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Mathematics and art are related in a variety of ways.Mathematics has itself been described as anartmotivated by beauty. Mathematics can be discerned in arts such asmusic,dance,painting,architecture,sculpture, andtextiles. This article focuses, however, on mathematics in the visual arts.
Mathematics and art have a long historical relationship.Artists have used mathematics since the 4th century BC when the GreeksculptorPolykleitos wrotehisCanon, prescribing proportionsconjectured to have been based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of thegolden ratio in ancient art and architecture, without reliable evidence. In the ItalianRenaissance,Luca Pacioli wrote the influential treatiseDe divina proportione (1509), illustrated with woodcuts byLeonardo da Vinci, on the use of the golden ratio in art. Another Italian painter,Piero della Francesca, developedEuclid's ideas onperspective in treatises such asDe Prospectiva Pingendi, and in his paintings. The engraverAlbrecht Dürer made many references to mathematics in his workMelencolia I. In modern times, thegraphic artistM. C. Escher made intensive use oftessellation andhyperbolic geometry, with the help of the mathematicianH. S. M. Coxeter, while theDe Stijl movement led byTheo van Doesburg andPiet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such asquilting,knitting,cross-stitch,crochet,embroidery,weaving,Turkish and othercarpet-making, as well askilim. InIslamic art, symmetries are evident in forms as varied as Persiangirih and Moroccanzellige tilework,Mughaljali pierced stone screens, and widespreadmuqarnas vaulting.
Mathematics has directly influenced art with conceptual tools such aslinear perspective, the analysis ofsymmetry, and mathematical objects such aspolyhedra and theMöbius strip.Magnus Wenninger creates colourfulstellated polyhedra, originally as models for teaching. Mathematical concepts such asrecursion and logical paradox can be seen in paintings byRené Magritte and in engravings by M. C. Escher.Computer art often makes use offractals including theMandelbrot set, and sometimes explores other mathematical objects such ascellular automata. Controversially, the artistDavid Hockney has argued that artists from the Renaissance onwards made use of thecamera lucida to draw precise representations of scenes; the architect Philip Steadman similarly argued thatVermeer used thecamera obscura in his distinctively observed paintings.
Other relationships include the algorithmic analysis of artworks byX-ray fluorescence spectroscopy, the finding that traditionalbatiks from different regions ofJava have distinctfractal dimensions, and stimuli to mathematics research, especiallyFilippo Brunelleschi's theory of perspective, which eventually led toGirard Desargues'sprojective geometry. A persistent view, based ultimately on thePythagorean notion of harmony in music, holds that everything was arranged by Number, that God is the geometer of the world, and that therefore the world'sgeometry is sacred.
Polykleitos the elder (c. 450–420 BC) was aGreeksculptor from the school ofArgos, and a contemporary ofPhidias. His works and statues consisted mainly of bronze and were of athletes. According to the philosopher and mathematicianXenocrates, Polykleitos is ranked as one of the most important sculptors ofclassical antiquity for his work on theDoryphorus and the statue ofHera in theHeraion of Argos.[3] While his sculptures may not be as famous as those of Phidias, they are much admired. InhisCanon, 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.[3]
TheCanon 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:√2 (about 1:1.4142).[4]
The influence of theCanon of Polykleitos is immense inClassical Greek,Roman, andRenaissance 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 thatPythagorean thought influenced theCanon of Polykleitos.[5] TheCanon applies the basic mathematical concepts of Greek geometry, such as the ratio, proportion, andsymmetria (Greek for "harmonious proportions") and turns it into a system capable of describing the human form through a series of continuousgeometric progressions.[4]
In classical times, rather than making distant figures smaller withlinear perspective, painters sized objects and figures according to their thematic importance. In the Middle Ages, some artists usedreverse perspective for special emphasis. The Muslim mathematicianAlhazen (Ibn al-Haytham) described a theory of optics in hisBook of Optics in 1021, but never applied it to art.[6] The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understandnature and thearts. 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.[7]
The rudiments of perspective arrived withGiotto (1266/7 – 1337), who attempted to draw in perspective using an algebraic method to determine the placement of distant lines. In 1415, the ItalianarchitectFilippo Brunelleschi and his friendLeon Battista Alberti demonstrated the geometrical method of applying perspective in Florence, usingsimilar triangles as formulated by Euclid, to find the apparent height of distant objects.[8][9] Brunelleschi's own perspective paintings are lost, butMasaccio's painting of the Holy Trinity shows his principles at work.[6][10][11]
