.html |archive-date=11 April 2009 }}</ref> His works are visual representations of mathematical objects; ''The Eightfold Way'' is based on the [[projective special linear group]] [[PSL(2,7)]], a finite group of 168 elements.<ref>{{cite book |url=http://www.msri.org/publications/books/Book35/files/thurston.pdf |contribution=The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson |author=Thurston, William P. |title=Volume 35: The Eightfold Way: The Beauty of Klein's Quartic Curve |publisher=MSRI Publications |year=1999 |pages=1–7 |editor=Levy, Silvio}}</ref><ref>{{cite web |url=http://www.maa.org/reviews/eightfold.html |title=MAA book review of ''The Eightfold Way: The Beauty of Klein's Quartic Curve'' |publisher=Maa.org |date=14 November 1993 |access-date=13 August 2009 |archive-date=21 December 2009 |archive-url=https://web.archive.org/web/20091221061724/http://www.maa.org/reviews/eightfold.html |url-status=dead }}</ref> The sculptor [[Bathsheba Grossman]] similarly bases her work on mathematical structures.<ref>{{cite magazine |url=http://blogs.scientificamerican.com/roots-of-unity/the-math-geek-holiday-gift-guide/ |title=The Math Geek Holiday Gift Guide |magazine=[[Scientific American]] |date=23 November 2014 |access-date=7 June 2015}}</ref><ref>{{cite web |last1=Hanna |first1=Raven |title=Gallery: Bathsheba Grossman |url=http://www.symmetrymagazine.org/article/september-2005/gallery-bathsheba-grossman |publisher=Symmetry Magazine |access-date=7 June 2015}}</ref> The artist [[Nelson Saiers]] incorporates mathematical concepts and theorems in his art from [[topos]]es and [[Scheme (mathematics)|schemes]] to the [[four color theorem]] and the irrationality of [[Pi|π]].<ref>{{cite web |last=Mastroianni |first=Brian |title=The perfect equation: Artist combines math and art |url=https://www.foxnews.com/science/the-perfect-equation-artist-combines-math-and-art |website=Fox News |date=26 May 2015 |access-date=28 January 2021}}</ref>

A liberal arts inquiry project examines connections between mathematics and art through the [[Möbius strip]], [[flexagon]]s, origami and [[panorama]] photography.<ref>{{cite book |last1=Fleron |first1=Julian F. |last2=Ecke |first2=Volker |last3=von Renesse |first3=Christine |last4=Hotchkiss |first4=Philip K. |title=Art and Sculpture: Mathematical Inquiry in the Liberal Arts |date=January 2015 |publisher=Discovering the Art of Mathematics project |edition=2nd |url=https://www.artofmathematics.org/books/art-and-sculpture}}</ref>

