Mathematics and art#mathekniticians

{{further|Artistic canons of body proportions}}

{{Anchor|Polykleitos}}

File:Doryphoros MAN Napoli Inv6011-2.jpg, originally a bronze by Polykleitos]]

{{further|Polykleitos}}

Polykleitos the elder (c. 450–420 BC) was a Greek sculptor from the school of 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.{{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 }} While his sculptures may not be as famous as those of Phidias, they are much admired. In 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.

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).{{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 }}

The influence of the Canon of Polykleitos is immense in Classical Greek, 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 Pythagorean thought influenced the Canon of Polykleitos.{{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 }} 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 progressions.

{{main|Perspective (graphical)}}

File:Brunelleschi's perspective experiment.jpg'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. 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.{{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}}

The rudiments of perspective arrived with 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.{{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 }}{{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/}} Brunelleschi's own perspective paintings are lost, but Masaccio's painting of the Holy Trinity shows his principles at work.{{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}}{{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}}

File:San Romano Battle (Paolo Uccello, London) 01.jpg made innovative use of perspective in The Battle of San Romano (c. 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.{{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}}{{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.}}

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, including De prospectiva pingendi (On Perspective for Painting), Trattato d'Abaco (Abacus Treatise), and De quinque corporibus regularibus (On the Five Regular Solids).{{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}}{{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}}{{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}} The historian Vasari in his Lives of the Painters calls Piero the "greatest geometer of his time, or perhaps of any time."{{cite book |last=Vasari |first=Giorgio |author-link=Giorgio Vasari |title=Le Opere, volume 2 |editor=G. Milanesi |date=1878 |page=490}} Piero's interest in perspective can be seen in his paintings including the Polyptych of Perugia,{{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]}} 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.{{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]}} He was taught commercial arithmetic in "abacus schools"; his writings are formatted like abacus school textbooks,{{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}} 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 of that pyramid.{{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}}

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".}} He uses deductive logic to lead the reader to the perspective representation of a three-dimensional body.{{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 }}

The artist David Hockney 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 Ingres, Van Eyck, and Caravaggio.{{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}} Critics disagree on whether Hockney was correct.{{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}}{{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}} Similarly, the architect Philip Steadman argued controversially{{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}} that Vermeer had used a different device, the camera obscura, to help him create his distinctively observed paintings.{{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 }}

In 1509, Luca Pacioli (c. 1447–1517) published De divina proportione on mathematical and artistic 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.{{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}} 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.{{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}} Leonardo studied Pacioli's Summa, from which he copied tables of proportions.{{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 }} In Mona Lisa and The Last Supper, Leonardo's work incorporated linear perspective with a vanishing point to provide apparent depth.{{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 }} 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.{{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}}{{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 }} 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.{{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}}

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,{{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 }} 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.{{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, 306–307}}

Three-dimensional space can be represented convincingly in art, as in technical drawing, by means other than perspective. Oblique projections, 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).{{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}}

File:Pacioli De Divina Proportione Head Equilateral Triangle 1509.jpg|Woodcut from Luca Pacioli's 1509 De divina proportione 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's Vitruvian Man, c. 1490

File:Masaccio, trinità.jpg|Brunelleschi's theory of perspective: Masaccio's 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, 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: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 and Ban-sugoroku board games. Painting by Torii Kiyonaga, Japan, c. 1780

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

The golden ratio (roughly equal to 1.618) was known to Euclid.{{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}} The golden ratio has persistently been claimed{{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 }}{{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}}{{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 }}{{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 }} in modern times to have been used in art and architecture by the ancients in Egypt, Greece and elsewhere, without reliable evidence.{{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 }} The claim may derive from confusion with "golden mean", which to the Ancient Greeks meant "avoidance of excess in either direction", not a ratio. 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').{{cite book |first=Socrates G. |last=Taseos |title=Back in Time 3104 B.C. to the Great Pyramid |publisher=SOC Publishers |date=1990}} 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.{{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 }}{{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}}}} The Parthenon, a 5th-century BC temple in Athens, has been claimed to use the golden ratio in its façade and floor plan,{{cite book |author=Huntley, H.E. |title=The Divine Proportion |url=https://archive.org/details/divineproportion0000hunt |url-access=registration |publisher=Dover |year=1970}}{{cite book |author=Hemenway, Priya |year=2005 |title=Divine Proportion: Phi In Art, Nature, and Science |publisher=Sterling |page=96}}{{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}} but these claims too are disproved by measurement. The Great Mosque of Kairouan in Tunisia has similarly been claimed to use the golden ratio in its design,{{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 }} but the ratio does not appear in the original parts of the mosque.{{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}} 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,{{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}} 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.{{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 }} 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 ratios quickly become hard to distinguish from the golden ratio.{{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 }} After Pacioli, the golden ratio is more definitely discernible in artworks including Leonardo's Mona Lisa.{{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}}