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The Italian painterPaolo Uccello (1397–1475) was fascinated by perspective, as shown in his paintings ofThe Battle of San Romano (c. 1435–1460): broken lances lie conveniently along perspective lines.[12][13]
The painterPiero della Francesca (c. 1415–1492) exemplified this new shift in Italian Renaissance thinking. He was an expertmathematician andgeometer, writing books onsolid geometry andperspective, includingDe prospectiva pingendi (On Perspective for Painting),Trattato d'Abaco (Abacus Treatise), andDe quinque corporibus regularibus (On the Five Regular Solids).[14][15][16] The historianVasari in hisLives of the Painters calls Piero the "greatest geometer of his time, or perhaps of any time."[17] Piero's interest in perspective can be seen in his paintings including thePolyptych of Perugia,[18] theSan Agostino altarpiece andThe Flagellation of Christ. His work on geometry influenced later mathematicians and artists includingLuca Pacioli in hisDe divina proportione andLeonardo da Vinci. Piero studied classical mathematics and the works ofArchimedes.[19] He was taught commercial arithmetic in "abacus schools"; his writings are formatted like abacus school textbooks,[20] perhaps including Leonardo Pisano (Fibonacci)'s 1202Liber Abaci.Linear perspective was just being introduced into the artistic world. Alberti explained in his 1435De pictura: "light rays travel in straight lines from points in the observed scene to the eye, forming a kind ofpyramid with the eye as vertex." A painting constructed with linear perspective is across-section of that pyramid.[21]
InDe 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".[a][7] He usesdeductive logic to lead the reader to the perspective representation of a three-dimensional body.[22]
The artistDavid Hockneyargued in his bookSecret Knowledge: Rediscovering the Lost Techniques of the Old Masters that artists started using acamera lucida from the 1420s, resulting in a sudden change in precision and realism, and that this practice was continued by major artists includingIngres,Van Eyck, andCaravaggio.[23] Critics disagree on whether Hockney was correct.[24][25] Similarly, the architect Philip Steadman argued controversially[26] thatVermeer had used a different device, thecamera obscura, to help him create his distinctively observed paintings.[27]
In 1509,Luca Pacioli (c. 1447–1517) publishedDe divina proportione onmathematical andartisticproportion, 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.[28] These, such as therhombicuboctahedron, 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 ofPiero della Francesca,Melozzo da Forlì, andMarco Palmezzano.[29] Leonardo studied Pacioli'sSumma, from which he copied tables of proportions.[30] InMona Lisa andThe Last Supper, Leonardo's work incorporated linear perspective with avanishing point to provide apparent depth.[31]The Last Supper is constructed in a tight ratio of 12:6:4:3, as isRaphael'sThe School of Athens, which includes Pythagoras with a tablet of ideal ratios, sacred to the Pythagoreans.[32][33] InVitruvian Man, Leonardo expressed the ideas of the Roman architectVitruvius, innovatively showing the male figure twice, and centring him in both a circle and a square.[34]
As early as the 15th century,curvilinear perspective found its way into paintings by artists interested in image distortions.Jan van Eyck's 1434Arnolfini Portrait contains a convex mirror with reflections of the people in the scene,[35] whileParmigianino'sSelf-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.[36]
Three-dimensional space can be represented convincingly in art, as intechnical 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 theUkiyo-e paintings ofTorii Kiyonaga (1752–1815).[37]
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Thegolden ratio (roughly equal to 1.618) was known toEuclid.[38] The golden ratio has persistently been claimed[39][40][41][42] in modern times to have been used in art andarchitecture by the ancients in Egypt, Greece and elsewhere, without reliable evidence.[43] The claim may derive from confusion with "golden mean", which to the Ancient Greeks meant "avoidance of excess in either direction", not a ratio.[43]Pyramidologists since the 19th century have argued on dubious mathematical grounds for the golden ratio in pyramid design.[b] TheParthenon, a 5th-century BC temple in Athens, has been claimed to use the golden ratio in itsfaçade and floor plan,[47][48][49] but these claims too are disproved by measurement.[43] TheGreat Mosque of Kairouan in Tunisia has similarly been claimed to use the golden ratio in its design,[50] but the ratio does not appear in the original parts of the mosque.[51] The historian of architectureFrederik Macody Lund argued in 1919 that theCathedral of Chartres (12th century),Notre-Dame of Laon (1157–1205) andNotre Dame de Paris (1160) are designed according to the golden ratio,[52] 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.[53] 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. SuchFibonacci ratios quickly become hard to distinguish from the golden ratio.[54] After Pacioli, the golden ratio is more definitely discernible in artworks including Leonardo'sMona Lisa.[55]