Mathematical objects including the [[Lorenz attractor|Lorenz manifold]] and the [[Hyperbolic manifold|hyperbolic plane]] have been crafted using [[Mathematics and fiber arts|fiber arts]] including crochet.{{efn|Images and videos of [[Hinke Osinga]]'s crocheted Lorenz manifold reached international television news, as can be seen in the linked website.<ref>{{cite web |last1=Osinga |first1=Hinke |title=Crocheting the Lorenz manifold |url=https://www.math.auckland.ac.nz/~hinke/crochet/ |publisher=University of Auckland |access-date=12 October 2015 |date=2005 |archive-url=https://web.archive.org/web/20150410054845/https://www.math.auckland.ac.nz/~hinke/crochet/ |archive-date=10 April 2015}}</ref>}}<ref>{{cite journal |last1=Osinga |first1=Hinke M. |author1-link=Hinke Osinga |last2=Krauskopf |first2=Bernd |doi=10.1007/BF02985416 |issue=4 |journal=[[The Mathematical Intelligencer]] |pages=25–37 |title=Crocheting the Lorenz manifold |url=http://www.enm.bris.ac.uk/anm/preprints/2004r03.html |volume=26 |year=2004 |citeseerx=10.1.1.108.4594 |s2cid=119728638 |access-date=2015-06-26 |archive-date=2013-04-19 |archive-url=https://web.archive.org/web/20130419094659/http://www.enm.bris.ac.uk/anm/preprints/2004r03.html |url-status=dead }}</ref> The American weaver [[Ada Dietz]] wrote a 1949 monograph ''Algebraic Expressions in Handwoven Textiles'', defining weaving patterns based on the expansion of multivariate [[polynomial]]s.<ref>{{cite book |last=Dietz |first=Ada K. |location=Louisville, Kentucky |publisher=The Little Loomhouse |title=Algebraic Expressions in Handwoven Textiles |url=http://www.cs.arizona.edu/patterns/weaving/monographs/dak_alge.pdf |year=1949 |access-date=2015-06-26 |archive-url=https://web.archive.org/web/20160222003421/http://www.cs.arizona.edu/patterns/weaving/monographs/dak_alge.pdf |archive-date=2016-02-22 |url-status=dead }}</ref> The mathematician [[Daina Taimiņa]] demonstrated features of the hyperbolic plane by crocheting in 2001.<ref>{{cite journal |last1=Henderson |first1=David |last2=Taimina |first2=Daina |author2-link=Daina Taimina |doi=10.1007/BF03026623 |issue=2 |journal=[[Mathematical Intelligencer]] |pages=17–28 |title=Crocheting the hyperbolic plane |url=http://www.math.cornell.edu/%7Edwh/papers/crochet/crochet.PDF |volume=23 |year=2001|s2cid=120271314 }}.</ref> This led [[Margaret Wertheim|Margaret and Christine Wertheim]] to crochet a [[coral reef]], consisting of many marine animals such as [[nudibranch]]s whose shapes are based on hyperbolic planes.<ref name="Barnett (TED) 2017">{{cite web |last1=Barnett |first1=Rebekah |title=Gallery: What happens when you mix math, coral and crochet? It's mind-blowing |url=https://ideas.ted.com/gallery-what-happens-when-you-mix-math-coral-and-crochet-its-mind-blowing/ |publisher=Ideas.TED.com |access-date=28 October 2019 |date=31 January 2017}}</ref> 
The mathematician [[J. C. P. Miller]] used the [[Rule 90]] [[cellular automaton]] to design [[tapestry|tapestries]] depicting both trees and abstract patterns of triangles.<ref>{{cite journal |first=J. C. P. |last=Miller |author-link=J. C. P. Miller |title=Periodic forests of stunted trees |journal=[[Philosophical Transactions of the Royal Society of London]] |series=Series A, Mathematical and Physical Sciences |volume=266 |issue=1172 |year=1970 |pages=63–111 |doi=10.1098/rsta.1970.0003 |jstor=73779|bibcode=1970RSPTA.266...63M |s2cid=123330469 }}</ref> The "{{visible anchor|mathekniticians}}"<ref>{{cite web |title=Pat Ashforth & Steve Plummer – Mathekniticians |url=http://www.woollythoughts.com/aboutus.html |website=Woolly Thoughts |access-date=4 October 2015}}</ref> Pat Ashforth and Steve Plummer use knitted versions of mathematical objects such as [[hexaflexagon]]s in their teaching, though their [[Menger sponge]] proved too troublesome to knit and was made of plastic canvas instead.<ref>{{cite news |last1=Ward |first1=Mark |title=Knitting reinvented: Mathematics, feminism and metal |url=https://www.bbc.co.uk/news/technology-19208292 |publisher=BBC |access-date=23 September 2015 |date=20 August 2012|work=BBC News }}</ref><ref>{{cite web |last1=Ashforth |first1=Pat |last2=Plummer |first2=Steve |title=Menger Sponge |url=http://www.woollythoughts.com/menger.html |website=Woolly Thoughts: In Pursuit of Crafty Mathematics |access-date=23 September 2015}}</ref> Their "mathghans" (Afghans for Schools) project introduced [[knitting]] into the British mathematics and technology curriculum.<ref>{{cite web |last1=Ashforth |first1=Pat |last2=Plummer |first2=Steve |title=Afghans for Schools |url=http://www.woollythoughts.com/schools/index.html |website=Woolly Thoughts: Mathghans |access-date=23 September 2015}}</ref><ref>{{cite magazine |title=Mathghans with a Difference |url=http://www.simplyknitting.co.uk/2008/07/01/mathghans-with-a-difference/ |magazine=Simply Knitting Magazine |access-date=23 September 2015 |date=1 July 2008 |archive-url=https://web.archive.org/web/20150925133153/http://www.simplyknitting.co.uk/2008/07/01/mathghans-with-a-difference/ |archive-date=25 September 2015 |url-status=dead }}</ref>

