# Gamwell Mathematics And Art Essay

*Mathematicians and artists have historically shared a common interest: inquiry and comprehension of the intricacies of the world around them, whether through numerical or aesthetic design. Illustrating the relationship between math and art from antiquity to present day, Lynn Gamwell‘s *Mathematics and Art

*highlights the significant impact these two linked worlds have on one another. Gamwell recently took the time to answer some questions about her book. Examining the modern disciplines of art and math, she reveals the profound philosophy of self-reflection that these two cultural and intellectual pursuits share. Don’t forget to check out the stunning slideshow following the Q&A.*

**What’s the basic idea of your book?**

LG: I started with the assumption that how people understand reality relates directly to the concepts of mathematics that develop in their culture. Mathematics is a search for patterns, and artists, in turn, create visualizations of the patterns discovered in their time. So I describe a general history of mathematics and the related artwork.

**Since you begin in Stone Age times, your book covers over 5000 years. Is there a historical focus to the book?**

LG: Yes, there are 13 chapters, and the first gives the background up to around 1800 AD. The other 12 chapters are on the modern and contemporary eras, although I occasionally dip back into pre-modern times to give the background of a topic. A central question that drove my exploration of the modern era was: where did abstract, non-objective art come from? Between around 1890 and 1915, many artists stopped depicting people and landscapes and start using pure color and form as the vocabulary of their art. Why? I argue that modern art is an expression of the scientific worldview. Beginning in the late nineteenth century and continuing today, researchers describe bacteria, cells, radiation, and pulsars that are invisible to the unaided eye, as well as mathematical patterns in nature.

**Can you give a few examples of the relation of math and art?**

LG: Italian Renaissance artists, such as Leonardo da Vinci, constructed the space in paintings such as *The Last Supper* using linear perspective, which is a geometric projection invented in the 1430s by the architect Filippo Brunelleschi. In the twentieth century, Swiss Constructivists such as Karl Gerstner created symmetrical patterns based on the mathematics of group theory, which measures the amount of symmetry in a system, such as atoms and sub-atomic particles. The contemporary America artist Jim Sanborn uses topology, which is the projection of geometric shapes onto surfaces that are stretched and distorted. For example in photographs of cliffs in Ireland, Jim first projected concentric circles onto the rocks and then took the photograph with a long exposure at moonrise. These artists are, of course, interested in many other things besides mathematics; aesthetic issues are their primary focus.

**The examples you give are artists who are inspired by math; are mathematicians ever influenced by art?**

LG: Mathematics are rarely inspired by a *particular* piece of art (since most artists use elementary arithmetic and geometry), but rather they aspire to include in their proofs general aesthetic qualities, such as purity, simplicity, and elegance.

**You mention Leonardo da Vinci; didn’t he use the Golden Ration?**

LG: No. It is a common misconception that a ratio described by Euclid as “mean and extreme ratio” has been used by artists throughout history because it holds the key to beautiful proportions. This myth was begun in the early nineteenth century by a German scholar who called Euclid’s ratio “golden.” The myth took a tenacious hold on Western intellectuals because, as science was beginning to take them off their privileged pedestal, it assured them that all beauty is based on a ratio embodied in human anatomy. There is no science supporting this claim.

**Your book is a global history; did you find that there is a difference between math in the East and West?**

LG: Yes, because a culture’s understanding of mathematics is based in its understanding of reality. In antiquity, Eastern mathematics in based in Taoism, the view that nature is composed of myriad parts that came together by self-assembly into a harmonious whole. Thus Chinese mathematicians discerned patterns in numbers, such as the Luoshu (magic square), in which numbers in the rows, columns, and diagonals have the same sum (the harmonious whole). On the other hand, Western cultures believed that a divine person (The Egyptian sun-god Ra, the God of Abraham, Plato’s carpenter) had imposed order on formless chaos. Thus Westerners went looking for this order, and they found it in the movement of the stars (the Babylonian zodiac), and the planets (Kepler’s Laws of Planetary Motion). Although there was a difference between Eastern and Western math when there was little contact, in today’s culture there is one global math.

**The book includes the diverse fields of art, philosophy, mathematics, and physics; what is your educational background?**

LG: I have a BA in philosophy and a PhD in art history. I’m self-taught in the history of science and math.

**At 576 pages, this is a long book with extensive endnotes and 500+ illustrations; how long did it take you?**

LG: 12 years of research and writing, plus one year in production.