Another ratio, the only other morphic number,[56] was named theplastic number[c] in 1928 by the Dutch architectHans van der Laan (originally namedle nombre radiant in French).[57] Its value is the solution of thecubic equation
an irrational number which is approximately 1.325. According to the architectRichard Padovan, this has characteristic ratios3/4 and1/7, which govern the limits of human perception in relating one physical size to another. Van der Laan used these ratios when designing the 1967St. Benedictusberg Abbey church in the Netherlands.[57]
Planar symmetries have for millennia been exploited in artworks such ascarpets, lattices, textiles and tilings.[59][60][61][62]
Many traditional rugs, whether pile carpets or flatweavekilims, 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.[59] In kilims fromAnatolia, themotifs 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 awallpaper group such as pmm, while the border may be laid out as a frieze offrieze 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.[59]The mathematician and architectural theoristNikos Salingaros suggests that the "powerful presence"[58] (aesthetic effect) of a "great carpet"[58] such as the best Konya two-medallion carpets of the 17th century is created by mathematical techniques related to the theories of the architectChristopher 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.[58]
Elaborate lattices are found in IndianJali work, carved in marble to adorn tombs and palaces.[60] 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.[63] Many Chinese lattices have been analysed mathematically by Daniel S. Dye; he identifiesSichuan as the centre of the craft.[64]
Symmetries are prominent intextile arts includingquilting,[61]knitting,[65]cross-stitch,crochet,[66]embroidery[67][68] andweaving,[69] where they may be purely decorative or may be marks of status.[70]Rotational symmetry is found in circular structures such asdomes; these are sometimes elaborately decorated with symmetric patterns inside and out, as at the 1619Sheikh Lotfollah Mosque inIsfahan.[71] Items of embroidery andlace work such as tablecloths and table mats, made using bobbins or bytatting, can have a wide variety of reflectional and rotational symmetries which are being explored mathematically.[72]
Islamic artexploits symmetries in many of its artforms, notably ingirih 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/5radians), offering fivefold and tenfold symmetries. The tiles are decorated withstrapwork lines (girih), generally more visible than the tile boundaries. In 2007, the physicistsPeter Lu andPaul Steinhardt argued that girih resembledquasicrystallinePenrose tilings.[73] Elaborate geometriczellige tilework is a distinctive element inMoroccan architecture.[62]Muqarnas vaults are three-dimensional but were designed in two dimensions with drawings of geometrical cells.[74]
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![Tupa Inca tunic from Peru, 1450 –1540, an Andean textile denoting high rank[70]](/mt/?noimg=&dark=on&url=https%3a%2f%2fen.wikipedia.org%2fwiki%2fFile%3aTupa-inca-tunic.png)
ThePlatonic solids and otherpolyhedra are a recurring theme in Western art. They are found, for instance, in a marble mosaic featuring thesmall stellated dodecahedron, attributed to Paolo Uccello, in the floor of theSan Marco Basilica in Venice;[12] in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations forLuca Pacioli's 1509 bookThe Divine Proportion;[12] as a glassrhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495;[12] in the truncated polyhedron (and various other mathematical objects) inAlbrecht Dürer's engravingMelencolia I;[12] and inSalvador Dalí's paintingThe Last Supper in which Christ and his disciples are pictured inside a giantdodecahedron.[75]
Albrecht Dürer (1471–1528) was aGerman Renaissanceprintmaker who made important contributions to polyhedral literature in his 1525 book,Underweysung der Messung (Education on Measurement), meant to teach the subjects oflinear perspective,geometry inarchitecture,Platonic solids, andregular polygons. Dürer was likely influenced by the works ofLuca Pacioli andPiero della Francesca during his trips toItaly.[76] While the examples of perspective inUnderweysung 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 ofpolyhedral nets, polyhedra unfolded to lie flat for printing.[77] Dürer published another influential book onhuman proportions calledVier Bücher von Menschlicher Proportion (Four Books on Human Proportion) in 1528.[78]
Dürer's well-known engravingMelencolia I depicts a frustrated thinker sitting by atruncated triangular trapezohedron and amagic square.[1] 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,[1][79][80] including a two-volume book by Peter-Klaus Schuster,[81] and an influential discussion inErwin Panofsky's monograph of Dürer.[1][82]
