Mathematics And Art
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Mathematics and art are related in a variety of ways.
Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
has itself been described as an
art Art is a diverse range of cultural activity centered around ''works'' utilizing creative or imaginative talents, which are expected to evoke a worthwhile experience, generally through an expression of emotional power, conceptual ideas, tec ...
motivated by beauty. Mathematics can be discerned in arts such as
music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
,
dance Dance is an The arts, art form, consisting of sequences of body movements with aesthetic and often Symbol, symbolic value, either improvised or purposefully selected. Dance can be categorized and described by its choreography, by its repertoir ...
,
painting Painting is a Visual arts, visual art, which is characterized by the practice of applying paint, pigment, color or other medium to a solid surface (called "matrix" or "Support (art), support"). The medium is commonly applied to the base with ...
,
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
,
sculpture Sculpture is the branch of the visual arts that operates in three dimensions. Sculpture is the three-dimensional art work which is physically presented in the dimensions of height, width and depth. It is one of the plastic arts. Durable sc ...
, and
textiles Textile is an Hyponymy and hypernymy, umbrella term that includes various Fiber, fiber-based materials, including fibers, yarns, Staple (textiles)#Filament fiber, filaments, Thread (yarn), threads, and different types of #Fabric, fabric. ...
. 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
sculptor Sculpture is the branch of the visual arts that operates in three dimensions. Sculpture is the three-dimensional art work which is physically presented in the dimensions of height, width and depth. It is one of the plastic arts. Durable sc ...
Polykleitos Polykleitos (; ) was an ancient Greek sculptor, active in the 5th century BCE. Alongside the Athenian sculptors Pheidias, Myron and Praxiteles, he is considered as one of the most important sculptors of classical antiquity. The 4th century B ...
wrote his ''Canon'', prescribing proportions conjectured to have been based on the ratio 1: for the ideal male nude. Persistent popular claims have been made for the use of the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
in ancient art and architecture, without reliable evidence. In the Italian
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
,
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
wrote the influential treatise ''
De divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'' (1509), illustrated with woodcuts by
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
, on the use of the golden ratio in art. Another Italian painter,
Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...
, developed
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's ideas on perspective in treatises such as ''De Prospectiva Pingendi'', and in his paintings. The engraver
Albrecht Dürer Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...
made many references to mathematics in his work '' Melencolia I''. In modern times, the
graphic artist A graphic designer is a practitioner who follows the discipline of graphic design, either within companies or organizations or independently. They are professionals in design and visual communication, with their primary focus on transforming l ...
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
made intensive use of
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
and
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
, with the help of the mathematician H. S. M. Coxeter, while the
De Stijl De Stijl (, ; 'The Style') was a Dutch art movement founded in 1917 by a group of artists and architects based in Leiden (Theo van Doesburg, Jacobus Oud, J.J.P. Oud), Voorburg (Vilmos Huszár, Jan Wils) and Laren, North Holland, Laren (Piet Mo ...
movement led by Theo van Doesburg and
Piet Mondrian Pieter Cornelis Mondriaan (; 7 March 1872 – 1 February 1944), known after 1911 as Piet Mondrian (, , ), was a Dutch Painting, painter and Theory of art, art theoretician who is regarded as one of the greatest artists of the 20th century. He w ...
explicitly embraced geometrical forms. Mathematics has inspired textile arts such as
quilting Quilting is the process of joining a minimum of three layers of textile, fabric together either through stitching manually using a Sewing needle, needle and yarn, thread, or mechanically with a sewing machine or specialised longarm quilting ...
,
knitting Knitting is a method for production of textile Knitted fabric, fabrics by interlacing yarn loops with loops of the same or other yarns. It is used to create many types of garments. Knitting may be done Hand knitting, by hand or Knitting machi ...
,
cross-stitch Cross-stitch is a form of sewing and a popular form of counted-thread embroidery in which X-shaped stitches (called cross stitches) in a tiled, raster graphics, raster-like pattern are used to form a picture. The stitcher counts the threads on a ...
,
crochet Crochet (; ) is a process of creating textiles by using a crochet hook to interlock loops of yarn, thread (yarn), thread, or strands of other materials. The name is derived from the French term ''crochet'', which means 'hook'. Hooks can be made ...
,
embroidery Embroidery is the art of decorating Textile, fabric or other materials using a Sewing needle, needle to stitch Yarn, thread or yarn. It is one of the oldest forms of Textile arts, textile art, with origins dating back thousands of years across ...
,
weaving Weaving is a method of textile production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. Other methods are knitting, crocheting, felting, and braiding or plaiting. The longitudinal ...
, Turkish and other
carpet A carpet is a textile floor covering typically consisting of an upper layer of Pile (textile), pile attached to a backing. The pile was traditionally made from wool, but since the 20th century synthetic fiber, synthetic fibres such as polyprop ...
-making, as well as
kilim A kilim ( ; ; ) is a flat tapestry-weaving, woven carpet or rug traditionally produced in countries of the former Persian Empire, including Iran and Turkey, but also in the Balkans and the Turkic countries. Kilims can be purely decorative ...
. In
Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslims, Muslim populations. Referring to characteristic traditions across ...
, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread
muqarnas Muqarnas (), also known in Iberian architecture as Mocárabe (from ), is a form of three-dimensional decoration in Islamic architecture in which rows or tiers of niche-like elements are projected over others below. It is an archetypal form of I ...
vaulting. Mathematics has directly influenced art with conceptual tools such as linear perspective, the analysis of
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
, and mathematical objects such as
polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
and the
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...
.
Magnus Wenninger Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to ...
creates colourful stellated polyhedra, originally as models for teaching. Mathematical concepts such as
recursion Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
and logical paradox can be seen in paintings by
René Magritte René François Ghislain Magritte (; 21 November 1898 – 15 August 1967) was a Belgium, Belgian surrealist artist known for his depictions of familiar objects in unfamiliar, unexpected contexts, which often provoked questions about the nature ...
and in engravings by M. C. Escher.
Computer art Computer art is art in which computers play a role in the production or display of the artwork. Such art can be an image, sound, animation, video, CD-ROM, DVD-ROM, video game, website, algorithm, performance or gallery installation. Many traditio ...
often makes use of
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
s including the
Mandelbrot set The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting ...
, and sometimes explores other mathematical objects such as
cellular automata A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
. Controversially, the artist
David Hockney David Hockney (born 9 July 1937) is an English Painting, painter, Drawing, draughtsman, Printmaking, printmaker, Scenic design, stage designer, and photographer. As an important contributor to the pop art movement of the 1960s, he is considere ...
has argued that artists from the Renaissance onwards made use of the
camera lucida A ''camera lucida'' is an optical device used as a drawing aid by artists and microscopy, microscopists. It projects an optics, optical superimposition of the subject being viewed onto the surface upon which the artist is drawing. The artist se ...
to draw precise representations of scenes; the architect Philip Steadman similarly argued that
Vermeer Johannes Vermeer ( , ; see below; also known as Jan Vermeer; October 1632 – 15 December 1675) was a Dutch painter who specialized in domestic interior scenes of middle-class life. He is considered one of the greatest painters of the Dutch ...
used the
camera obscura A camera obscura (; ) is the natural phenomenon in which the rays of light passing through a aperture, small hole into a dark space form an image where they strike a surface, resulting in an inverted (upside down) and reversed (left to right) ...
in his distinctively observed paintings. Other relationships include the algorithmic analysis of artworks by X-ray fluorescence spectroscopy, the finding that traditional
batik Batik is a dyeing technique using wax Resist dyeing, resist. The term is also used to describe patterned textiles created with that technique. Batik is made by drawing or stamping wax on a cloth to prevent colour absorption during the dyein ...
s from different regions of
Java Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...
have distinct
fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...
s, and stimuli to mathematics research, especially Filippo Brunelleschi's theory of perspective, which eventually led to
Girard Desargues Girard Desargues (; 21 February 1591September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named i ...
's
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
. 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