Another ratio, the only other morphic number,{{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}} was named the plastic number{{efn|'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).{{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}} Its value is the solution of the cubic equation

:$x^3=x+1\backslash ,$,

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.

File:Mathematical Pyramid.svg|Base:hypotenuse(b:a) ratios for the Pyramid of Khufu could be: 1:φ (Kepler triangle), 3:5 (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.

{{further|Planar symmetry|Wallpaper group|Islamic geometric patterns|Kilim}}

File:Carpet with Double Medallion.jpg (Konya – Karapınar), circa 1600. Alâeddin Mosque]]

Planar symmetries have for millennia been exploited in artworks such as carpets, lattices, textiles and tilings.

Many traditional rugs, whether pile carpets or flatweave kilims, 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.{{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}} In kilims from Anatolia, the 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.

The mathematician and architectural theorist Nikos Salingaros suggests that the "powerful presence" (aesthetic effect) of a "great carpet" 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.{{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}}

Elaborate lattices are found in Indian Jali work, carved in marble to adorn tombs and palaces.{{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}} 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.{{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}} Many Chinese lattices have been analysed mathematically by Daniel S. Dye; he identifies Sichuan as the centre of the craft.{{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 }}

File:Girih tiles.svg]]

Symmetries are prominent in textile arts including quilting,{{cite book |last1=Ellison |first1=Elaine |last2=Venters |first2=Diana |publisher=Key Curriculum |title=Mathematical Quilts: No Sewing Required |year=1999}} knitting,{{cite journal |last1=belcastro |first1=sarah-marie |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 }} cross-stitch, crochet,{{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}} embroiderySnook, Barbara. Florentine Embroidery. Scribner, Second edition 1967.Williams, Elsa S. Bargello: Florentine Canvas Work. Van Nostrand Reinhold, 1967. and weaving,{{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 }} where they may be purely decorative or may be marks of status.{{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}} Rotational symmetry is found in circular structures such as domes; these are sometimes elaborately decorated with symmetric patterns inside and out, as at the 1619 Sheikh Lotfollah Mosque in Isfahan.{{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}} 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.{{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 }}

Islamic art exploits symmetries in many of its artforms, notably in 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 radians), 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 tilings.{{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 |volume=315 |pages=1106–1110 |doi=10.1126/science.1135491 |pmid=17322056 |issue=5815 |bibcode=2007Sci...315.1106L|s2cid=10374218 }} Elaborate geometric zellige tilework is a distinctive element in Moroccan architecture.{{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}} Muqarnas vaults are three-dimensional but were designed in two dimensions with drawings of geometrical cells.{{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}}

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 chamfered hexagonal lattice pattern

File:Salim Chishti Tomb-2.jpg|Jaali marble lattice at tomb of Salim Chishti, Fatehpur Sikri, India

File:Florentine Bargello Pattern.png|Symmetries: Florentine 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|Tessellations: 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|Tupa Inca tunic from Peru, 1450 –1540, an Andean textile denoting high rank

The Platonic solids and other 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 Basilica in Venice; in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's 1509 book The Divine Proportion; as a glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495; in the truncated polyhedron (and various other mathematical objects) in Albrecht Dürer's engraving Melencolia I; and in Salvador Dalí's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.{{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}}

Albrecht Dürer (1471–1528) was a 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.{{cite book |last=Panofsky |first=E. |title=The Life and Art of Albrecht Durer |publisher=Princeton |year=1955}} 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 polyhedral nets, polyhedra unfolded to lie flat for printing.{{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}} Dürer published another influential book on human proportions called Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion) in 1528.{{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}}