Salvador Dalí's 1954 paintingCorpus Hypercubus uniquely depicts the cross of Christ as an unfolded three-dimensional net for ahypercube, also known as atesseract: 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.[83][84] 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 mathematicianThomas Banchoff states that Dalí was trying to go beyond the three-dimensional world, while the poet and art criticKelly 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."[85]
Traditional Indonesian wax-resistbatik designs on cloth combinerepresentational 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 afractal dimension between 1 and 2, varying in different regional styles. For example, the batik ofCirebon has a fractal dimension of 1.1; the batiks ofYogyakarta andSurakarta (Solo) in CentralJava have a fractal dimension of 1.2 to 1.5; and the batiks ofLasem on the north coast of Java and ofTasikmalaya in West Java have a fractal dimension between 1.5 and 1.7.[86]
Thedrip painting works of the modern artistJackson Pollock are similarly distinctive in their fractal dimension. His 1948Number 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.[87]
The astronomerGalileo Galilei in hisIl Saggiatore wrote that "[The universe] is written inthe language of mathematics, and its characters are triangles, circles, and other geometric figures."[88] 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 mathematicianFelipe Cucker suggests that mathematics, and especially geometry, is a source of rules for "rule-driven artistic creation", though not the only one.[89] Some of the many strands of the resulting complex relationship[90] are described below.
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.[91] King cites the mathematicianG. H. Hardy's 1940 essayA Mathematician's Apology. In it, Hardy discusses why he finds two theorems ofclassical times as first rate, namelyEuclid's proof there are infinitely manyprime numbers, and the proof that the square root of 2 isirrational. King evaluates this last against Hardy's criteria formathematical elegance: "seriousness, depth, generality, unexpectedness, inevitability, andeconomy" (King's italics), and describes the proof as "aesthetically pleasing".[92] The Hungarian mathematicianPaul Erdős agreed that mathematics possessed beauty but considered the reasons beyond explanation: "Why are numbers beautiful? It's like asking why isBeethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. Iknow numbers are beautiful."[93]
Mathematics can be discerned in many of the arts, such asmusic,dance,[94]painting,architecture, andsculpture. Each of these is richly associated with mathematics.[95] Among the connections to the visual arts, mathematics can provide tools for artists, such as the rules oflinear perspective as described byBrook Taylor andJohann Lambert, or the methods ofdescriptive geometry, now applied in software modelling of solids, dating back to Albrecht Dürer andGaspard Monge.[96] Artists from Luca Pacioli in theMiddle Ages and Leonardo da Vinci and Albrecht Dürer in theRenaissance have made use of and developed mathematical ideas in the pursuit of their artistic work.[95][97] The use of perspective began, despite some embryonic usages in the architecture of Ancient Greece, with Italian painters such asGiotto in the 13th century; rules such as thevanishing point were first formulated byBrunelleschi in about 1413,[6] his theory influencing Leonardo and Dürer.Isaac Newton's work on theoptical spectrum influencedGoethe'sTheory of Colours and in turn artists such asPhilipp Otto Runge,J. M. W. Turner,[98] thePre-Raphaelites andWassily Kandinsky.[99][100] Artists may also choose to analyse thesymmetry of a scene.[101] Tools may be applied by mathematicians who are exploring art, or artists inspired by mathematics, such asM. C. Escher (inspired byH. S. M. Coxeter) and the architectFrank Gehry, who more tenuously argued thatcomputer aided design enabled him to express himself in a wholly new way.[102]
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 thatfractals 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 theMandelbrot set, an image generated by acellular automaton algorithm, and acomputer-rendered image, and discusses, with reference to theTuring test, whetheralgorithmic products can be art.[103] Sasho Kalajdzievski'sMath and Art: An Introduction to Visual Mathematics takes a similar approach, looking at suitably visual mathematics topics such as tilings, fractals and hyperbolic geometry.[104]
Some of the first works of computer art were created byDesmond Paul Henry's "Drawing Machine 1", ananalogue machine based on abombsight computer and exhibited in 1962.[105][106] The machine was capable of creating complex, abstract, asymmetrical, curvilinear, but repetitive line drawings.[105][107] 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.[108][109][110] Artists such as Mikael Hvidtfeldt Christensen create works ofgenerative oralgorithmic art by writing scripts for a software system such asStructure Synth: the artist effectively directs the system to apply a desired combination of mathematical operations to a chosen set of data.[111][112]