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

Polykleitos Polykleitos (; ) was an ancient Greek sculptor, active in the 5th century BCE. Alongside the Athenian sculptors Pheidias, Myron and Praxiteles, he is considered as one of the most important sculptors of classical antiquity. The 4th century B ...
the elder (c. 450–420 BC) was a
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
sculptor Sculpture is the branch of the visual arts that operates in three dimensions. Sculpture is the three-dimensional art work which is physically presented in the dimensions of height, width and depth. It is one of the plastic arts. Durable sc ...
from the school of Argos, and a contemporary of
Phidias Phidias or Pheidias (; , ''Pheidias''; ) was an Ancient Greek sculptor, painter, and architect, active in the 5th century BC. His Statue of Zeus at Olympia was one of the Seven Wonders of the Ancient World. Phidias also designed the statues of ...
. His works and statues consisted mainly of bronze and were of athletes. According to the philosopher and mathematician
Xenocrates Xenocrates (; ; c. 396/5314/3 BC) of Chalcedon was a Greek philosopher, mathematician, and leader ( scholarch) of the Platonic Academy from 339/8 to 314/3 BC. His teachings followed those of Plato, which he attempted to define more closely, of ...
, Polykleitos is ranked as one of the most important sculptors of
classical antiquity Classical antiquity, also known as the classical era, classical period, classical age, or simply antiquity, is the period of cultural History of Europe, European history between the 8th century BC and the 5th century AD comprising the inter ...
for his work on the '' Doryphorus'' and the statue of
Hera In ancient Greek religion, Hera (; ; in Ionic Greek, Ionic and Homeric Greek) is the goddess of marriage, women, and family, and the protector of women during childbirth. In Greek mythology, she is queen of the twelve Olympians and Mount Oly ...
in the
Heraion of Argos The Heraion of Argos () is an ancient sanctuary in the Argolid, Greece, dedicated to Hera, whose epithet "Argive Hera" (Ἥρη Ἀργείη ''Here Argeie'') appears in Homer's works. Hera herself claims to be the protector of Ancient Argos, A ...
. 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 Body proportions is the study of artistic anatomy, which attempts to explore the relation of the elements of the human body to each other and to the whole. These ratios are used in depictions of the human figure and may become part of an artisti ...
of the male nude, Polykleitos gives us a mathematical approach towards sculpturing the human body. The ''Canon'' itself has been lost but it is conjectured that Polykleitos used a sequence of proportions where each length is that of the diagonal of a square drawn on its predecessor, 1: (about 1:1.4142). The influence of the ''Canon'' of Polykleitos is immense in
Classical Greek Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archa ...
, Roman, and
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
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. 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.


