Mathematics and art

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truncated rhombohedron, while measurement is indicated by the scales and hourglass.[1]
Wireframe drawing[2] of a vase as a solid of revolution[2] by Paolo Uccello. 15th century
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.

Mathematics and art have a long historical relationship.

zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas
vaulting.

Mathematics has directly influenced art with conceptual tools such as

Vermeer used the camera obscura
in his distinctively observed paintings.

Other relationships include the algorithmic analysis of artworks by

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

Origins: from ancient Greece to the Renaissance

Further information: Artistic canons of body proportions

Polykleitos's Canon and symmetria

Roman copy in marble of Doryphoros, originally a bronze by Polykleitos
Further information: Polykleitos

Doryphorus and the statue of Hera in the Heraion of Argos.[3] 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.[3]

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:√2 (about 1:1.4142).[4]

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.[5] 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.[4]

Perspective and proportion

Main article: Perspective (graphical)
linear perspective

In classical times, rather than making distant figures smaller with

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.[7]

The rudiments of perspective arrived with

similar triangles as formulated by Euclid, to find the apparent height of distant objects.[8][9] Brunelleschi's own perspective paintings are lost, but Masaccio's painting of the Holy Trinity shows his principles at work.[6][10][11]

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.[12][13]

The painter

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.[21]

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".

deductive logic to lead the reader to the perspective representation of a three-dimensional body.[22]

The artist

Vermeer had used a different device, the camera obscura, to help him create his distinctively observed paintings.[27]

In 1509,

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.[32][33] 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.[34]

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,[35] 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.[36]

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).[37]

Golden ratio

Further information: List of works designed with the golden ratio

The

Fibonacci ratios quickly become hard to distinguish from the golden ratio.[54] After Pacioli, the golden ratio is more definitely discernible in artworks including Leonardo's Mona Lisa.[55]

Another ratio, the only other morphic number,[56] was named the plastic number[c] in 1928 by the Dutch architect Hans van der Laan (originally named le nombre radiant in French).[57] Its value is the solution of the cubic equation

,

an irrational number which is approximately 1.325. According to the architect Richard Padovan, this has characteristic ratios 3/4 and 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.[57]

Planar symmetries

Further information:
Planar symmetry, Wallpaper group, Islamic geometric patterns, and Kilim
Powerful presence:[58] carpet with double medallion. Central Anatolia (Konya – Karapınar), turn of the 16th/17th centuries. Alâeddin Mosque

Planar symmetries have for millennia been exploited in artworks such as carpets, lattices, textiles and tilings.[59][60][61][62]

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.[59] 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.[59] The mathematician and architectural theorist Nikos 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 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.[58]

Elaborate lattices are found in Indian Jali 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 identifies Sichuan as the centre of the craft.[64]

Girih tiles

Symmetries are prominent in textile arts including quilting,[61] knitting,[65] cross-stitch, crochet,[66] embroidery[67][68] and weaving,[69] where they may be purely decorative or may be marks of status.[70] 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.[71] 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.[72]

zellige tilework is a distinctive element in Moroccan architecture.[62] Muqarnas vaults are three-dimensional but were designed in two dimensions with drawings of geometrical cells.[74]

Polyhedra

The

San Marco Basilica in Venice;[12] in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's 1509 book The Divine Proportion;[12] as a glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495;[12] in the truncated polyhedron (and various other mathematical objects) in Albrecht Dürer's engraving Melencolia I;[12] and in Salvador Dalí's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.[75]

human proportions called Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion) in 1528.[78]

Dürer's well-known engraving Melencolia I depicts a frustrated thinker sitting by a truncated triangular trapezohedron and a magic 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 in Erwin Panofsky's monograph of Dürer.[1][82]

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.[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 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."[85]

Fractal dimensions

Batiks 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.[86]

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.[87]

A complex relationship

The astronomer

Il Saggiatore wrote that "[The universe] is written in the 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 mathematician Felipe 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 G. H. Hardy defined a set of criteria for mathematical beauty.