The mathematician andtheoretical physicistHenri Poincaré'sScience and Hypothesis was widely read by theCubists, includingPablo Picasso andJean Metzinger.[114][115] Being thoroughly familiar withBernhard Riemann's work on non-Euclidean geometry, Poincaré was more than aware thatEuclidean geometry is just one of many possible geometric configurations, rather than as an absolute objective truth. The possible existence of afourth dimension inspired artists to question classicalRenaissance perspective:non-Euclidean geometry became a valid alternative.[116][117][118] The concept that painting could be expressed mathematically, in colour and form, contributed to Cubism, the art movement that led toabstract art.[119] Metzinger, in 1910, wrote that: "[Picasso] lays out a free, mobile perspective, from which that ingenious mathematicianMaurice Princet has deduced a whole geometry".[120] 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 invokedn-dimensional continuums. He loved to get the artists interested in thenew views on space that had been opened up bySchlegel and some others. He succeeded at that.[121]
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 theDadaistsMan Ray,[122]Marcel Duchamp[123] andMax Ernst,[124][125] and following Man Ray,Hiroshi Sugimoto.[126]
Man Ray photographed some of the mathematical models in theInstitut Henri Poincaré in Paris, includingObjet mathematique (Mathematical object). He noted that this representedEnneper surfaces with constantnegative curvature, derived from thepseudo-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 onShakespeare's plays, such as his 1934 paintingAntony and Cleopatra.[127] The art reporter Jonathan Keats, writing inForbesLife, argues that Man Ray photographed "the elliptic paraboloids and conic points in the same sensual light as his pictures ofKiki de Montparnasse", and "ingeniously repurposes the cool calculations of mathematics to reveal the topology of desire".[128] Twentieth century sculptors such asHenry Moore,Barbara Hepworth andNaum Gabo took inspiration from mathematical models.[129] Moore wrote of his 1938Stringed Mother and Child: "Undoubtedly the source of my stringed figures was theScience 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."[130]
The artistsTheo van Doesburg andPiet Mondrian founded theDe Stijl movement, which they wanted to "establish a visual vocabulary comprised of elementary geometrical forms comprehensible by all and adaptable to any discipline".[131][132] 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.[131] After the breakup of De Stijl, Van Doesburg founded theAvant-gardeArt Concret movement, describing his 1929–1930Arithmetic Composition, a series of four black squares on the diagonal of a squared background, as "a structure that can be controlled, adefinite surface without chance elements or individual caprice", yet "not lacking in spirit, not lacking the universal and not ... empty as there iseverything 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.[133]
The mathematics oftessellation, polyhedra, shaping of space, and self-reference provided the graphic artistM. C. Escher (1898—1972) with a lifetime's worth of materials for his woodcuts.[134][135] In theAlhambra 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, andtranslations 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'sAscending and Descending is based on the "impossible staircase" created by the medical scientistLionel Penrose and his son the mathematicianRoger Penrose.[136][137][138]
Some of Escher's many tessellation drawings were inspired by conversations with the mathematicianH. S. M. Coxeter onhyperbolic geometry.[139] Escher was especially interested in five specific polyhedra, which appear many times in his work. ThePlatonic solids—tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons—are especially prominent inOrder and Chaos andFour Regular Solids.[140] 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.[141]
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 ofpolyhedra and sculpts objects inspired by them;Magnus Wenninger makes "especially beautiful" models ofcomplex stellated polyhedra.[142]
The distorted perspectives ofanamorphosis have been explored in art since the sixteenth century, whenHans Holbein the Younger incorporated a severely distorted skull in his 1533 paintingThe Ambassadors. Many artists since then, including Escher, have make use of anamorphic tricks.[143]
The mathematics oftopology has inspired several artists in modern times. The sculptorJohn Robinson (1935–2007) created works such asGordian Knot andBands of Friendship, displayingknot theory in polished bronze.[7] Other works by Robinson explore the topology oftoruses.Genesis is based onBorromean rings – a set of three circles, no two of which link but in which the whole structure cannot be taken apart without breaking.[144] The sculptorHelaman Ferguson creates complexsurfaces and othertopological objects.[145] His works are visual representations of mathematical objects;The Eightfold Way is based on theprojective special linear groupPSL(2,7), a finite group of 168 elements.[146][147] The sculptorBathsheba Grossman similarly bases her work on mathematical structures.[148][149] The artistNelson Saiers incorporates mathematical concepts and theorems in his art fromtoposes andschemes to thefour color theorem and the irrationality ofπ.[150]