<gallery mode="nolines" heights="180px" widths="180px">
File:Jouffret.gif|[[Four-dimensional space]] to [[Cubism]]: [[Esprit Jouffret]]'s 1903 ''Traité élémentaire de géométrie à quatre dimensions''.<ref>{{cite book|last=Jouffret|first=Esprit|title=Traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n dimensions|url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=04810001|access-date=26 September 2015|year=1903 |publisher=Gauthier-Villars|location=Paris|language=fr |oclc=1445172}}</ref>{{efn|[[Maurice Princet]] gave a copy to [[Pablo Picasso]], whose sketchbooks for ''[[Les Demoiselles d'Avignon]]'' illustrate Jouffret's influence.<ref name="Miller">{{cite book|last=Miller|first=Arthur I. |title=Einstein, Picasso: Space, Time, and the Beauty That Causes Havoc |url=https://archive.org/details/einsteinpicassos00mill|url-access=registration|year=2001 |publisher=Basic Books|location=New York |isbn=978-0-465-01860-4 |page=[https://archive.org/details/einsteinpicassos00mill/page/171 171]}}</ref><ref>{{cite book |last=Seckel |first=Hélène|editor=William Rubin|editor2=Hélène Seckel|editor3=Judith Cousins |title=Les Demoiselles d'Avignon|year=1994|publisher=Museum of Modern Art|location=New York |isbn=978-0-87070-162-7 |page=264 |chapter=Anthology of Early Commentary on ''Les Demoiselles d'Avignon''}}</ref>}}<!--close efn-->
File:Theo van Doesburg Composition I.jpg|[[De Stijl]]: [[Theo van Doesburg]]'s geometric ''Composition I (Still Life)'', 1916
File:Magnus Wenninger polyhedral models.jpg|[[Pedagogy]] to art: [[Magnus Wenninger]] with some of his [[stellation|stellated polyhedra]], 2009
<!--File:Escher Circle Limit III.jpg|[[Tessellation]] and [[Hyperbolic geometry]] in art: ''[[Circle Limit III]]'' by M.C. Escher (1959)-->
File:Moebiusstripscarf.jpg|A [[Möbius strip]] scarf in [[crochet]], 2007
File:Hans Holbein the Younger - The Ambassadors - Google Art Project.jpg|[[Anamorphism]]: ''[[The Ambassadors (Holbein)|The Ambassadors]]'' by [[Hans Holbein the Younger]], 1533, with severely distorted skull in foreground
File:The Föhr Reef in Tübingen.JPG|[[Crochet]]ed [[coral reef]]: many animals modelled as [[hyperbolic plane]]s with varying parameters by [[Margaret Wertheim|Margaret and Christine Wertheim]]. ''Föhr Reef'', Tübingen, 2013
</gallery>

[[File:René Magritte The Human Condition.jpg|thumb|upright=0.8|[[Semiotic]] joke: [[René Magritte]]'s ''[[The Human Condition (painting)|La condition humaine]]'' 1933]]

===Illustrating mathematics===
[[File:Giotto. The Stefaneschi Triptych (verso) c.1330 220x245cm. Pinacoteca, Vatican..jpg|thumb|left|Front face of [[Giotto]]'s ''[[Stefaneschi Triptych]]'', 1320 illustrates [[recursion]].]]
[[File:Giotto di Bondone - The Stefaneschi Triptych - St Peter Enthroned (detail) - WGA09356.jpg|thumb|left|upright=0.5|Detail of Cardinal Stefaneschi holding the triptych]]

Modelling is far from the only possible way to illustrate mathematical concepts. Giotto's ''[[Stefaneschi Triptych]]'', 1320, illustrates [[recursion]] in the form of ''[[mise en abyme]]''; the central panel of the triptych contains, lower left, the kneeling figure of Cardinal Stefaneschi, holding up the triptych as an offering.<ref>{{cite web |title=Giotto di Bondone and assistants: Stefaneschi triptych |url=http://mv.vatican.va/3_EN/pages/PIN/PIN_Sala02_03.html |publisher=The Vatican |access-date=16 September 2015}}</ref> [[Giorgio de Chirico]]'s [[metaphysics|metaphysical]] paintings such as his 1917 ''Great Metaphysical Interior'' explore the question of levels of representation in art by depicting paintings within his paintings.<ref>{{cite book |last1=Gamwell |first1=Lynn |title=Mathematics and Art: A Cultural History |date=2015 |publisher=Princeton University Press |pages=337–338 |isbn=978-0-691-16528-8}}</ref>