Mathematics as an art

Main article: 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.[91] 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".[92] 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."[93]

Mathematical tools for art

Further information: List of mathematical artists, fractal art, and computer art

Mathematics can be discerned in many of the arts, such as

computer aided design enabled him to express himself in a wholly new way.[102]

Octopod by Mikael Hvidtfeldt Christensen. Algorithmic art 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 "

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.[103] 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.[104]

Some of the first works of computer art were created by

analogue machine based on a bombsight 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 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.[111][112]

From mathematics to art

Proto-Cubism: Pablo Picasso's 1907 painting Les Demoiselles d'Avignon uses a fourth dimension projection to show a figure both full face and in profile.[113]
Further information: Proto-Cubism, tessellation, M. C. Escher, Mathematics of paper folding, and 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.[114][115] 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.[116][117][118] The concept that painting could be expressed mathematically, in colour and form, contributed to Cubism, the art movement that led to abstract art.[119] Metzinger, in 1910, wrote that: "[Picasso] lays out a free, mobile perspective, from which that ingenious mathematician Maurice 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 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.[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 the

Dadaists Man Ray,[122] Marcel Duchamp[123] and Max Ernst,[124][125] and following Man Ray, Hiroshi Sugimoto.[126]

Dadaism: Man Ray
's 1934 Objet mathematique

Man Ray photographed some of the mathematical models in the

Kiki de Montparnasse", and "ingeniously repurposes the cool calculations of mathematics to reveal the topology of desire".[128] Twentieth century sculptors such as Henry Moore, Barbara Hepworth and Naum Gabo took inspiration from mathematical models.[129] 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."[130]

Theo van Doesburg's Six Moments in the Development of Plane to Space, 1926 or 1929

The artists Theo van Doesburg and Piet Mondrian founded the De Stijl movement, which they wanted to "establish a visual vocabulary 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 the Avant-garde Art Concret movement, describing his 1929–1930 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.[133]

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.[134][135] 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.[136][137][138]

Some of Escher's many tessellation drawings were inspired by conversations with the mathematician

Platonic solids—tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons—are especially prominent in Order and Chaos and Four 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.

polyhedra and sculpts objects inspired by them; Magnus Wenninger makes "especially beautiful" models of complex stellated polyhedra.[142]

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.[143]

The mathematics of

projective special linear group PSL(2,7), a finite group of 168 elements.[146][147] The sculptor Bathsheba Grossman similarly bases her work on mathematical structures.[148][149] 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.[150]

A liberal arts inquiry project examines connections between mathematics and art through the Möbius strip, flexagons, origami and panorama photography.[151]

Mathematical objects including the

Lorenz manifold and the hyperbolic plane have been crafted using fiber arts including crochet.[d][153] The American weaver Ada Dietz wrote a 1949 monograph Algebraic Expressions in Handwoven Textiles, defining weaving patterns based on the expansion of multivariate polynomials.[154] The mathematician Daina Taimiņa demonstrated features of the hyperbolic plane by crocheting in 2001.[155] 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.[156]
The mathematician
hexaflexagons in their teaching, though their Menger sponge proved too troublesome to knit and was made of plastic canvas instead.[159][160] Their "mathghans" (Afghans for Schools) project introduced knitting into the British mathematics and technology curriculum.[161][162]
La condition humaine
1933

Illustrating mathematics

Front face of Giotto's Stefaneschi Triptych, 1320 illustrates recursion.
Detail of Cardinal Stefaneschi holding the triptych

Modelling is far from the only possible way to illustrate mathematical concepts. Giotto's Stefaneschi Triptych, 1320, illustrates recursion in the form of mise en abyme; the central panel of the triptych contains, lower left, the kneeling figure of Cardinal Stefaneschi, holding up the triptych as an offering.[165] 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.[166]

Art can exemplify logical paradoxes, as in some paintings by the

recursively contains the picture, and so ad 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 1931 Mental Arithmetic as if they were children's building blocks, but house-sized.[168] 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.[169]