A liberal arts inquiry project examines connections between mathematics and art through theMöbius strip,flexagons, origami andpanorama photography.[151]
Mathematical objects including theLorenz manifold and thehyperbolic plane have been crafted usingfiber arts including crochet.[d][153] The American weaverAda Dietz wrote a 1949 monographAlgebraic Expressions in Handwoven Textiles, defining weaving patterns based on the expansion of multivariatepolynomials.[154] The mathematicianDaina Taimiņa demonstrated features of the hyperbolic plane by crocheting in 2001.[155] This ledMargaret and Christine Wertheim to crochet acoral reef, consisting of many marine animals such asnudibranchs whose shapes are based on hyperbolic planes.[156] The mathematicianJ. C. P. Miller used theRule 90cellular automaton to designtapestries depicting both trees and abstract patterns of triangles.[157] The "mathekniticians"[158] Pat Ashforth and Steve Plummer use knitted versions of mathematical objects such ashexaflexagons in their teaching, though theirMenger sponge proved too troublesome to knit and was made of plastic canvas instead.[159][160] Their "mathghans" (Afghans for Schools) project introducedknitting into the British mathematics and technology curriculum.[161][162]
![Four-dimensional space to Cubism: Esprit Jouffret's 1903 Traité élémentaire de géométrie à quatre dimensions.[163][e]](/mt/?noimg=&dark=on&url=https%3a%2f%2fen.wikipedia.org%2fwiki%2fFile%3aJouffret.gif)
Modelling is far from the only possible way to illustrate mathematical concepts. Giotto'sStefaneschi Triptych, 1320, illustratesrecursion in the form ofmise en abyme; the central panel of the triptych contains, lower left, the kneeling figure of Cardinal Stefaneschi, holding up the triptych as an offering.[165]Giorgio de Chirico'smetaphysical paintings such as his 1917Great Metaphysical Interior explore the question of levels of representation in art by depicting paintings within his paintings.[166]
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Art can exemplify logical paradoxes, as in some paintings by thesurrealistRené Magritte, which can be read assemiotic jokes about confusion between levels. InLa 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.[167] Earlier artists such asRembrandt had already explored the question of framing, in multiple paintings such as his 1646The Holy Family with a Curtain. That work depicts both the scene and its frame (both wood and curtain), introducing a level confusion between reality and depiction of reality.[168] Alberti, in his discussion of perspective, had likened paintings to open windows on to the scenes depicted.[169][168] InLa condition humaine, Magritte, in the words of the art historian András Rényi, "escalates the subversive power of the iconic difference that is already manifest in Rembrandt, and heightens it to an open paradox."[168] In Rényi's view, the resulting visual joke is the entire purpose of the painting, subverting the art of painting. He comments thatLa condition humaine is "more of a painted, meta-painterly philosopheme than a painterly work".[168]
Another approach to mathematical paradox is taken in Escher'sPrint Gallery (1956); this is a print which depicts a distorted city which contains a gallery whichrecursively contains the picture, and soad infinitum.[167] Magritte made use of spheres and cuboids to distort reality in a different way, painting them alongside an assortment of houses in his 1931Mental Arithmetic as if they were children's building blocks, but house-sized.[170]The Guardian observed that the "eerie toytown image" prophesiedModernism's usurpation of "cosy traditional forms", but also plays with the human tendency to seekpatterns in nature.[171]
Salvador Dalí's last painting,The Swallow's Tail (1983), was part of a series inspired byRené Thom'scatastrophe theory.[173] The Spanish painter and sculptorPablo 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 asAngular I andAutomnes, Palazuelo expressed himself in geometric transformations.[7]
The artist Adrian Gray practisesstone balancing, exploitingfriction and thecentre of gravity to create striking and seemingly impossible compositions.[174]
Artists, however, do not necessarily take geometry literally. AsDouglas 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 andnon-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 seeminglyparadoxical lithographPrint 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.[172] The image's central void has also attracted the interest of mathematicians Bart de Smit andHendrik Lenstra, who propose that it could contain aDroste effect copy of itself, rotated and shrunk; this would be a further illustration of recursion beyond that noted by Hofstadter.[175][176]