Perspective and proportion

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 Reverse perspective, also called inverse perspective,. inverted perspective, divergent perspective, or Byzantine perspective, is a form of perspective (graphical), perspective drawing where the objects depicted in a scene are placed between the ...
for special emphasis. The Muslim mathematician
Alhazen Ḥasan Ibn al-Haytham ( Latinized as Alhazen; ; full name ; ) was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the princ ...
(Ibn al-Haytham) described a theory of optics in his ''
Book of Optics The ''Book of Optics'' (; or ''Perspectiva''; ) is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen or Alhacen (965–c. 1040 AD). The ''Book ...
'' in 1021, but never applied it to art. The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand
nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
and the
arts The arts or creative arts are a vast range of human practices involving creativity, creative expression, storytelling, and cultural participation. The arts encompass diverse and plural modes of thought, deeds, and existence in an extensive ...
. 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. The rudiments of perspective arrived with
Giotto Giotto di Bondone (; – January 8, 1337), known mononymously as Giotto, was an List of Italian painters, Italian painter and architect from Florence during the Late Middle Ages. He worked during the International Gothic, Gothic and Italian Ren ...
(1266/7 – 1337), who attempted to draw in perspective using an algebraic method to determine the placement of distant lines. In 1415, the Italian
architect An architect is a person who plans, designs, and oversees the construction of buildings. To practice architecture means to provide services in connection with the design of buildings and the space within the site surrounding the buildings that h ...
Filippo Brunelleschi and his friend
Leon Battista Alberti Leon Battista Alberti (; 14 February 1404 – 25 April 1472) was an Italian Renaissance humanist author, artist, architect, poet, Catholic priest, priest, linguistics, linguist, philosopher, and cryptography, cryptographer; he epitomised the natu ...
demonstrated the geometrical method of applying perspective in Florence, using
similar triangles In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly ...
as formulated by Euclid, to find the apparent height of distant objects. Brunelleschi's own perspective paintings are lost, but
Masaccio Masaccio (, ; ; December 21, 1401 – summer 1428), born Tommaso di Ser Giovanni di Simone, was a Florentine artist who is regarded as the first great List of Italian painters, Italian painter of the Quattrocento period of the Italian Renaiss ...
's painting of the Holy Trinity shows his principles at work. The Italian painter
Paolo Uccello Paolo Uccello ( , ; 1397 – 10 December 1475), born Paolo di Dono, was an Italian Renaissance painter and mathematician from Florence who was notable for his pioneering work on visual Perspective (graphical), perspective in art. In his book ''Liv ...
(1397–1475) was fascinated by perspective, as shown in his paintings of ''
The Battle of San Romano ''The Battle of San Romano'' is a set of three paintings by the Florence, Florentine painter Paolo Uccello depicting events that took place at the Battle of San Romano between Florentine and Sienese forces in 1432. They are significant as reveal ...
'' (c. 1435–1460): broken lances lie conveniently along perspective lines. The painter
Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...
(c. 1415–1492) exemplified this new shift in Italian Renaissance thinking. He was an expert
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
and geometer, writing books on
solid geometry Solid geometry or stereometry is the geometry of Three-dimensional space, three-dimensional Euclidean space (3D space). A solid figure is the region (mathematics), region of 3D space bounded by a two-dimensional closed surface; for example, a ...
and perspective, including '' De prospectiva pingendi (On Perspective for Painting)'', ''Trattato d'Abaco (Abacus Treatise)'', and '' De quinque corporibus regularibus (On the Five Regular Solids)''. The historian
Vasari Giorgio Vasari (30 July 1511 – 27 June 1574) was an Italian Renaissance painter, architect, art historian, and biographer who is best known for his work '' Lives of the Most Excellent Painters, Sculptors, and Architects'', considered the ide ...
in his '' Lives of the Painters'' calls Piero the "greatest geometer of his time, or perhaps of any time." Piero's interest in perspective can be seen in his paintings including the Polyptych of Perugia, the ''San Agostino altarpiece'' and '' The Flagellation of Christ''. His work on geometry influenced later mathematicians and artists including
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
in his ''De divina proportione'' and
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
. Piero studied classical mathematics and the works of
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
. He was taught commercial arithmetic in "abacus schools"; his writings are formatted like abacus school textbooks, perhaps including Leonardo Pisano (
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
)'s 1202 ''
Liber Abaci The or (Latin for "The Book of Calculation") was a 1202 Latin work on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. It is primarily famous for introducing both base-10 positional notation and the symbols known as Arabic n ...
''. 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 A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...
with the eye as vertex." A painting constructed with linear perspective is a
cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture and engineering 3D * Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) ...
of that pyramid. 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". He uses
deductive logic Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the ...
to lead the reader to the perspective representation of a three-dimensional body. The artist
David Hockney David Hockney (born 9 July 1937) is an English Painting, painter, Drawing, draughtsman, Printmaking, printmaker, Scenic design, stage designer, and photographer. As an important contributor to the pop art movement of the 1960s, he is considere ...
argued in his book '' Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters'' that artists started using a
camera lucida A ''camera lucida'' is an optical device used as a drawing aid by artists and microscopy, microscopists. It projects an optics, optical superimposition of the subject being viewed onto the surface upon which the artist is drawing. The artist se ...
from the 1420s, resulting in a sudden change in precision and realism, and that this practice was continued by major artists including
Ingres Jean-Auguste-Dominique Ingres ( ; ; 29 August 1780 – 14 January 1867) was a French Neoclassicism, Neoclassical Painting, painter. Ingres was profoundly influenced by past artistic traditions and aspired to become the guardian of academic ...
,
Van Eyck Van Eyck or Van Eijk () is a Dutch language, Dutch toponymic surname. ''Eijck'', ''Eyck'', ''Eyk'' and ''Eijk'' are all archaic spellings of modern Dutch ("oak") and the surname literally translates as "from/of oak". However, in most cases, the fam ...
, and
Caravaggio Michelangelo Merisi da Caravaggio (also Michele Angelo Merigi or Amerighi da Caravaggio; 29 September 1571 – 18 July 1610), known mononymously as Caravaggio, was an Italian painter active in Rome for most of his artistic life. During the fina ...
. Critics disagree on whether Hockney was correct. Similarly, the architect Philip Steadman argued controversially that
Vermeer Johannes Vermeer ( , ; see below; also known as Jan Vermeer; October 1632 – 15 December 1675) was a Dutch painter who specialized in domestic interior scenes of middle-class life. He is considered one of the greatest painters of the Dutch ...
had used a different device, the
camera obscura A camera obscura (; ) is the natural phenomenon in which the rays of light passing through a aperture, small hole into a dark space form an image where they strike a surface, resulting in an inverted (upside down) and reversed (left to right) ...
, to help him create his distinctively observed paintings. In 1509,
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
(c. 1447–1517) published ''
De divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'' on
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
artistic Art is a diverse range of culture, cultural activity centered around works of art, ''works'' utilizing Creativity, creative or imagination, imaginative talents, which are expected to evoke a worthwhile experience, generally through an express ...
proportion, including in the human face.
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
(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. These, such as the
rhombicuboctahedron In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for ''truncated cuboctahedral rhombus'', w ...
, 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 Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...
,
Melozzo da Forlì Melozzo da Forlì ( – 8 November 1494) was an Italian Renaissance painter and architect. His fresco paintings are notable for the use of foreshortening. He was the most important member of the Forlì painting school. Biography Melozzo was s ...
, and
Marco Palmezzano Marco Palmezzano (1460–1539) was an Italian painter and architect, belonging to the Forlì painting school, who painted in a style recalling earlier Northern Renaissance models. He was mostly active near Forlì. Biography Palmezzano was ...
. Leonardo studied Pacioli's ''Summa'', from which he copied tables of proportions. In ''
Mona Lisa The ''Mona Lisa'' is a half-length portrait painting by the Italian artist Leonardo da Vinci. Considered an archetypal masterpiece of the Italian Renaissance, it has been described as "the best known, the most visited, the most written about, ...
'' and '' The Last Supper'', Leonardo's work incorporated linear perspective with a
vanishing point A vanishing point is a point (geometry), point on the projection plane, image plane of a graphical perspective, perspective rendering where the two-dimensional perspective projections of parallel (geometry), parallel lines in three-dimensional ...
to provide apparent depth. ''The Last Supper'' is constructed in a tight ratio of 12:6:4:3, as is
Raphael Raffaello Sanzio da Urbino (; March 28 or April 6, 1483April 6, 1520), now generally known in English as Raphael ( , ), was an Italian painter and architect of the High Renaissance. List of paintings by Raphael, His work is admired for its cl ...
's ''
The School of Athens ''The School of Athens'' () is a fresco by the Italian Renaissance artist Raphael. It was painted between 1509 and 1511 as part of a commission by Pope Julius II to decorate the rooms now called the in the Apostolic Palace in Vatican City. ...
'', which includes Pythagoras with a tablet of ideal ratios, sacred to the Pythagoreans. In ''
Vitruvian Man The ''Vitruvian Man'' (; ) is a drawing by the Italian Renaissance artist and scientist Leonardo da Vinci, dated to . Inspired by the writings of the ancient Roman architect Vitruvius, the drawing depicts a nude man in two superimposed positions ...
'', Leonardo expressed the ideas of the Roman architect
Vitruvius Vitruvius ( ; ; –70 BC – after ) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work titled . As the only treatise on architecture to survive from antiquity, it has been regarded since the Renaissan ...
, innovatively showing the male figure twice, and centring him in both a circle and a square. As early as the 15th century,
curvilinear perspective Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces, for which (straight) lines on the 3D object are projected to curves on the 2D surface that are typically not straight (hence ...
found its way into paintings by artists interested in image distortions.
Jan van Eyck Jan van Eyck ( ; ; – 9 July 1441) was a Flemish people, Flemish painter active in Bruges who was one of the early innovators of what became known as Early Netherlandish painting, and one of the most significant representatives of Early Nort ...
's 1434 ''
Arnolfini Portrait ''The Arnolfini Portrait'' (or ''The Arnolfini Wedding'', ''The Arnolfini Marriage'', the ''Portrait of Giovanni Arnolfini and his Wife'', or other titles) is an oil painting on oak panel by the Early Netherlandish painter Jan van Eyck, dated 14 ...
'' contains a convex mirror with reflections of the people in the scene, while
Parmigianino Girolamo Francesco Maria Mazzola (11 January 150324 August 1540), also known as Francesco Mazzola or, more commonly, as Parmigianino (, , ; "the little one from Parma"), was an Italian Mannerist painter and printmaker active in Florence, Rome, ...
'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. Three-dimensional space can be represented convincingly in art, as in
technical drawing Technical drawing, drafting or drawing, is the act and discipline of composing drawings that visually communicate how something functions or is constructed. Technical drawing is essential for communicating ideas in industry and engineering. ...
, by means other than perspective.
Oblique projection Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects. The objects are not in perspective and so do not correspond to any view of an ...
s, including cavalier perspective (used by French military artists to depict fortifications in the 18th century), were used continuously and ubiquitously by Chinese artists from the first or second centuries until the 18th century. The Chinese acquired the technique from India, which acquired it from Ancient Rome. Oblique projection is seen in Japanese art, such as in the
Ukiyo-e is a genre of Japanese art that flourished from the 17th through 19th centuries. Its artists produced woodblock printing, woodblock prints and Nikuhitsu-ga, paintings of such subjects as female beauties; kabuki actors and sumo wrestlers; scenes ...
paintings of
Torii Kiyonaga Torii Kiyonaga (; 1752 – June 28, 1815) was a Japanese ukiyo-e artist of the Torii school. Originally Sekiguchi Shinsuke, the son of an Edo bookseller, from Motozaimokuchō Itchōme in Edo, he took on Torii Kiyonaga as an art name. Altho ...
(1752–1815). File:Pacioli De Divina Proportione Head Equilateral Triangle 1509.jpg, Woodcut from
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
's 1509 ''
De divina proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'' with an
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
on a human face File:Camera Lucida in use drawing small figurine.jpg,
Camera lucida A ''camera lucida'' is an optical device used as a drawing aid by artists and microscopy, microscopists. It projects an optics, optical superimposition of the subject being viewed onto the surface upon which the artist is drawing. The artist se ...
in use. ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...
'', 1879 File:Camera obscura2.jpg, Illustration of an artist using a
camera obscura A camera obscura (; ) is the natural phenomenon in which the rays of light passing through a aperture, small hole into a dark space form an image where they strike a surface, resulting in an inverted (upside down) and reversed (left to right) ...
. 17th century File:Da Vinci Vitruve Luc Viatour.jpg, Proportion: Leonardo's ''
Vitruvian Man The ''Vitruvian Man'' (; ) is a drawing by the Italian Renaissance artist and scientist Leonardo da Vinci, dated to . Inspired by the writings of the ancient Roman architect Vitruvius, the drawing depicts a nude man in two superimposed positions ...
'', c. 1490 File:Masaccio, trinità.jpg,
Brunelleschi Filippo di ser Brunellesco di Lippo Lapi (1377 – 15 April 1446), commonly known as Filippo Brunelleschi ( ; ) and also nicknamed Pippo by Leon Battista Alberti, was an Italian architect, designer, goldsmith and sculptor. He is considered to ...
's theory of perspective:
Masaccio Masaccio (, ; ; December 21, 1401 – summer 1428), born Tommaso di Ser Giovanni di Simone, was a Florentine artist who is regarded as the first great List of Italian painters, Italian painter of the Quattrocento period of the Italian Renaiss ...
's ''Trinità'', c. 1426–1428, in the
Basilica of Santa Maria Novella Santa Maria Novella is a church in Florence, Italy, situated opposite, and lending its name to, the city's main railway station. Chronologically, it is the first great basilica in Florence, and is the city's principal Dominican church. The ch ...
File:Della Pittura Alberti perspective pillars on grid.jpg, Diagram from
Leon Battista Alberti Leon Battista Alberti (; 14 February 1404 – 25 April 1472) was an Italian Renaissance humanist author, artist, architect, poet, Catholic priest, priest, linguistics, linguist, philosopher, and cryptography, cryptographer; he epitomised the natu ...
's 1435 '' Della Pittura'', with pillars in perspective on a grid File:Piero - The Flagellation.jpg, Linear perspective in
Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...
's ''
Flagellation of Christ The Flagellation of Christ, in art sometimes known as Christ at the Column or the Scourging at the Pillar, is an episode from the Passion of Jesus as presented in the Gospels. As such, it is frequently shown in Christian art, in cycles of the Pas ...
'', c. 1455–1460 File:The Arnolfini Portrait, détail (2).jpg,
Curvilinear perspective Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces, for which (straight) lines on the 3D object are projected to curves on the 2D surface that are typically not straight (hence ...
:
convex mirror A curved mirror is a mirror with a curved reflecting surface. The surface may be either ''convex'' (bulging outward) or ''concave'' (recessed inward). Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are ...
in
Jan van Eyck Jan van Eyck ( ; ; – 9 July 1441) was a Flemish people, Flemish painter active in Bruges who was one of the early innovators of what became known as Early Netherlandish painting, and one of the most significant representatives of Early Nort ...
's ''
Arnolfini Portrait ''The Arnolfini Portrait'' (or ''The Arnolfini Wedding'', ''The Arnolfini Marriage'', the ''Portrait of Giovanni Arnolfini and his Wife'', or other titles) is an oil painting on oak panel by the Early Netherlandish painter Jan van Eyck, dated 14 ...
'', 1434 File:Parmigianino Selfportrait.jpg,
Parmigianino Girolamo Francesco Maria Mazzola (11 January 150324 August 1540), also known as Francesco Mazzola or, more commonly, as Parmigianino (, , ; "the little one from Parma"), was an Italian Mannerist painter and printmaker active in Florence, Rome, ...
, '' Self-portrait in a Convex Mirror'', c. 1523–1524 File:Pythagoras with tablet of ratios.jpg, Pythagoras with tablet of ratios, in
Raphael Raffaello Sanzio da Urbino (; March 28 or April 6, 1483April 6, 1520), now generally known in English as Raphael ( , ), was an Italian painter and architect of the High Renaissance. List of paintings by Raphael, His work is admired for its cl ...
's ''
The School of Athens ''The School of Athens'' () is a fresco by the Italian Renaissance artist Raphael. It was painted between 1509 and 1511 as part of a commission by Pope Julius II to decorate the rooms now called the in the Apostolic Palace in Vatican City. ...
'', 1509 File:Xu Yang - Entrance and yard of a yamen.jpg,
Oblique projection Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects. The objects are not in perspective and so do not correspond to any view of an ...
: ''Entrance and yard of a yamen''. Detail of scroll about
Suzhou Suzhou is a major prefecture-level city in southern Jiangsu province, China. As part of the Yangtze Delta megalopolis, it is a major economic center and focal point of trade and commerce. Founded in 514 BC, Suzhou rapidly grew in size by the ...
by Xu Yang, ordered by the
Qianlong Emperor The Qianlong Emperor (25 September 17117 February 1799), also known by his temple name Emperor Gaozong of Qing, personal name Hongli, was the fifth Emperor of China, emperor of the Qing dynasty and the fourth Qing emperor to rule over China pr ...
. 18th century File:3 Brettspiele.jpg,
Oblique projection Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects. The objects are not in perspective and so do not correspond to any view of an ...
: women playing
Shogi , also known as Japanese chess, is a Strategy game, strategy board game for two players. It is one of the most popular board games in Japan and is in the same family of games as chess, Western chess, chaturanga, xiangqi, Indian chess, and janggi. ...
, Go and Ban-sugoroku board games. Painting by
Torii Kiyonaga Torii Kiyonaga (; 1752 – June 28, 1815) was a Japanese ukiyo-e artist of the Torii school. Originally Sekiguchi Shinsuke, the son of an Edo bookseller, from Motozaimokuchō Itchōme in Edo, he took on Torii Kiyonaga as an art name. Altho ...
, Japan, c. 1780