Art can exemplify logical paradoxes, as in some paintings by the [[surrealist]] [[René Magritte]], which can be read as [[semiotic]] jokes about confusion between levels. In ''[[The Human Condition (painting)|La condition humaine]]'' (1933), Magritte depicts an easel (on the real canvas), seamlessly supporting a view through a window which is framed by "real" curtains in the painting. Similarly, Escher's ''[[Print Gallery (M. C. Escher)|Print Gallery]]'' (1956) is a print which depicts a distorted city which contains a gallery which [[recursive]]ly contains the picture, and so ''[[ad infinitum]]''.<ref>{{cite web |last1=Cooper |first1=Jonathan |title=Art and Mathematics |url=http://www.doctordada.com/art/art-and-mathematics/ |access-date=5 September 2015 |date=5 September 2007}}</ref> Magritte made use of spheres and cuboids to distort reality in a different way, painting them alongside an assortment of houses in his 1931 ''Mental Arithmetic'' as if they were children's building blocks, but house-sized.<ref>{{cite book |last=Hofstadter |first=Douglas R. |author-link=Douglas Hofstadter |title=Gödel, Escher, Bach: An Eternal Golden Braid |title-link=Gödel, Escher, Bach |date=1980 |publisher=Penguin |isbn=978-0-14-028920-6 |page=627}}</ref> ''[[The Guardian]]'' observed that the "eerie toytown image" prophesied [[Modernism]]'s usurpation of "cosy traditional forms", but also plays with the human tendency to seek [[patterns in nature]].<ref>{{cite news |last1=Hall |first1=James |title=René Magritte: The Pleasure Principle – exhibition |url=https://www.theguardian.com/artanddesign/2011/jun/10/rene-magritte-pleasure-principle-exhibition |newspaper=The Guardian |access-date=5 September 2015 |date=10 June 2011}}</ref>

[[File:Escher Paradox Diagram.png|thumb|upright=1.25<!--fmt for readability of small text-->|Diagram of the apparent paradox embodied in M. C. Escher's 1956 lithograph ''Print Gallery'', as discussed by [[Douglas Hofstadter]] in his 1980 book ''[[Gödel, Escher, Bach]]''<ref name=GEB/>]]

Salvador Dalí's last painting, ''[[The Swallow's Tail]]'' (1983), was part of a series inspired by [[René Thom]]'s [[catastrophe theory]].<ref name="King">{{cite book |last1=King |first1=Elliott |editor1-last=Ades |editor1-first=Dawn |title=Dali |date=2004 |publisher=Bompiani Arte |location=Milan |pages=418–421}}</ref> The Spanish painter and sculptor [[Pablo Palazuelo]] (1916–2007) focused on the investigation of form. He developed a style that he described as the geometry of life and the geometry of all nature. Consisting of simple geometric shapes with detailed patterning and coloring, in works such as ''Angular I'' and ''Automnes'', Palazuelo expressed himself in geometric transformations.<ref name=Emmer/>

The artist Adrian Gray practises [[Rock balancing|stone balancing]], exploiting [[friction]] and the [[centre of gravity]] to create striking and seemingly impossible compositions.<ref>{{cite journal |title=Stone balancing |url=http://mei.org.uk/files/pdf/MM_July_2013.pdf |journal=Monthly Maths |issue=29 |date=July 2013 |access-date=10 June 2017 |archive-date=16 April 2021 |archive-url=https://web.archive.org/web/20210416080503/https://mei.org.uk/files/pdf/MM_July_2013.pdf |url-status=dead }}</ref>

[[File:Print Gallery by M. C. Escher.jpg|thumb|upright|Lithograph ''[[Print Gallery (M. C. Escher)|Print Gallery]]'' by [[M. C. Escher]], 1956]]

Artists, however, do not necessarily take geometry literally. As [[Douglas Hofstadter]] writes in his 1980 reflection on human thought, ''[[Gödel, Escher, Bach]]'', by way of (among other things) the mathematics of art: "The difference between an Escher drawing and [[non-Euclidean geometry]] is that in the latter, comprehensible interpretations can be found for the undefined terms, resulting in a comprehensible total system, whereas for the former, the end result is not reconcilable with one's conception of the world, no matter how long one stares at the pictures." Hofstadter discusses the seemingly [[paradox]]ical lithograph ''Print Gallery'' by M. C. Escher; it depicts a seaside town containing an art gallery which seems to contain a painting of the seaside town, there being a "strange loop, or tangled hierarchy" to the levels of reality in the image. The artist himself, Hofstadter observes, is not seen; his reality and his relation to the lithograph are not paradoxical.<ref name=GEB>{{cite book |last1=Hofstadter |first1=Douglas R. |author-link=Douglas Hofstadter |title=Gödel, Escher, Bach: An Eternal Golden Braid |title-link=Gödel, Escher, Bach |date=1980 |publisher=Penguin |isbn=978-0-394-74502-2 |pages=98–99, 690–717}}</ref> The image's central void has also attracted the interest of mathematicians Bart de Smit and [[Hendrik Lenstra]], who propose that it could contain a [[Droste effect]] copy of itself, rotated and shrunk; this would be a further illustration of recursion beyond that noted by Hofstadter.<ref>{{Cite journal |last=de Smit |first=B. |title=The Mathematical Structure of Escher's Print Gallery |year=2003 |journal=[[Notices of the American Mathematical Society]] |volume=50 |issue=4 |pages=446–451}}</ref><ref name=LenstraWebsite>{{cite web |last1=Lenstra |first1=Hendrik |last2=De Smit |first2=Bart |title=Applying mathematics to Escher's Print Gallery |url=http://escherdroste.math.leidenuniv.nl/index.php?menu=intro |publisher=Leiden University |access-date=10 November 2015 |archive-date=14 January 2018 |archive-url=https://web.archive.org/web/20180114013908/http://escherdroste.math.leidenuniv.nl/index.php?menu=intro |url-status=dead }}</ref>