Dürer's well-known engraving Melencolia I depicts a frustrated thinker sitting by a truncated triangular trapezohedron and a magic square. 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,{{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 }}{{cite book |first=Campbell |last=Dodgson |title=Albrecht Dürer |location= London |publisher=Medici Society |year=1926 |page=94}} including a two-volume book by Peter-Klaus Schuster,{{cite book |last=Schuster |first=Peter-Klaus |title=Melencolia I: Dürers Denkbild |pages=17–83 |location=Berlin |publisher=Gebr. Mann Verlag |year=1991}} and an influential discussion in Erwin Panofsky's monograph of Dürer.{{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}}{{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}}

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.{{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 }}{{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}} 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."{{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}}

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

File:Batik pedalaman - parang klithik.JPGs 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 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 Lasem on the north coast of Java and of Tasikmalaya in West Java have a fractal dimension between 1.5 and 1.7.{{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 }}

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.{{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}}

The astronomer Galileo Galilei in his Il Saggiatore wrote that "[The universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures."{{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 }} 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.{{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}} Some of the many strands of the resulting complex relationship{{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}} are described below.

File:Ghhardy@72.jpg defined a set of criteria for mathematical beauty.]]

{{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.{{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}} 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 times as first rate, namely Euclid's proof there are infinitely many prime numbers, and the proof that the square root of 2 is irrational. King evaluates this last against Hardy's criteria for mathematical elegance: "seriousness, depth, generality, unexpectedness, inevitability, and economy" (King's italics), and describes the proof as "aesthetically pleasing".{{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}} 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 Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful."{{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}}

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

Mathematics can be discerned in many of the arts, such as music, dance,{{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}} painting, architecture, and sculpture. Each of these is richly associated with mathematics.{{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}} 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.{{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}} 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.{{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 }} 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,{{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}} his theory influencing Leonardo and Dürer. Isaac Newton's work on the optical spectrum influenced Goethe's Theory of Colours and in turn artists such as Philipp Otto Runge, J. M. W. Turner,{{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}} the Pre-Raphaelites and Wassily Kandinsky.{{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}}{{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}} Artists may also choose to analyse the symmetry of a scene.{{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}} 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.{{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}}

File:Octopod by syntopia.jpg produced with the software Structure Synth]]

The artist Richard 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 computer-rendered image, and discusses, with reference to the Turing test, whether algorithmic products can be art.{{cite journal |last=Wright |first=Richard |title=Some Issues in the Development of Computer Art as a Mathematical Art Form |journal=Leonardo |date=1988 |volume=1 |issue=Electronic Art, supplemental issue |pages=103–110 |doi=10.2307/1557919 |jstor=1557919}} 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.{{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}}

Some of the first works of computer art were created by Desmond Paul Henry's "Drawing Machine 1", an analogue machine based on a bombsight computer and exhibited in 1962.{{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 }}{{cite news |title=Computer Does Drawings: Thousands of lines in each |agency=The Guardian |date=17 September 1962}} in Beddard, 2015. The machine was capable of creating complex, abstract, asymmetrical, curvilinear, but repetitive line drawings.{{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. 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.{{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}}{{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}}{{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}} Artists such as Mikael Hvidtfeldt Christensen create works of 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.{{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].{{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}}

File:Bathsheba Grossman geometric art.jpg|Mathematical sculpture by Bathsheba Grossman, 2007

File:Hartmut Skerbisch.jpg|Fractal sculpture: 3D Fraktal 03/H/dd by 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.

File:Les Demoiselles d'Avignon.jpg: Pablo Picasso's 1907 painting Les Demoiselles d'Avignon uses a fourth dimension projection to show a figure both full face and in profile.{{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}}]]

{{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 Cubists, including Pablo Picasso and Jean Metzinger.{{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}} 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 inspired artists to question classical Renaissance perspective: non-Euclidean geometry became a valid alternative.{{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}}{{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 }}{{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 }} The concept that painting could be expressed mathematically, in colour and form, contributed to Cubism, the art movement that led to abstract art.{{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}} Metzinger, in 1910, wrote that: "[Picasso] lays out a free, mobile perspective, from which that ingenious mathematician Maurice Princet has deduced a whole geometry".{{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 }} Later, Metzinger wrote in his memoirs:

Maurice Princet joined us often ... 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 new views on space that had been opened up by Schlegel and some others. He succeeded at that.{{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 }}

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 Dadaists Man Ray,{{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}} Marcel Duchamp{{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}} and Max Ernst,{{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}}{{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}} and following Man Ray, Hiroshi Sugimoto.{{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}}