Golden ratio

The
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
(roughly equal to 1.618) was known to
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
. The golden ratio has persistently been claimed in modern times to have been used in art and
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
by the ancients in Egypt, Greece and elsewhere, without reliable evidence. The claim may derive from confusion with "golden mean", which to the Ancient Greeks meant "avoidance of excess in either direction", not a ratio. Pyramidologists since the 19th century have argued on dubious mathematical grounds for the golden ratio in pyramid design. The
Parthenon The Parthenon (; ; ) is a former Ancient Greek temple, temple on the Acropolis of Athens, Athenian Acropolis, Greece, that was dedicated to the Greek gods, goddess Athena. Its decorative sculptures are considered some of the high points of c ...
, a 5th-century BC temple in Athens, has been claimed to use the golden ratio in its
façade A façade or facade (; ) is generally the front part or exterior of a building. It is a loanword from the French language, French (), which means "frontage" or "face". In architecture, the façade of a building is often the most important asp ...
and floor plan, but these claims too are disproved by measurement. The
Great Mosque of Kairouan The Great Mosque of Kairouan (), also known as the Mosque of Uqba (), is a mosque situated in the UNESCO World Heritage town of Kairouan, Tunisia and is one of the largest Islamic monuments in North Africa. Established by the Arab general U ...
in Tunisia has similarly been claimed to use the golden ratio in its design, but the ratio does not appear in the original parts of the mosque. The historian of architecture Frederik Macody Lund argued in 1919 that the Cathedral of Chartres (12th century),
Notre-Dame of Laon Laon Cathedral () is a Roman Catholic church architecture, church located in Laon, Aisne, Hauts-de-France, France. Built in the twelfth and thirteenth centuries, it is one of the most important and stylistically unified examples of early Gothi ...
(1157–1205) and
Notre Dame de Paris Notre-Dame de Paris ( ; meaning "Cathedral of Our Lady of Paris"), often referred to simply as Notre-Dame, is a Medieval architecture, medieval Catholic cathedral on the Île de la Cité (an island in the River Seine), in the 4th arrondissemen ...
(1160) are designed according to the golden ratio, 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. 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. After Pacioli, the golden ratio is more definitely discernible in artworks including Leonardo's ''
Mona Lisa The ''Mona Lisa'' is a half-length portrait painting by the Italian artist Leonardo da Vinci. Considered an archetypal masterpiece of the Italian Renaissance, it has been described as "the best known, the most visited, the most written about, ...
''. Another ratio, the only other morphic number, was named the
plastic number Plastic number may refer to: * Plastic ratio, a mathematical ratio * Resin identification code The Resin Identification Code (RIC) is a technical standard with a set of symbols appearing on plastic products that identify the Synthetic resin, ...
in 1928 by the Dutch architect
Hans van der Laan Dom Hans van der Laan (29 December 1904 – 19 August 1991) was a Dutch Benedictine monk and architect. He was a leading figure in the Bossche School. His theories on numerical ratios in architecture, in particular regarding the plastic ratio, ...
(originally named ''le nombre radiant'' in French). Its value is the solution of the
cubic equation In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
:x^3=x+1\,, an irrational number which is approximately 1.325. According to the architect Richard Padovan, this has characteristic ratios and , which govern the limits of human perception in relating one physical size to another. Van der Laan used these ratios when designing the 1967 St. Benedictusberg Abbey church in the Netherlands. File:Mathematical Pyramid.svg, Base:hypotenuse(b:a) ratios for the
Pyramid of Khufu The Great Pyramid of Giza is the largest Egyptian pyramid. It served as the tomb of pharaoh Khufu, who ruled during the Fourth Dynasty of the Old Kingdom. Built , over a period of about 26 years, the pyramid is the oldest of the Seven Wond ...
could be: 1:φ ( Kepler triangle), 3:5 ( 3-4-5 Triangle), or 1:4/π File:Laon Cathedral's regulator lines.jpg, Supposed ratios:
Notre-Dame of Laon Laon Cathedral () is a Roman Catholic church architecture, church located in Laon, Aisne, Hauts-de-France, France. Built in the twelfth and thirteenth centuries, it is one of the most important and stylistically unified examples of early Gothi ...
File:Mona Lisa Golden Ratio.jpg, Golden rectangles superimposed on the
Mona Lisa The ''Mona Lisa'' is a half-length portrait painting by the Italian artist Leonardo da Vinci. Considered an archetypal masterpiece of the Italian Renaissance, it has been described as "the best known, the most visited, the most written about, ...
File:Interieur bovenkerk, zicht op de middenbeuk met koorbanken voor de monniken - Mamelis - 20536587 - RCE.jpg, The 1967 St. Benedictusberg Abbey church by
Hans van der Laan Dom Hans van der Laan (29 December 1904 – 19 August 1991) was a Dutch Benedictine monk and architect. He was a leading figure in the Bossche School. His theories on numerical ratios in architecture, in particular regarding the plastic ratio, ...
has
plastic ratio In mathematics, the plastic ratio is a geometrical aspect ratio, proportion, given by the unique real polynomial root, solution of the equation Its decimal expansion begins as . The adjective ''plastic'' does not refer to Plastic, the artifici ...
proportions.