===Analysis of art history===

Algorithmic analysis of images of artworks, for example using [[X-ray fluorescence spectroscopy]], can reveal information about art. Such techniques can uncover images in layers of paint later covered over by an artist; help art historians to visualize an artwork before it cracked or faded; help to tell a copy from an original, or distinguish the brushstroke style of a master from those of his apprentices.<ref>{{cite magazine |last1=Stanek |first1=Becca |title=Van Gogh and the Algorithm: How Math Can Save Art |url=https://time.com/2884058/math-art-van-gogh-duke/ |magazine=Time Magazine |access-date=4 September 2015 |date=16 June 2014}}</ref><ref>{{cite web |last1=Sipics |first1=Michelle |title=The Van Gogh Project: Art Meets Mathematics in Ongoing International Study |url=https://www.siam.org/news/news.php?id=1568 |publisher=Society for Industrial and Applied Mathematics |access-date=4 September 2015 |date=18 May 2009 |archive-url=https://web.archive.org/web/20150907222302/http://www.siam.org/news/news.php?id=1568 |archive-date=7 September 2015 |url-status=dead |df=dmy-all }}</ref>

[[File:Max Ernst making Lissajous Figures 1942.jpg|thumb|upright|[[Max Ernst]] making [[Lissajous figures]], New York, 1942]]
[[Jackson Pollock]]'s [[drip painting]] style<ref name="Emmerling">{{cite book |last=Emmerling
|first=Leonhard |title=Jackson Pollock, 1912–1956 |date=2003 |page=63 |publisher=Taschen |isbn=978-3-8228-2132-9 |url=https://books.google.com/books?id=K-ZZmvjJ_-IC&pg=PA63}}</ref> has a definite [[fractal dimension]];<ref>{{cite journal |author1=Taylor, Richard P. |author2=Micolich, Adam P. |author3=Jonas, David |title=Fractal analysis of Pollock's drip paintings |journal=[[Nature (journal)|Nature]] |volume=399 |issue=6735 |page=422 |date=June 1999 |doi=10.1038/20833 |bibcode=1999Natur.399..422T |s2cid=204993516 |doi-access=free }}</ref> among the artists who may have influenced Pollock's controlled [[chaos theory|chaos]],<ref>{{cite journal |last1=Taylor |first1=Richard |last2=Micolich |first2=Adam P. |last3=Jonas |first3=David |title=Fractal Expressionism: Can Science Be Used To Further Our Understanding Of Art? |url=http://phys.unsw.edu.au/phys_about/PHYSICS!/FRACTAL_EXPRESSIONISM/fractal_taylor.html |archive-url=https://archive.today/20120805084052/http://phys.unsw.edu.au/phys_about/PHYSICS!/FRACTAL_EXPRESSIONISM/fractal_taylor.html |url-status=dead |archive-date=2012-08-05 |journal=[[Physics World]] |date=October 1999 |quote=Pollock died in 1956, before chaos and fractals were discovered. It is highly unlikely, therefore, that Pollock consciously understood the fractals he was painting. Nevertheless, his introduction of fractals was deliberate. For example, the colour of the anchor layer was chosen to produce the sharpest contrast against the canvas background and this layer also occupies more canvas space than the other layers, suggesting that Pollock wanted this highly fractal anchor layer to visually dominate the painting. Furthermore, after the paintings were completed, he would dock the canvas to remove regions near the canvas edge where the pattern density was less uniform. |doi=10.1088/2058-7058/12/10/21 |volume=12 |issue=10 |pages=25–28 }}</ref> Max Ernst painted [[Lissajous figures]] directly by swinging a punctured bucket of paint over a canvas.<ref>{{cite web |url=https://www.herts.ac.uk/__data/assets/pdf_file/0013/12307/WPIAAD_vol2_king.pdf |title=From Max Ernst to Ernst Mach: epistemology in art and science. |last=King |first=M. |date=2002 |access-date=17 September 2015}}</ref>