**Did you make any discoveries about art that especially surprised you?**

LG: Yes. When I started my research I thought that artists during the modern era (the twentieth- and twenty-first centuries) would have only a vague knowledge of the math of their times, because of the famed “two cultures” divide. But I found specific historical evidence (an artist’s essay, manifesto, interview, or letter), which demonstrated that the artist had direct knowledge of a particular piece of mathematics and had embodied it in his or her art. Examples include: Aleksandr Rodchenko, Henry Moore, Piet Mondrian, Max Bill, Dorothea Rockburne, as well as musicians, such as Arnold Schoenberg, and poets, such as T. S. Eliot and James Joyce. Again, I would stress that for such artists mathematics is a secondary interest at best, and they are concerned with materials, expressive content, and purely aesthetic issues.

**Any surprising discoveries about math and science?**

LG: Yes, here are two. Much of what is taught as physics is really *philosophy* (interpretation) of physical data. An example is the Copenhagen interpretation of quantum physics, which was taught as THE gospel truth from its announcement in 1927 to around 1960. In fact, there are other ways to interpret the same laboratory data, which were largely ignored. I’m used to such dogmatism in the art world, where artists and critics are known to proclaim what art IS, but I expected to find a more cool-headed rationalism in the laboratory. Alas, we’re all human beings, driven by our passions. Another example is the strong resistance to Platonism (the view that abstract objects exist outside time and space) in modern culture, even though Platonism is the view held by most working mathematicians (i.e., they believe they are discovering patterns not creating them). While doing research, I found myself viewed with suspicion of being a religious missionary (disguised as a scholar) because I gave a sympathetic reading of historical religious documents (in other words, I tried to describe reality from their point of view). In fact, my outlook is completely secular. I came to realize that many secularists are unable to separate Platonism from its long association with religious doctrine, which touches a nerve in certain otherwise dispassionate academics.

**Are you planning another project? What are you going to do next?**

LG: I’m going to take some time off and regroup. I’ve started to think about writing something for children.

Check out the slideshow highlighting just a few of the book’s stunning images:

**Lynn Gamwell** is lecturer in the history of art, science, and mathematics at the School of Visual Arts in New York. She is the author of *Exploring the Invisible: Art, Science, and the Spiritual* (Princeton).

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

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

Other relationships include the algorithmic analysis of artworks by X-ray fluorescence spectroscopy, the finding that traditional 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, as seen in artworks such as William Blake's *The Ancient of Days*.

## Origins: from Ancient Greece to the Renaissance[edit]

### Polykleitos's *Canon* and *symmetria*[edit]

Further information: Polykleitos

Polykleitos the elder (c.450–420 BC) was a Greeksculptor 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.^{[2]} While his sculptures may not be as famous as those of Phidias, they are much admired. In the *Canon* of Polykleitos, a treatise he wrote designed to document the "perfect" anatomical proportions of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body.^{[2]}

Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body.^{[3]} Polykleitos multiplies the length of the distal phalanx by the square root of two (√2) to get the distance of the second phalanges and multiplies the length again by √2 to get the length of the third phalanges. Next, he takes the finger length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, and so on.^{[4]}

The influence of the *Canon* of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, 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.^{[3]}

### Perspective and proportion[edit]

Main article: Perspective (graphical)

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.^{[6]} 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.^{[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 architectFilippo 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.^{[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]}

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 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 corporibus regularibus (On Regular Solids)*.^{[14]}^{[15]}^{[16]} The historian Vasari in his *Lives 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 the Polyptych of Perugia,^{[18]} 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.^{[19]} He was taught commercial arithmetic in "abacus schools"; his writings are formatted like abacus school textbooks,^{[20]} 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.^{[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".^{[a]}^{[7]} He uses deductive logic to lead the reader to the perspective representation of a three-dimensional body.^{[22]}

The artist David Hockneyargued 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.^{[23]} Critics disagree on whether Hockney was correct.^{[24]}^{[25]} Similarly, the architect Philip Steadman argued controversially^{[26]} that Vermeer had used a different device, the camera obscura, to help him create his distinctively observed paintings.^{[27]}