Planar symmetries

Planar symmetries have for millennia been exploited in artworks such as
carpet A carpet is a textile floor covering typically consisting of an upper layer of Pile (textile), pile attached to a backing. The pile was traditionally made from wool, but since the 20th century synthetic fiber, synthetic fibres such as polyprop ...
s, lattices, textiles and tilings. Many traditional rugs, whether pile carpets or flatweave
kilim A kilim ( ; ; ) is a flat tapestry-weaving, woven carpet or rug traditionally produced in countries of the former Persian Empire, including Iran and Turkey, but also in the Balkans and the Turkic countries. Kilims can be purely decorative ...
s, are divided into a central field and a framing border; both can have symmetries, though in handwoven carpets these are often slightly broken by small details, variations of pattern and shifts in colour introduced by the weaver. In kilims from
Anatolia Anatolia (), also known as Asia Minor, is a peninsula in West Asia that makes up the majority of the land area of Turkey. It is the westernmost protrusion of Asia and is geographically bounded by the Mediterranean Sea to the south, the Aegean ...
, 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 A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...
such as pmm, while the border may be laid out as a frieze of
frieze group In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. The term is derived from friezes in architecture and decorative arts, where such repeating patterns are often used. Frieze patterns can be classif ...
pm11, pmm2 or pma2. Turkish and Central Asian kilims often have three or more borders in different frieze groups. Weavers certainly had the intention of symmetry, without explicit knowledge of its mathematics. The mathematician and architectural theorist Nikos Salingaros suggests that the "powerful presence" (aesthetic effect) of a "great carpet" such as the best Konya two-medallion carpets of the 17th century is created by mathematical techniques related to the theories of the architect
Christopher Alexander Christopher Wolfgang John Alexander (4 October 1936 – 17 March 2022) was an Austrian-born British-American architect and Design theory, design theorist. He was an Professors in the United States#Professor emeritus and emerita, emeritus profes ...
. 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. Reprinted in Elaborate lattices are found in Indian
Jali A ''jali'' or ''jaali'' (''jālī'', meaning "net") is the term for a perforated stone or latticed screen, usually with an ornamental pattern constructed through the use of calligraphy, geometry or natural patterns. This form of architectu ...
work, carved in marble to adorn tombs and palaces. 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. Many Chinese lattices have been analysed mathematically by Daniel S. Dye; he identifies
Sichuan Sichuan is a province in Southwestern China, occupying the Sichuan Basin and Tibetan Plateau—between the Jinsha River to the west, the Daba Mountains to the north, and the Yunnan–Guizhou Plateau to the south. Its capital city is Cheng ...
as the centre of the craft. Symmetries are prominent in
textile arts Textile arts are arts and crafts that use fiber crop, plant, Animal fiber, animal, or synthetic fibers to construct practical or decorative Physical object, objects. Textiles have been a fundamental part of human life since the beginning of ...
including
quilting Quilting is the process of joining a minimum of three layers of textile, fabric together either through stitching manually using a Sewing needle, needle and yarn, thread, or mechanically with a sewing machine or specialised longarm quilting ...
,
knitting Knitting is a method for production of textile Knitted fabric, fabrics by interlacing yarn loops with loops of the same or other yarns. It is used to create many types of garments. Knitting may be done Hand knitting, by hand or Knitting machi ...
,
cross-stitch Cross-stitch is a form of sewing and a popular form of counted-thread embroidery in which X-shaped stitches (called cross stitches) in a tiled, raster graphics, raster-like pattern are used to form a picture. The stitcher counts the threads on a ...
,
crochet Crochet (; ) is a process of creating textiles by using a crochet hook to interlock loops of yarn, thread (yarn), thread, or strands of other materials. The name is derived from the French term ''crochet'', which means 'hook'. Hooks can be made ...
,
embroidery Embroidery is the art of decorating Textile, fabric or other materials using a Sewing needle, needle to stitch Yarn, thread or yarn. It is one of the oldest forms of Textile arts, textile art, with origins dating back thousands of years across ...
and
weaving Weaving is a method of textile production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. Other methods are knitting, crocheting, felting, and braiding or plaiting. The longitudinal ...
, where they may be purely decorative or may be marks of status.
Rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
is found in circular structures such as
dome A dome () is an architectural element similar to the hollow upper half of a sphere. There is significant overlap with the term cupola, which may also refer to a dome or a structure on top of a dome. The precise definition of a dome has been a m ...
s; these are sometimes elaborately decorated with symmetric patterns inside and out, as at the 1619 Sheikh Lotfollah Mosque in
Isfahan Isfahan or Esfahan ( ) is a city in the Central District (Isfahan County), Central District of Isfahan County, Isfahan province, Iran. It is the capital of the province, the county, and the district. It is located south of Tehran. The city ...
. Items of embroidery and
lace Lace is a delicate fabric made of yarn or thread in an open weblike pattern, made by machine or by hand. Generally, lace is split into two main categories, needlelace and bobbin lace, although there are other types of lace, such as knitted o ...
work such as tablecloths and table mats, made using bobbins or by
tatting Tatting is a technique for handcrafting a particularly durable lace from a series of knots and loops. Tatting can be used to make lace edging as well as doilies, collars, accessories such as earrings, necklaces, waist beads, and other decor ...
, can have a wide variety of reflectional and rotational symmetries which are being explored mathematically.
Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslims, Muslim populations. Referring to characteristic traditions across ...
exploits symmetries in many of its artforms, notably in
girih ''Girih'' (, "knot", also written ''gereh'') are decorative Islamic geometric patterns used in architecture and handicraft objects, consisting of angled lines that form an interlaced strapwork pattern. ''Girih'' decoration is believed to have b ...
tilings. These are formed using a set of five tile shapes, namely a regular decagon, an elongated hexagon, a bow tie, a rhombus, and a regular pentagon. All the sides of these tiles have the same length; and all their angles are multiples of 36° (π/5
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s), offering fivefold and tenfold symmetries. The tiles are decorated with
strapwork In the history of art and design, strapwork is the use of stylised representations in ornament of ribbon-like forms. These may loosely imitate leather straps, parchment or metal cut into elaborate shapes, with piercings, and often interwoven in ...
lines (girih), generally more visible than the tile boundaries. In 2007, the physicists Peter Lu and
Paul Steinhardt Paul Joseph Steinhardt (born December 25, 1952) is an American theoretical physicist whose principal research is in cosmology and condensed matter physics. He is currently the Albert Einstein Professorship in Science, Albert Einstein Professor in ...
argued that girih resembled quasicrystalline
Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and a tiling is ''aperiodic'' if it does not contain arbitrarily large Perio ...
s. Elaborate geometric zellige tilework is a distinctive element in Moroccan architecture.
Muqarnas Muqarnas (), also known in Iberian architecture as Mocárabe (from ), is a form of three-dimensional decoration in Islamic architecture in which rows or tiers of niche-like elements are projected over others below. It is an archetypal form of I ...
vaults are three-dimensional but were designed in two dimensions with drawings of geometrical cells. File:Hotamis Kilim.jpg, Hotamis kilim (detail), central
Anatolia Anatolia (), also known as Asia Minor, is a peninsula in West Asia that makes up the majority of the land area of Turkey. It is the westernmost protrusion of Asia and is geographically bounded by the Mediterranean Sea to the south, the Aegean ...
, early 19th century File:Ming flower brocade (cropped)2.jpg, Detail of a
Ming Dynasty The Ming dynasty, officially the Great Ming, was an Dynasties of China, imperial dynasty of China that ruled from 1368 to 1644, following the collapse of the Mongol Empire, Mongol-led Yuan dynasty. The Ming was the last imperial dynasty of ...
brocade, using a chamfered hexagonal lattice pattern File:Salim Chishti Tomb-2.jpg, '' Jaali'' marble lattice at tomb of Salim Chishti,
Fatehpur Sikri Fatehpur Sikri () is a town in the Agra District of Uttar Pradesh, India. Situated from the district headquarters of Agra, Fatehpur Sikri itself was founded as the capital of the Mughal Empire in 1571 by Mughal emperors, Emperor Akbar, servin ...
,
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
File:Florentine Bargello Pattern.png,
Symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
: Florentine Bargello pattern tapestry work File:Isfahan Lotfollah mosque ceiling symmetric.jpg, Ceiling of the Sheikh Lotfollah Mosque,
Isfahan Isfahan or Esfahan ( ) is a city in the Central District (Isfahan County), Central District of Isfahan County, Isfahan province, Iran. It is the capital of the province, the county, and the district. It is located south of Tehran. The city ...
, 1619 File:Frivolité.jpg,
Rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
in
lace Lace is a delicate fabric made of yarn or thread in an open weblike pattern, made by machine or by hand. Generally, lace is split into two main categories, needlelace and bobbin lace, although there are other types of lace, such as knitted o ...
:
tatting Tatting is a technique for handcrafting a particularly durable lace from a series of knots and loops. Tatting can be used to make lace edging as well as doilies, collars, accessories such as earrings, necklaces, waist beads, and other decor ...
work File:Darb-i Imam shrine spandrel.JPG, Girih tiles: patterns at large and small scales on a
spandrel A spandrel is a roughly triangular space, usually found in pairs, between the top of an arch and a rectangular frame, between the tops of two adjacent arches, or one of the four spaces between a circle within a square. They are frequently fil ...
from the Darb-i Imam shrine, Isfahan, 1453 File:Fes Medersa Bou Inania Mosaique2.jpg,
Tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
s: zellige mosaic tiles at Bou Inania Madrasa,
Fes,
Morocco Morocco, officially the Kingdom of Morocco, is a country in the Maghreb region of North Africa. It has coastlines on the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to Algeria–Morocc ...
File:Mezquita Shah, Isfahán, Irán, 2016-09-20, DD 64 (detail).jpg, The complex geometry and tilings of the
muqarnas Muqarnas (), also known in Iberian architecture as Mocárabe (from ), is a form of three-dimensional decoration in Islamic architecture in which rows or tiers of niche-like elements are projected over others below. It is an archetypal form of I ...
vaulting in the Sheikh Lotfollah Mosque, Isfahan File:Topkapi Scroll p294 muqarnas.JPG, Architect's plan of a muqarnas quarter vault.
Topkapı Scroll The Topkapı Scroll () is a Timurid dynasty patterned scroll in the collection of the Topkapı Palace museum. The scroll is a valuable source of information, consisting of 114 patterns that may have been used both indirectly and directly by arch ...
File:Tupa-inca-tunic.png, Tupa Inca tunic from
Peru Peru, officially the Republic of Peru, is a country in western South America. It is bordered in the north by Ecuador and Colombia, in the east by Brazil, in the southeast by Bolivia, in the south by Chile, and in the south and west by the Pac ...
, 1450 –1540, an Andean textile denoting high rank