The computer scientist [[Neil Dodgson]] investigated whether [[Bridget Riley]]'s stripe paintings could be characterised mathematically, concluding that while separation distance could "provide some characterisation" and global [[entropy]] worked on some paintings, [[autocorrelation]] failed as Riley's patterns were irregular. Local entropy worked best, and correlated well with the description given by the art critic Robert Kudielka.<ref>{{cite journal |last1=Dodgson |first1=N. A. |title=Mathematical characterisation of Bridget Riley's stripe paintings |journal=[[Journal of Mathematics and the Arts]] |date=2012 |volume=5 |issue=2–3 |pages=89–106 |url=http://www.cl.cam.ac.uk/~nad10/pubs/jma12.pdf |quote=over the course [of] the early 1980s, Riley's patterns moved from more regular to more random (as characterised by global entropy), without losing their rhythmic structure (as characterised by local entropy). This reflects Kudielka's description of her artistic development. |doi=10.1080/17513472.2012.679468|s2cid=10349985 |doi-access=free }}</ref>

The American mathematician [[George Birkhoff]]'s 1933 ''Aesthetic Measure'' proposes a quantitative metric of the [[Aesthetics|aesthetic quality]] of an artwork. It does not attempt to measure the connotations of a work, such as what a painting might mean, but is limited to the "elements of order" of a polygonal figure. Birkhoff first combines (as a sum) five such elements: whether there is a vertical axis of symmetry; whether there is optical equilibrium; how many rotational symmetries it has; how wallpaper-like the figure is; and whether there are unsatisfactory features such as having two vertices too close together. This metric, ''O'', takes a value between −3 and 7. The second metric, ''C'', counts elements of the figure, which for a polygon is the number of different straight lines containing at least one of its sides. Birkhoff then defines his aesthetic measure of an object's beauty as ''O/C''. This can be interpreted as a balance between the pleasure looking at the object gives, and the amount of effort needed to take it in. Birkhoff's proposal has been criticized in various ways, not least for trying to put beauty in a formula, but he never claimed to have done that.<ref name=Cucker116>{{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=116–120}}</ref>

===Stimuli to mathematical research===

{{further|Projective geometry|Mathematics of paper folding}}

Art has sometimes stimulated the development of mathematics, as when Brunelleschi's theory of perspective in architecture and painting started a cycle of research that led to the work of [[Brook Taylor]] and [[Johann Heinrich Lambert]] on the mathematical foundations of perspective drawing,<ref>{{cite web |last1=Treibergs |first1=Andrejs |title=The Geometry of Perspective Drawing on the Computer |url=http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm |publisher=University of Utah |access-date=5 September 2015 |date=24 July 2001}}</ref> and ultimately to the mathematics of [[projective geometry]] of [[Girard Desargues]] and [[Jean-Victor Poncelet]].<ref>{{cite book |last1=Gamwell |first1=Lynn |title=Mathematics and Art: A Cultural History |date=2015 |publisher=Princeton University Press |page=xviii |isbn=978-0-691-16528-8}}</ref>

The Japanese paper-folding art of [[origami]] has been reworked mathematically by [[Tomoko Fusé]] [[modular origami|using modules]], congruent pieces of paper such as squares, and making them into polyhedra or tilings.<ref>{{cite web |last=Malkevitch |first=Joseph |title=Mathematics and Art. 6. Origami |url=https://www.ams.org/samplings/feature-column/fcarc-art6 |publisher=American Mathematical Society |access-date=1 September 2015}}</ref> Paper-folding was used in 1893 by T. Sundara Rao in his ''[[Geometric Exercises in Paper Folding]]'' to demonstrate geometrical proofs.<ref>{{cite book |last=Rao |first=T. Sundara |title=Geometric Exercises in Paper Folding |title-link=Geometric Exercises in Paper Folding |publisher=Addison |year=1893}}</ref> The [[mathematics of paper folding]] has been explored in [[Maekawa's theorem]],<ref>{{cite journal |last=Justin |first=J. |title=Mathematics of Origami, part 9 |journal=British Origami |date=June 1986 |pages=28–30}}.</ref> [[Kawasaki's theorem]],<ref>{{cite book |first1=Claudi |last1=Alsina |first2=Roger |last2=Nelsen |title=Charming Proofs: A Journey Into Elegant Mathematics |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-348-1 |series=Dolciani Mathematical Expositions |volume=42 |year=2010 |page=57 |url=https://books.google.com/books?id=mIT5-BN_L0oC&pg=PA57}}</ref> and the [[Huzita–Hatori axioms]].<ref>{{cite journal |last1=Alperin |first1=Roger C. |author1-link=Roger C. Alperin |last2=Lang |first2=Robert J. |title=One-, Two-, and Multi-Fold Origami Axioms |url=http://www.math.sjsu.edu/~alperin/AlperinLang.pdf |journal=4OSME |year=2009 |access-date=2015-10-27 |archive-date=2022-02-13 |archive-url=https://web.archive.org/web/20220213123946/http://www.math.sjsu.edu/~alperin/AlperinLang.pdf |url-status=dead }}</ref>