File:Objet mathematique by Man Ray.jpgs 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 surfaces 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.{{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}} 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".{{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}} Twentieth century sculptors such as Henry Moore, Barbara Hepworth and Naum Gabo took inspiration from mathematical models.{{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}} Moore wrote of his 1938 Stringed Mother and Child: "Undoubtedly the source of my stringed figures was the Science Museum ... I was fascinated by the mathematical models I saw there ... 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."{{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}}

File:Theo van Doesburg 122.jpg'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".{{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}}{{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}} 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. 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 ... 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.{{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}}

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.{{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 }}{{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}} 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 reflections, and 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 "impossible staircase" created by the medical scientist Lionel Penrose and his son the mathematician Roger Penrose.{{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}}{{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 }}{{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}}

Some of Escher's many tessellation drawings were inspired by conversations with the mathematician H. S. M. Coxeter on hyperbolic geometry.{{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}} 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.{{cite book |author=Escher, M.C. |title=The World of MC Escher |publisher=Random House |date=1988}} 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.{{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}}

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 complex stellated polyhedra.{{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}}

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. Many artists since then, including Escher, have make use of anamorphic tricks.{{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}}

The mathematics of topology has inspired several artists in modern times. The sculptor John Robinson (1935–2007) created works such as Gordian Knot and Bands of Friendship, displaying knot theory in polished bronze. Other works by Robinson explore the topology of toruses. 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.{{cite web |url=http://www.bradshawfoundation.com/jr/ |title=John Robinson |date=2007 |publisher=Bradshaw Foundation |access-date=13 August 2009}} The sculptor Helaman Ferguson creates complex surfaces and other topological objects.{{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 }} 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.{{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}}{{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 }} The sculptor Bathsheba Grossman similarly bases her work on mathematical structures.{{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}}{{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}} The artist Nelson Saiers incorporates mathematical concepts and theorems in his art from toposes and schemes to the four color theorem and the irrationality of π.{{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}}

A liberal arts inquiry project examines connections between mathematics and art through the Möbius strip, flexagons, origami and panorama photography.{{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}}

Mathematical objects including the Lorenz manifold and the hyperbolic plane have been crafted using 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.{{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}}}}{{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 }} The American weaver Ada Dietz wrote a 1949 monograph Algebraic Expressions in Handwoven Textiles, defining weaving patterns based on the expansion of multivariate polynomials.{{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 }} The mathematician Daina Taimiņa demonstrated features of the hyperbolic plane by crocheting in 2001.{{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 }}. This led Margaret and Christine Wertheim to crochet a coral reef, consisting of many marine animals such as nudibranchs whose shapes are based on hyperbolic planes.{{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}}

The mathematician J. C. P. Miller used the Rule 90 cellular automaton to design tapestries depicting both trees and abstract patterns of triangles.{{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 }} The "{{visible anchor|mathekniticians}}"{{cite web |title=Pat Ashforth & Steve Plummer – Mathekniticians |url=http://www.woollythoughts.com/aboutus.html |website=Woolly Thoughts |access-date=4 October 2015}} Pat Ashforth and Steve Plummer use knitted versions of mathematical objects such as hexaflexagons in their teaching, though their Menger sponge proved too troublesome to knit and was made of plastic canvas instead.{{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 }}{{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}} Their "mathghans" (Afghans for Schools) project introduced knitting into the British mathematics and technology curriculum.{{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}}{{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 }}

File:Jouffret.gif|Four-dimensional space to Cubism: Esprit Jouffret's 1903 Traité élémentaire de géométrie à quatre dimensions.{{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}}{{efn|Maurice Princet gave a copy to Pablo Picasso, whose sketchbooks for Les Demoiselles d'Avignon illustrate Jouffret's influence.{{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]}}{{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}}}}

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 stellated polyhedra, 2009

File:Moebiusstripscarf.jpg|A Möbius strip scarf in crochet, 2007

File:Hans Holbein the Younger - The Ambassadors - Google Art Project.jpg|Anamorphism: The Ambassadors by Hans Holbein the Younger, 1533, with severely distorted skull in foreground

File:The Föhr Reef in Tübingen.JPG|Crocheted coral reef: many animals modelled as hyperbolic planes with varying parameters by Margaret and Christine Wertheim. Föhr Reef, Tübingen, 2013

File:René Magritte The Human Condition.jpg joke: René Magritte's La condition humaine 1933]]

File:Giotto. The Stefaneschi Triptych (verso) c.1330 220x245cm. Pinacoteca, Vatican..jpg's Stefaneschi Triptych, 1320 illustrates recursion.]]