Polyhedra

The
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s and other
polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
are a recurring theme in Western art. They are found, for instance, in a marble mosaic featuring the
small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...
, attributed to Paolo Uccello, in the floor of the
San Marco Basilica The Patriarchal Cathedral Basilica of Saint Mark (), commonly known as St Mark's Basilica (; ), is the cathedral church of the Patriarchate of Venice; it became the episcopal seat of the Patriarch of Venice in 1807, replacing the earlier cathed ...
in Venice; in Leonardo da Vinci's diagrams of regular polyhedra drawn as illustrations for
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
's 1509 book ''The Divine Proportion''; as a glass
rhombicuboctahedron In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for ''truncated cuboctahedral rhombus'', w ...
in Jacopo de Barbari's portrait of Pacioli, painted in 1495; in the truncated polyhedron (and various other mathematical objects) in
Albrecht Dürer Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...
's engraving Melencolia I; and in
Salvador Dalí Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (11 May 190423 January 1989), known as Salvador Dalí ( ; ; ), was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, ...
's painting ''The Last Supper'' in which Christ and his disciples are pictured inside a giant
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
.
Albrecht Dürer Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...
(1471–1528) was a
German German(s) may refer to: * Germany, the country of the Germans and German things **Germania (Roman era) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizenship in Germany, see also Ge ...
Renaissance
printmaker Printmaking is the process of creating artworks by printing, normally on paper, but also on fabric, wood, metal, and other surfaces. "Traditional printmaking" normally covers only the process of creating prints using a hand processed technique ...
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 Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
in
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
,
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...
, and
regular polygons In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
. Dürer was likely influenced by the works of
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
and
Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...
during his trips to
Italy Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
. 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. Dürer published another influential book on human proportions called ''Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion)'' in 1528. Dürer's well-known engraving '' Melencolia I'' depicts a frustrated thinker sitting by a truncated triangular trapezohedron and a
magic square In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
. 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, including a two-volume book by Peter-Klaus Schuster, and an influential discussion in
Erwin Panofsky Erwin Panofsky (March 30, 1892 – March 14, 1968) was a German-Jewish art historian whose work represents a high point in the modern academic study of iconography, including his hugely influential ''Renaissance and Renascences in Western Art ...
's monograph of Dürer.
Salvador Dalí Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (11 May 190423 January 1989), known as Salvador Dalí ( ; ; ), was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, ...
's 1954 painting '' Corpus Hypercubus'' uniquely depicts the cross of Christ as an unfolded three-dimensional net for a
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
, also known as a
tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...
: 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. 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." File:Leonardo polyhedra.png, The first printed illustration of a
rhombicuboctahedron In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for ''truncated cuboctahedral rhombus'', w ...
, by
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
, published in ''
De Divina Proportione ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'', 1509 File:Icosahedron-spinoza.jpg,
Icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
as a part of the monument to
Baruch Spinoza Baruch (de) Spinoza (24 November 163221 February 1677), also known under his Latinized pen name Benedictus de Spinoza, was a philosopher of Portuguese-Jewish origin, who was born in the Dutch Republic. A forerunner of the Age of Enlightenmen ...
, Amsterdam


Fractal dimensions

Traditional Indonesian wax-resist
batik Batik is a dyeing technique using wax Resist dyeing, resist. The term is also used to describe patterned textiles created with that technique. Batik is made by drawing or stamping wax on a cloth to prevent colour absorption during the dyein ...
designs on cloth combine representation (arts), representational motifs (such as floral and vegetal elements) with abstract and somewhat chaotic elements, including imprecision in applying the wax resist, and random variation introduced by cracking of the wax. Batik designs have a
fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...
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 Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...
have a fractal dimension of 1.2 to 1.5; and the batiks of Rembang Regency, Lasem on the north coast of Java and of Tasikmalaya in West Java have a fractal dimension between 1.5 and 1.7. 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.


A complex relationship

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


Mathematics as an art

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. King cites the mathematician G. H. Hardy's 1940 essay ''A Mathematician's Apology''. In it, Hardy discusses why he finds two theorems of classical antiquity, classical times as first rate, namely
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's proof there are infinitely many prime numbers, and the proof that the square root of 2 is irrational number, irrational. King evaluates this last against Hardy's criteria for mathematical beauty, mathematical elegance: "''seriousness, depth, generality, unexpectedness, inevitability'', and ''economy''" (King's italics), and describes the proof as "aesthetically pleasing". The Hungarian mathematician Paul Erdős agreed that mathematics possessed beauty but considered the reasons beyond explanation: "Why are numbers beautiful? It's like asking why is Symphony No. 9 (Beethoven), Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I ''know'' numbers are beautiful."


Mathematical tools for art

Mathematics can be discerned in many of the arts, such as music,
dance Dance is an The arts, art form, consisting of sequences of body movements with aesthetic and often Symbol, symbolic value, either improvised or purposefully selected. Dance can be categorized and described by its choreography, by its repertoir ...
,
painting Painting is a Visual arts, visual art, which is characterized by the practice of applying paint, pigment, color or other medium to a solid surface (called "matrix" or "Support (art), support"). The medium is commonly applied to the base with ...
,
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
, and
sculpture Sculpture is the branch of the visual arts that operates in three dimensions. Sculpture is the three-dimensional art work which is physically presented in the dimensions of height, width and depth. It is one of the plastic arts. Durable sc ...
. Each of these is richly associated with mathematics. 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. Artists from Luca Pacioli in the Middle Ages and Leonardo da Vinci and Albrecht Dürer in the
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
have made use of and developed mathematical ideas in the pursuit of their artistic work. 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 A vanishing point is a point (geometry), point on the projection plane, image plane of a graphical perspective, perspective rendering where the two-dimensional perspective projections of parallel (geometry), parallel lines in three-dimensional ...
were first formulated by
Brunelleschi Filippo di ser Brunellesco di Lippo Lapi (1377 – 15 April 1446), commonly known as Filippo Brunelleschi ( ; ) and also nicknamed Pippo by Leon Battista Alberti, was an Italian architect, designer, goldsmith and sculptor. He is considered to ...
in about 1413, his theory influencing Leonardo and Dürer. Isaac Newton's work on the visible spectrum, optical spectrum influenced Johann Wolfgang Goethe, Goethe's ''Theory of Colours'' and in turn artists such as Philipp Otto Runge, J. M. W. Turner, the Pre-Raphaelites and Wassily Kandinsky. Artists may also choose to analyse the
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
of a scene. Tools may be applied by mathematicians who are exploring art, or artists inspired by mathematics, such as
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
(inspired by H. S. M. Coxeter) and the architect Frank Gehry, who more tenuously argued that computer aided design enabled him to express himself in a wholly new way. The artist Richard Wright argues that mathematical objects that can be constructed can be seen either "as processes to simulate phenomena" or as works of "computer art". He considers the nature of mathematical thought, observing that fractals were known to mathematicians for a century before they were recognised as such. Wright concludes by stating that it is appropriate to subject mathematical objects to any methods used to "come to terms with cultural artifacts like art, the tension between objectivity and subjectivity, their metaphorical meanings and the character of representational systems." He gives as instances an image from the
Mandelbrot set The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting ...
, an image generated by a cellular automaton algorithm, and a rendering (computer graphics), computer-rendered image, and discusses, with reference to the Turing test, whether algorithmic products can be art. 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. Some of the first works of computer art were created by Desmond Paul Henry's "Drawing Machine 1", an analogue computer, analogue machine based on a bombsight computer and exhibited in 1962. The machine was capable of creating complex, abstract, asymmetrical, curvilinear, but repetitive line drawings. 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. Artists such as Mikael Hvidtfeldt Christensen create works of generative art, generative or algorithmic art by writing scripts for a software system such as ''Structure Synth'': the artist effectively directs the system to apply a desired combination of mathematical operations to a chosen set of data. File:Bathsheba Grossman geometric art.jpg, Mathematical sculpture by Bathsheba Grossman, 2007 File:Hartmut Skerbisch.jpg, Fractal sculpture: ''3D Fraktal 03/H/dd'' by :de:Hartmut Skerbisch, Hartmut Skerbisch, 2003 File:FWF Samuel Monnier détail.jpg, Fibonacci word: detail of artwork by Samuel Monnier, 2009 File:Wiki.picture by drawing machine 1.jpg,
Computer art Computer art is art in which computers play a role in the production or display of the artwork. Such art can be an image, sound, animation, video, CD-ROM, DVD-ROM, video game, website, algorithm, performance or gallery installation. Many traditio ...
image produced by Desmond Paul Henry's "Drawing Machine 1", exhibited 1962 File:A Bird in Flight by Hamid Naderi Yeganeh 2016.jpg, ''A Bird in Flight'', by Hamid Naderi Yeganeh, 2016, constructed with a family of mathematical curves.