<gallery mode="nolines" heights="180px" widths="180px">
File:Della Pittura Alberti perspective circle to ellipse.jpg|Stimulus to [[projective geometry]]: [[Leon Battista Alberti|Alberti]]'s diagram showing a circle seen in perspective as an [[ellipse]]. ''Della Pittura'', 1435–1436
File:Origami spring.jpg|Mathematical [[origami]]: ''[[Spring (device)|Spring]] Into Action'', by Jeff Beynon, made from a single paper rectangle.<ref>[http://www1.ttcn.ne.jp/~a-nishi/ The World of Geometric Toys], ''[http://www1.ttcn.ne.jp/~a-nishi/spring/z_spring.html Origami Spring]'', August, 2007.</ref>
</gallery>

===Illusion to op art===
{{further|Op art}}

[[File:Fraser spiral.svg|thumb|upright|The [[Fraser spiral illusion]], named for Sir James Fraser who discovered it in 1908.]]

[[Optical illusion]]s such as the [[Fraser spiral illusion|Fraser spiral]] strikingly demonstrate limitations in human visual perception, creating what the [[art historian]] [[Ernst Gombrich]] called a "baffling trick." The black and white ropes that appear to form [[spiral]]s are in fact [[concentric circle]]s. The mid-twentieth century [[Op art|op art or optical art]] style of painting and graphics exploited such effects to create the impression of movement and flashing or vibrating patterns seen in the work of artists such as [[Bridget Riley]]<!--slight overlink seems sensible here-->, Spyros Horemis,<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=163–166}}</ref> and [[Victor Vasarely]].<ref>{{cite book |last1=Gamwell |first1=Lynn |title=Mathematics and Art: A Cultural History |date=2015 |publisher=Princeton University Press |pages=406–410 |isbn=978-0-691-16528-8}}</ref>