Diagram of the apparent paradox embodied in M. C. Escher's 1956 lithograph Print Gallery, as discussed by Douglas Hofstadter in his 1980 book Gödel, Escher, Bach[170]

Salvador Dalí's last painting, The Swallow's Tail (1983), was part of a series inspired by René Thom's catastrophe theory.[171] 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.[7]

The artist Adrian Gray practises

centre of gravity to create striking and seemingly impossible compositions.[172]

Lithograph Print Gallery by M. C. Escher, 1956

Artists, however, do not necessarily take geometry literally. As Douglas Hofstadter writes in his 1980 reflection on human thought, Gödel, Escher, Bach, by way of (among other things) the mathematics of art: "The difference between an Escher drawing and non-Euclidean geometry is that in the latter, comprehensible interpretations can be found for the undefined terms, resulting in a comprehensible total system, whereas for the former, the end result is not reconcilable with one's conception of the world, no matter how long one stares at the pictures." Hofstadter discusses the seemingly 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.[170] 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.[173][174]

Analysis of art history

Algorithmic analysis of images of artworks, for example using

X-ray fluorescence spectroscopy, can reveal information about art. Such techniques can uncover images in layers of paint later covered over by an artist; help art historians to visualize an artwork before it cracked or faded; help to tell a copy from an original, or distinguish the brushstroke style of a master from those of his apprentices.[175][176]

Lissajous figures
, New York, 1942

Lissajous figures directly by swinging a punctured bucket of paint over a canvas.[180]

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.[181]

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.[182]

Stimuli to mathematical research

Further information: Projective geometry and 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,[183] and ultimately to the mathematics of projective geometry of Girard Desargues and Jean-Victor Poncelet.[184]

The Japanese paper-folding art of

Tomoko Fusé using modules, congruent pieces of paper such as squares, and making them into polyhedra or tilings.[185] Paper-folding was used in 1893 by T. Sundara Rao in his Geometric Exercises in Paper Folding to demonstrate geometrical proofs.[186] The mathematics of paper folding has been explored in Maekawa's theorem,[187] Kawasaki's theorem,[188] and the Huzita–Hatori axioms.[189]

Illusion to Op art

Further information: Op art
The Fraser spiral illusion, named for Sir James Fraser who discovered it in 1908.

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,[191] and Victor Vasarely.[192]

Sacred geometry

Further information:
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.

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), .[195] 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.[195] 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.[196][197]
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.[84] In Dalí's The Sacrament of the Last Supper (1955) Christ and his disciples are pictured inside a giant dodecahedron.[198]

See also

Notes

  1. ^ In Piero's Italian: "una cosa tanto picholina quanto e possible ad ochio comprendere".
  2. ^ 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').[43][44] 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.[45][46]
  3. ^ 'Plastic' named the ability to take on a chosen three-dimensional shape.
  4. ^ Images and videos of Hinke Osinga's crocheted Lorenz manifold reached international television news, as can be seen in the linked website.[152]
  5. ^ Maurice Princet gave a copy to Pablo Picasso, whose sketchbooks for Les Demoiselles d'Avignon illustrate Jouffret's influence.[114][164]

References

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  3. ^
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  4. ^
    S2CID 191362470
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  6. ^ a b c O'Connor, J. J.; Robertson, E. F. (January 2003). "Mathematics and art – perspective". University of St Andrews. Retrieved 1 September 2015.
  7. ^
    ISBN 978-0-262-05048-7
    .
  8. Lives of the Artists
    . Torrentino. p. Chapter on Brunelleschi.
  9. ^ Alberti, Leon Battista; Spencer, John R. (1956) [1435]. On Painting. Yale University Press.
  10. ISBN 978-0-19-852394-9
    .
  11. ^ Witcombe, Christopher L. C. E. "Art History Resources". Retrieved 5 September 2015.
  12. ^ a b c d e Hart, George W. "Polyhedra in Art". Retrieved 24 June 2015.
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    . 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.
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