From mathematics to art

The mathematician and theoretical physicist Henri Poincaré's ''Science and Hypothesis'' was widely read by the Cubism, Cubists, including Pablo Picasso and Jean Metzinger. Being thoroughly familiar with Bernhard Riemann's work on non-Euclidean geometry, Poincaré was more than aware that Euclidean geometry is just one of many possible geometric configurations, rather than as an absolute objective truth. The possible existence of a Fourth dimension in art, fourth dimension inspired artists to question classical Perspective (graphical)#Renaissance : Mathematical basis, Renaissance perspective: non-Euclidean geometry became a valid alternative. The concept that painting could be expressed mathematically, in colour and form, contributed to Cubism, the art movement that led to abstract art. Metzinger, in 1910, wrote that: "[Picasso] lays out a free, mobile perspective, from which that ingenious mathematician Maurice Princet has deduced a whole geometry". 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 Schlegel diagram, new views on space that had been opened up by Victor Schlegel, Schlegel and some others. He succeeded at that.
The impulse to make teaching or research models of mathematical forms naturally creates objects that have symmetries and surprising or pleasing shapes. Some of these have inspired artists such as the Dadaism, Dadaists Man Ray, Marcel Duchamp and Max Ernst, and following Man Ray, Hiroshi Sugimoto. Man Ray photographed some of the mathematical models in the Institut Henri Poincaré in Paris, including ''Objet mathematique'' (Mathematical object). He noted that this represented Enneper surfaces with constant negative curvature, derived from the pseudo-sphere. This mathematical foundation was important to him, as it allowed him to deny that the object was "abstract", instead claiming that it was as real as the urinal that Duchamp made into a work of art. Man Ray admitted that the object's [Enneper surface] formula "meant nothing to me, but the forms themselves were as varied and authentic as any in nature." He used his photographs of the mathematical models as figures in his series he did on Shakespeare's plays, such as his 1934 painting ''Antony and Cleopatra''. The art reporter Jonathan Keats, writing in ''ForbesLife'', argues that Man Ray photographed "the elliptic paraboloids and conic points in the same sensual light as his pictures of Kiki de Montparnasse", and "ingeniously repurposes the cool calculations of mathematics to reveal the topology of desire". Twentieth century sculptors such as Henry Moore, Barbara Hepworth and Naum Gabo took inspiration from mathematical models. Moore wrote of his 1938 ''Stringed Mother and Child'': "Undoubtedly the source of my stringed figures was the Science Museum, London, 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." The artists Theo van Doesburg and
Piet Mondrian Pieter Cornelis Mondriaan (; 7 March 1872 – 1 February 1944), known after 1911 as Piet Mondrian (, , ), was a Dutch Painting, painter and Theory of art, art theoretician who is regarded as one of the greatest artists of the 20th century. He w ...
founded the
De Stijl De Stijl (, ; 'The Style') was a Dutch art movement founded in 1917 by a group of artists and architects based in Leiden (Theo van Doesburg, Jacobus Oud, J.J.P. Oud), Voorburg (Vilmos Huszár, Jan Wils) and Laren, North Holland, Laren (Piet Mo ...
movement, which they wanted to "establish a visual vocabulary elementary geometrical forms comprehensible by all and adaptable to any discipline". Many of their artworks visibly consist of ruled squares and triangles, sometimes also with circles. De Stijl artists worked in painting, furniture, interior design and architecture. After the breakup of De Stijl, Van Doesburg founded the Avant-garde Art Concret movement, describing his 1929–1930
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. The mathematics of
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
, polyhedra, shaping of space, and self-reference provided the graphic artist
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
(1898—1972) with a lifetime's worth of materials for his woodcuts. 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 Translation (geometry), translations to obtain further patterns. Many of his works contain impossible constructions, made using geometrical objects which set up a contradiction between perspective projection and three dimensions, but are pleasant to the human sight. Escher's ''Ascending and Descending'' is based on the "Penrose stairs, impossible staircase" created by the medical scientist Lionel Penrose and his son the mathematician Roger Penrose. Some of Escher's many tessellation drawings were inspired by conversations with the mathematician H. S. M. Coxeter on
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
. Escher was especially interested in five specific polyhedra, which appear many times in his work. The
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...
—tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons—are especially prominent in ''Order and Chaos'' and ''Four Regular Solids''. 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. The visual intricacy of mathematical structures such as tessellations and polyhedra have inspired a variety of mathematical artworks. Stewart Coffin makes polyhedral puzzles in rare and beautiful woods; George W. Hart works on the theory of polyhedra and sculpts objects inspired by them;
Magnus Wenninger Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to ...
makes "especially beautiful" models of List of Wenninger polyhedron models, complex stellated polyhedra. The distorted perspectives of anamorphosis have been explored in art since the sixteenth century, when Hans Holbein the Younger incorporated a severely distorted skull in his 1533 painting ''The Ambassadors (Holbein), The Ambassadors''. Many artists since then, including Escher, have make use of anamorphic tricks. The mathematics of topology has inspired several artists in modern times. The sculptor John Robinson (sculptor), John Robinson (1935–2007) created works such as ''Gordian Knot'' and ''Bands of Friendship'', displaying knot theory in polished bronze. Other works by Robinson explore the topology of toruses. ''Genesis'' is based on Borromean rings – a set of three circles, no two of which link but in which the whole structure cannot be taken apart without breaking. The sculptor Helaman Ferguson creates complex Surface (topology), surfaces and other topological space, topological objects. His works are visual representations of mathematical objects; ''The Eightfold Way'' is based on the projective special linear group PSL(2,7), a finite group of 168 elements. The sculptor Bathsheba Grossman similarly bases her work on mathematical structures. The artist Nelson Saiers incorporates mathematical concepts and theorems in his art from toposes and Scheme (mathematics), schemes to the four color theorem and the irrationality of Pi, π. A liberal arts inquiry project examines connections between mathematics and art through the
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...
, flexagons, origami and panorama photography. Mathematical objects including the Lorenz attractor, Lorenz manifold and the Hyperbolic manifold, hyperbolic plane have been crafted using Mathematics and fiber arts, fiber arts including crochet. The American weaver Ada Dietz wrote a 1949 monograph ''Algebraic Expressions in Handwoven Textiles'', defining weaving patterns based on the expansion of multivariate polynomials. The mathematician Daina Taimiņa demonstrated features of the hyperbolic plane by crocheting in 2001. This led Margaret Wertheim, Margaret and Christine Wertheim to crochet a coral reef, consisting of many marine animals such as nudibranchs whose shapes are based on hyperbolic planes. The mathematician J. C. P. Miller used the Rule 90 cellular automaton to design tapestry, tapestries depicting both trees and abstract patterns of triangles. The "" Pat Ashforth and Steve Plummer use knitted versions of mathematical objects such as hexaflexagons in their teaching, though their Menger sponge proved too troublesome to knit and was made of plastic canvas instead. Their "mathghans" (Afghans for Schools) project introduced
knitting Knitting is a method for production of textile Knitted fabric, fabrics by interlacing yarn loops with loops of the same or other yarns. It is used to create many types of garments. Knitting may be done Hand knitting, by hand or Knitting machi ...
into the British mathematics and technology curriculum. File:Jouffret.gif, Four-dimensional space to Cubism: Esprit Jouffret's 1903 ''Traité élémentaire de géométrie à quatre dimensions''. File:Theo van Doesburg Composition I.jpg,
De Stijl De Stijl (, ; 'The Style') was a Dutch art movement founded in 1917 by a group of artists and architects based in Leiden (Theo van Doesburg, Jacobus Oud, J.J.P. Oud), Voorburg (Vilmos Huszár, Jan Wils) and Laren, North Holland, Laren (Piet Mo ...
: Theo van Doesburg's geometric ''Composition I (Still Life)'', 1916 File:Magnus Wenninger polyhedral models.jpg, Pedagogy to art:
Magnus Wenninger Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to ...
with some of his stellated polyhedra, 2009 File:Moebiusstripscarf.jpg, A
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...
scarf in
crochet Crochet (; ) is a process of creating textiles by using a crochet hook to interlock loops of yarn, thread (yarn), thread, or strands of other materials. The name is derived from the French term ''crochet'', which means 'hook'. Hooks can be made ...
, 2007 File:Hans Holbein the Younger - The Ambassadors - Google Art Project.jpg, Anamorphism: ''The Ambassadors (Holbein), The Ambassadors'' by Hans Holbein the Younger, 1533, with severely distorted skull in foreground File:The Föhr Reef in Tübingen.JPG, Crocheted coral reef: many animals modelled as hyperbolic planes with varying parameters by Margaret Wertheim, Margaret and Christine Wertheim. ''Föhr Reef'', Tübingen, 2013