===Sacred geometry===
{{further|Sacred geometry|Mathematics and music}}

A strand of art from Ancient Greece onwards sees God as the geometer of the world, and the world's geometry therefore as sacred. The belief that God created the universe according to a geometric plan has ancient origins. [[Plutarch]] attributed the belief to [[Plato]], writing that "Plato said God geometrizes continually" (''Convivialium disputationum'', liber 8,2). This image has influenced Western thought ever since. The Platonic concept derived in its turn from a [[Pythagoras|Pythagorean]] notion of harmony in music, where the notes were spaced in perfect proportions, corresponding to the lengths of the lyre's strings; indeed, the Pythagoreans held that everything was arranged by Number. In the same way, in Platonic thought, the [[Platonic solid|regular or Platonic solids]] dictate the proportions found in nature, and in art.<ref>{{cite book |last1=Ghyka |first1=Matila |title=The Geometry of Art and Life |date=2003 |publisher=Dover |isbn=978-0-486-23542-4 |pages=ix–xi |url=https://archive.org/details/geometryofartlif00mati |url-access=registration }}</ref><ref>{{cite book |last1=Lawlor |first1=Robert |title=Sacred Geometry: Philosophy and Practice |date=1982 |publisher=Thames & Hudson |isbn=978-0-500-81030-9 |url=https://archive.org/details/sacredgeometryph00lawl }}</ref> An illumination in the 13th-century ''[[Codex Vindobonensis]]'' shows God drawing out the universe with a pair of compasses, which may refer to a verse in the Old Testament: "When he established the heavens I was there: when he set a compass upon the face of the deep" (Proverbs 8:27), .<ref name=Calter>{{cite web |last1=Calter |first1=Paul |title=Celestial Themes in Art & Architecture |url=https://www.dartmouth.edu/~matc/math5.geometry/unit10/unit10.html |publisher=Dartmouth College |access-date=5 September 2015 |date=1998 |archive-date=23 June 2015 |archive-url=https://web.archive.org/web/20150623104029/http://www.dartmouth.edu/~matc/math5.geometry/unit10/unit10.html |url-status=dead }}</ref> In 1596, the mathematical astronomer [[Johannes Kepler]] modelled the universe as a set of nested Platonic solids, determining the relative sizes of the orbits of the planets.<ref name=Calter/> [[William Blake]]'s ''Ancient of Days'' (depicting [[Urizen]], Blake's embodiment of reason and law) and his painting of the physicist [[Isaac Newton]], naked, hunched and drawing with a compass, use the symbolism of compasses to critique conventional reason and materialism as narrow-minded.<ref>{{cite news |last=Maddocks |first=Fiona |date=21 Nov 2014 |title=The 10 best works by William Blake |url=https://www.theguardian.com/culture/2014/nov/21/the-10-best-works-by-william-blake |newspaper=The Guardian |access-date=25 December 2019}}</ref><ref>{{cite web |date=October 2018 |archive-url=https://web.archive.org/web/20190328134113/https://www.tate.org.uk/art/artworks/blake-newton-n05058 |archive-date=28 March 2019 |url=https://www.tate.org.uk/art/artworks/blake-newton-n05058 |title=William Blake, Newton, 1795–c.1805 |website=Tate }}</ref>
[[Salvador Dalí]]'s 1954 ''[[Crucifixion (Corpus Hypercubus)]]'' depicts the cross as a [[hypercube]], representing the divine perspective with four dimensions rather than the usual three.<ref name=CorpusHypercubus /> In Dalí's ''[[The Sacrament of the Last Supper]]'' (1955) Christ and his disciples are pictured inside a giant [[dodecahedron]].<ref>{{cite web |last=Livio |first=Mario |author-link=Mario Livio |title=The golden ratio and aesthetics |url=https://plus.maths.org/content/golden-ratio-and-aesthetics |access-date=26 June 2015|date=November 2002 }}</ref>

<gallery mode="nolines" heights="180px" widths="180px">
File:God the Geometer.jpg|God the geometer. ''[[Codex Vindobonensis]]'', c. 1220
File:Bible moralisée de Tolède - Dieu pantocrator.jpg|The creation, with the [[Pantocrator]] bearing. [[Bible of St Louis]], c. 1220–1240
File:Kepler-solar-system-2.png|[[Johannes Kepler]]'s [[Platonic solid]] model of planetary spacing in the [[Solar System]] from ''[[Mysterium Cosmographicum]]'', 1596
File:The Ancient of Days.jpg|[[William Blake]]'s ''[[The Ancient of Days]]'', 1794
File:William Blake - Newton.png|William Blake's ''[[Newton (Blake)|Newton]]'', c. 1800
</gallery>

==See also==
* [[Mathematics and architecture]]
<!--* [[Mathematics and dance]]-->
* [[Music and mathematics]]

==Notes==
{{notelist}}

==References==
{{reflist|28em}}

==External links==
{{Commons category |Mathematics in art}}
* [http://www.bridgesmathart.org/ Bridges Organization] conference on connections between art and mathematics
* [http://www.scientificamerican.com/slideshow/bridging-the-gap/ Bridging the Gap Between Math and Art] – Slide Show from ''[[Scientific American]]''
* [https://www.artofmathematics.org/ Discovering the Art of Mathematics]
* [https://www.ams.org/samplings/feature-column/fcarc-art1 Mathematics and Art] – [[American Mathematical Society|AMS]]
* [http://www.cut-the-knot.org/ctk/ArtMath.shtml Mathematics and Art] – [[Cut-the-Knot]]
* [https://www.ams.org/mathimagery/ Mathematical Imagery] – [[American Mathematical Society]]
* [https://web.archive.org/web/20150507151115/http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml Mathematics in Art and Architecture] – National University of Singapore
* [http://virtualmathmuseum.org/mathart/MathematicalArt.html Mathematical Art] – Virtual Math Museum
* [https://www.sciencenews.org/article/when-art-and-math-collide When art and math collide] – [[Science News]]
* [https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/dec/02/why-the-history-of-maths-is-also-the-history-of-art Why the history of maths is also the history of art]: Lynn Gamwell in ''[[The Guardian]]''

{{Mathematics and art|}}
{{Aesthetics}}
{{Areas of mathematics}}

{{DEFAULTSORT:Mathematics and Art}}
[[Category:Mathematics and art| ]]<!--please leave the empty space as standard-->
[[Category:Art history]]
[[Category:Mathematics and culture|Art]]
[[Category:Visual arts]]
[[Category:Applied mathematics]]