File:Giotto di Bondone - The Stefaneschi Triptych - St Peter Enthroned (detail) - WGA09356.jpg

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.{{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}} Giorgio de Chirico's metaphysical paintings such as his 1917 Great Metaphysical Interior explore the question of levels of representation in art by depicting paintings within his paintings.{{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}}

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 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 (1956) is a print which depicts a distorted city which contains a gallery which recursively contains the picture, and so ad infinitum.{{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}} 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.{{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}} 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.{{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}}

File:Escher Paradox Diagram.png in his 1980 book Gödel, Escher, Bach]]

Salvador Dalí's last painting, The Swallow's Tail (1983), was part of a series inspired by René Thom's catastrophe theory.{{cite book |last1=King |first1=Elliott |editor1-last=Ades |editor1-first=Dawn |title=Dali |date=2004 |publisher=Bompiani Arte |location=Milan |pages=418–421}} 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.

The artist Adrian Gray practises stone balancing, exploiting friction and the centre of gravity to create striking and seemingly impossible compositions.{{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 }}

File:Print Gallery by M. C. Escher.jpg 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 paradoxical 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.{{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}} 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.{{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}}{{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 }}

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.{{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}}{{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 }}

File:Max Ernst making Lissajous Figures 1942.jpg making Lissajous figures, New York, 1942]]

Jackson Pollock's drip painting style{{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}} has a definite fractal dimension;{{cite journal |author1=Taylor, Richard P. |author2=Micolich, Adam P. |author3=Jonas, David |title=Fractal analysis of Pollock's drip paintings |journal=Nature |volume=399 |issue=6735 |page=422 |date=June 1999 |doi=10.1038/20833 |bibcode=1999Natur.399..422T |s2cid=204993516 |doi-access=free }} among the artists who may have influenced Pollock's controlled chaos,{{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 }} Max Ernst painted Lissajous figures directly by swinging a punctured bucket of paint over a canvas.{{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}}

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.{{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 }}

The American mathematician George Birkhoff's 1933 Aesthetic Measure proposes a quantitative metric of the 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.{{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}}

{{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,{{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}} and ultimately to the mathematics of projective geometry of Girard Desargues and Jean-Victor Poncelet.{{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}}

The Japanese paper-folding art of origami has been reworked mathematically by Tomoko Fusé using modules, congruent pieces of paper such as squares, and making them into polyhedra or tilings.{{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}} Paper-folding was used in 1893 by T. Sundara Rao in his Geometric Exercises in Paper Folding to demonstrate geometrical proofs.{{cite book |last=Rao |first=T. Sundara |title=Geometric Exercises in Paper Folding |title-link=Geometric Exercises in Paper Folding |publisher=Addison |year=1893}} The mathematics of paper folding has been explored in Maekawa's theorem,{{cite journal |last=Justin |first=J. |title=Mathematics of Origami, part 9 |journal=British Origami |date=June 1986 |pages=28–30}}. Kawasaki's theorem,{{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}} and the Huzita–Hatori axioms.{{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 }}

File:Della Pittura Alberti perspective circle to ellipse.jpg|Stimulus to projective geometry: Alberti's diagram showing a circle seen in perspective as an ellipse. Della Pittura, 1435–1436

File:Origami spring.jpg|Mathematical origami: Spring Into Action, by Jeff Beynon, made from a single paper rectangle.[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.

{{further|Op art}}

File:Fraser spiral.svg, named for Sir James Fraser who discovered it in 1908.]]

Optical illusions such as the 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 spirals are in fact concentric circles. The mid-twentieth century 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, Spyros Horemis,{{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}} and Victor Vasarely.{{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}}

{{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 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 regular or Platonic solids dictate the proportions found in nature, and in art.{{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 }}{{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 }} 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), .{{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 }} 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. 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.{{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}}{{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 }}

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. In Dalí's The Sacrament of the Last Supper (1955) Christ and his disciples are pictured inside a giant dodecahedron.{{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 }}

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, c. 1800

• Mathematics and architecture

• Music and mathematics
• List of mathematical art software

{{notelist}}

{{reflist|28em}}

{{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] – 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:History of art

Art

Category:Visual arts

Category:Applied mathematics