In 1509, Luca Pacioli (c. 1447–1517) published *De divina proportione* on mathematical and artisticproportion, 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 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.^{[29]} Da Vinci studied Pacioli's *Summa*, from which he copied tables of proportions.^{[30]} In *Mona Lisa* and *The Last Supper*, Da Vinci's work incorporated linear perspective with a vanishing point to provide apparent depth.^{[31]}*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[edit]

Further information: List of works designed with the golden ratio

The golden ratio (roughly equal to 1.618) was known to Euclid.^{[38]} The golden ratio has persistently been claimed^{[39]}^{[40]}^{[41]}^{[42]} in modern times to have been used in art and architecture 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 nineteenth century have argued on dubious mathematical grounds for the golden ratio in pyramid design.^{[b]} The Parthenon, a 5th-century BC temple in Athens, has been claimed to use the golden ratio in its façade and floor plan,^{[46]}^{[47]}^{[48]} but these claims too are disproved by measurement.^{[43]} The Great Mosque of Kairouan in Tunisia has similarly been claimed to use the golden ratio in its design,^{[49]} but the ratio does not appear in the original parts of the mosque.^{[50]} 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,^{[51]} 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.^{[52]} 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.^{[53]} After Pacioli, the golden ratio is more definitely discernible in artworks including Leonardo's *Mona Lisa*.^{[54]}

Another ratio, the only other morphic number,^{[55]} was named the plastic number^{[c]} in 1928 by the Dutch architect Hans van der Laan (originally named *le nombre radiant* in French).^{[56]} 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.^{[56]}

### Planar symmetries[edit]

Further information: Planar symmetry, Wallpaper group, Islamic geometric patterns, and Kilim

Planar symmetries have for millennia been exploited in artworks such as carpets, lattices, textiles and tilings.^{[58]}^{[59]}^{[60]}^{[61]}

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.^{[58]} 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.^{[58]} The mathematician and architectural theorist Nikos Salingaros suggests that the "powerful presence"^{[57]} (aesthetic effect) of a "great carpet"^{[57]} 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.^{[57]}

Elaborate lattices are found in Indian Jaali work, carved in marble to adorn tombs and palaces.^{[59]} 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.^{[62]} Many Chinese lattices have been analysed mathematically by Daniel S. Dye; he identifies Sichuan as the centre of the craft.^{[63]}

Symmetries are prominent in textile arts including quilting,^{[60]}knitting,^{[64]}cross-stitch, crochet,^{[65]}embroidery^{[66]}^{[67]} and weaving,^{[68]} where they may be purely decorative or may be marks of status.^{[69]}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.^{[70]} 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.^{[71]}

Islamic artexploits 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 quasicrystallinePenrose tilings.^{[72]} Elaborate geometric zellige tilework is a distinctive element in Moroccan architecture.^{[61]}Muqarnas vaults are three-dimensional but were designed in two dimensions with drawings of geometrical cells.^{[73]}

Hotamis kilim (detail), central Anatolia, early 19th century

The complex geometry and tilings of the muqarnas vaulting in the Sheikh Lotfollah Mosque, Isfahan

### Polyhedra[edit]

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;^{[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.

Albrecht Dürer (1471–1528) was a GermanRenaissanceprintmaker 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.^{[74]} 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.^{[75]} Dürer published another influential book on human proportions called *Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion)* in 1528.^{[76]}

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]}^{[77]}^{[78]} including a two-volume book by Peter-Klaus Schuster,^{[79]} and an influential discussion in Erwin Panofsky's monograph of Dürer.^{[1]}^{[80]}Salvador Dalí's *Corpus Hypercubus* depicts an unfolded three-dimensional net for a hypercube, a four-dimensional regular polyhedron.^{[81]}

### Fractal dimensions[edit]

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.^{[82]}

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.^{[83]}

## A complex relationship[edit]

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."^{[84]} 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.^{[85]} Some of the many strands of the resulting complex relationship^{[86]} are described below.

### Mathematics as an art[edit]

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.^{[87]} 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".^{[88]} 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." ^{[89]}

### Mathematical tools for art[edit]

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

Mathematics can be discerned in many of the arts, such as music, dance,^{[90]}painting, architecture, and sculpture. Each of these is richly associated with mathematics.^{[91]} 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.^{[92]} 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.^{[91]}^{[93]} 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,^{[6]} his theory influencing Leonardo and Dürer. Isaac Newton

^{[57]}carpet with double medallion. Central Anatolia (Konya – Karapınar), turn of the 16th/17th centuries. Alâeddin Mosque

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