Algorithmic analysis of images of artworks, for example usingX-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.[177][178]
The 20th centuryDadaist artist Max Ernst paintedLissajous figures directly by swinging a punctured bucket of paint over a canvas. He had himself photographed in the act of making the mathematical figures in New York in 1942. He suspended the paint container by a string from a second string, which was in turn attached at two points to a rod. He allowed the container to swing freely over a square sheet to create the artwork, while he sat nearby, dressed in a suit and tie, watching the process.[179][f] Ernst was the first to introduce the use of this semi-automatic[180] form ofdrip painting (also called "oscillation"[180]), which he popularised.[181] He applied the technique to multiple paintings during his artistic career, including his 1942 worksThe Bewildered Planet,Surrealism and Painting, andYoung Man Intrigued by the Flight of a Non-Euclidean Fly, and his 1970 workGreen Zone.[182][183] Lissajous figures later featured repeatedly in earlycomputer art.[182] Ernst's use of the mathematical technique likely influencedJackson Pollock's drip painting style.[184] Pollock's paintings have a definitefractal dimension[185] of controlledchaos.[186]
The computer scientistNeil Dodgson investigated whetherBridget Riley's stripe paintings could be characterised mathematically, concluding that while separation distance could "provide some characterisation" and globalentropy 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.[187]
The American mathematicianGeorge Birkhoff's 1933Aesthetic Measure proposes a quantitative metric of theaesthetic 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 asO/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.[188]
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 ofBrook Taylor andJohann Heinrich Lambert on the mathematical foundations of perspective drawing,[189] and ultimately to the mathematics ofprojective geometry ofGirard Desargues andJean-Victor Poncelet.[190]
The Japanese paper-folding art oforigami has been reworked mathematically byTomoko Fuséusing modules, congruent pieces of paper such as squares, and making them into polyhedra or tilings.[191] Paper-folding was used in 1893 by T. Sundara Rao in hisGeometric Exercises in Paper Folding to demonstrate geometrical proofs.[192] Themathematics of paper folding has been explored inMaekawa's theorem,[193]Kawasaki's theorem,[194] and theHuzita–Hatori axioms.[195]
![Mathematical origami: Spring Into Action, by Jeff Beynon, made from a single paper rectangle.[196]](/mt/?noimg=&dark=on&url=https%3a%2f%2fen.wikipedia.org%2fwiki%2fFile%3aOrigami_spring.jpg)
Optical illusions such as theFraser spiral strikingly demonstrate limitations in human visual perception, creating what theart historianErnst Gombrich called a "baffling trick." The black and white ropes that appear to formspirals are in factconcentric circles. The mid-twentieth centuryop 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 asBridget Riley, Spyros Horemis,[197] andVictor Vasarely.[198]
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 toPlato, 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 aPythagorean 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, theregular or Platonic solids dictate the proportions found in nature, and in art.[199][200] An illumination in the 13th-centuryCodex 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).[201] In 1596, the mathematical astronomerJohannes Kepler modelled the universe as a set of nested Platonic solids, determining the relative sizes of the orbits of the planets.[201]William Blake'sAncient of Days (depictingUrizen, Blake's embodiment of reason and law) and his painting of the physicistIsaac Newton, naked, hunched and drawing with a compass, use the symbolism of compasses to critique conventional reason and materialism as narrow-minded.[202][203]Salvador Dalí's 1954Crucifixion (Corpus Hypercubus) depicts the cross as ahypercube, representing the divine perspective with four dimensions rather than the usual three.[84] In Dalí'sThe Sacrament of the Last Supper (1955) Christ and his disciples are pictured inside a giantdodecahedron.[204]
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.
{{cite book}}: CS1 maint: location missing publisher (link){{cite book}}: CS1 maint: location missing publisher (link)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.
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.
{{cite book}}: CS1 maint: ignored ISBN errors (link)hen describing his method of perspective in his treatise De Pictura, Leon Battista Alberti compared a painting with an aperta finestra, an open window.
Oscillation In around 1942, while an exile in the USA, Max Ernst started developing the technique of oscillation. He let paint drip out of a tin perforated with a number of holes, which he attached to a long string and swung to and fro over the canvas.
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.
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.
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