Illustrating mathematics

Modelling is far from the only possible way to illustrate mathematical concepts. Giotto's ''Stefaneschi Triptych'', 1320, illustrates
recursion Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
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. Giorgio de Chirico's metaphysics, metaphysical paintings such as his 1917 ''Great Metaphysical Interior'' explore the question of levels of representation in art by depicting paintings within his paintings. File:Giotto. The Stefaneschi Triptych (verso) c.1330 220x245cm. Pinacoteca, Vatican..jpg, Front face of Giotto's ''Stefaneschi Triptych'', 1320 illustrates
recursion Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
. File:Giotto di Bondone - The Stefaneschi Triptych - St Peter Enthroned (detail) - WGA09356.jpg, Detail of Cardinal Stefaneschi holding the triptych
Art can exemplify logical paradoxes, as in some paintings by the surrealist
René Magritte René François Ghislain Magritte (; 21 November 1898 – 15 August 1967) was a Belgium, Belgian surrealist artist known for his depictions of familiar objects in unfamiliar, unexpected contexts, which often provoked questions about the nature ...
, which can be read as semiotic jokes about confusion between levels. In ''The Human Condition (painting), La condition humaine'' (1933), Magritte depicts an easel (on the real canvas), seamlessly supporting a view through a window which is framed by "real" curtains in the painting. Earlier artists such as Rembrandt had already explored the question of framing, in multiple paintings such as his 1646 ''The 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. Alberti, in his discussion of perspective, had likened paintings to open windows on to the scenes depicted. In ''La 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." In Rényi's view, the resulting visual joke is the entire purpose of the painting, subverting the art of painting. He comments that ''La condition humaine'' is "more of a painted, meta-painterly philosopheme than a painterly work". Another approach to mathematical paradox is taken in Escher's ''Print Gallery (M. C. Escher), Print Gallery'' (1956); this is a print which depicts a distorted city which contains a gallery which recursively contains the picture, and so ''ad infinitum''. 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. ''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. Salvador Dalí's last painting, ''The Swallow's Tail'' (1983), was part of a series inspired by René Thom's catastrophe theory. The Spanish painter and sculptor Pablo Palazuelo (1916–2007) focused on the investigation of form. He developed a style that he described as the geometry of life and the geometry of all nature. Consisting of simple geometric shapes with detailed patterning and coloring, in works such as ''Angular I'' and ''Automnes'', Palazuelo expressed himself in geometric transformations. The artist Adrian Gray practises Rock balancing, stone balancing, exploiting friction and the centre of gravity to create striking and seemingly impossible compositions. 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. 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.


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. The 20th century Dadaist artist Max Ernst painted Lissajous 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. Ernst was the first to introduce the use of this semi-automatic form of drip painting (also called "oscillation"), which he popularised. He applied the technique to multiple paintings during his artistic career, including his 1942 works ''The Bewildered Planet'', ''Surrealism and Painting'', and ''Young Man Intrigued by the Flight of a Non-Euclidean Fly'', and his 1970 work ''Green Zone''. Lissajous figures later featured repeatedly in early computer art. Ernst's use of the mathematical technique likely influenced Jackson Pollock's drip painting style. Pollock's paintings have a definite
fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...
of controlled chaos theory, chaos. 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. The American mathematician George Birkhoff's 1933 ''Aesthetic Measure'' proposes a quantitative metric of the Aesthetics, aesthetic quality of an artwork. It does not attempt to measure the connotations of a work, such as what a painting might mean, but is limited to the "elements of order" of a polygonal figure. Birkhoff first combines (as a sum) five such elements: whether there is a vertical axis of symmetry; whether there is optical equilibrium; how many rotational symmetries it has; how wallpaper-like the figure is; and whether there are unsatisfactory features such as having two vertices too close together. This metric, ''O'', takes a value between −3 and 7. The second metric, ''C'', counts elements of the figure, which for a polygon is the number of different straight lines containing at least one of its sides. Birkhoff then defines his aesthetic measure of an object's beauty as ''O/C''. This can be interpreted as a balance between the pleasure looking at the object gives, and the amount of effort needed to take it in. Birkhoff's proposal has been criticized in various ways, not least for trying to put beauty in a formula, but he never claimed to have done that.


Stimuli to mathematical research

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, and ultimately to the mathematics of
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
of
Girard Desargues Girard Desargues (; 21 February 1591September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named i ...
and Jean-Victor Poncelet. The Japanese paper-folding art of origami has been reworked mathematically by Tomoko Fusé modular origami, using modules, congruent pieces of paper such as squares, and making them into polyhedra or tilings. Paper-folding was used in 1893 by T. Sundara Rao in his ''Geometric Exercises in Paper Folding'' to demonstrate geometrical proofs. The mathematics of paper folding has been explored in Maekawa's theorem, Kawasaki's theorem, and the Huzita–Hatori axioms. File:Della Pittura Alberti perspective circle to ellipse.jpg, Stimulus to
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
: Leon Battista Alberti, Alberti's diagram showing a circle seen in perspective as an ellipse. ''Della Pittura'', 1435–1436 File:Origami spring.jpg, Mathematical origami: ''Spring (device), Spring Into Action'', by Jeff Beynon, made from a single paper rectangle.


Illusion to op art

Optical illusions such as the Fraser spiral illusion, Fraser spiral strikingly demonstrate limitations in human visual perception, creating what the art historian Ernst Gombrich called a "baffling trick." The black and white ropes that appear to form spirals are in fact concentric circles. The mid-twentieth century Op art, op art or optical art style of painting and graphics exploited such effects to create the impression of movement and flashing or vibrating patterns seen in the work of artists such as Bridget Riley, Spyros Horemis, and Victor Vasarely.


Sacred geometry

A strand of art from Ancient Greece onwards sees God as the geometer of the world, and the world's geometry therefore as sacred. The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing that "Plato said God geometrizes continually" (''Convivialium disputationum'', liber 8,2). This image has influenced Western thought ever since. The Platonic concept derived in its turn from a Pythagoras, Pythagorean notion of harmony in music, where the notes were spaced in perfect proportions, corresponding to the lengths of the lyre's strings; indeed, the Pythagoreans held that everything was arranged by Number. In the same way, in Platonic thought, the Platonic solid, regular or Platonic solids dictate the proportions found in nature, and in art. An illumination in the 13th-century ''Codex Vindobonensis'' shows God drawing out the universe with a pair of compasses, which may refer to a verse in the Old Testament: "When he established the heavens I was there: when he set a compass upon the face of the deep" (Proverbs 8:27). In 1596, the mathematical astronomer Johannes Kepler modelled the universe as a set of nested Platonic solids, determining the relative sizes of the orbits of the planets. William Blake's ''Ancient of Days'' (depicting Urizen, Blake's embodiment of reason and law) and his painting of the physicist Isaac Newton, naked, hunched and drawing with a compass, use the symbolism of compasses to critique conventional reason and materialism as narrow-minded.
Salvador Dalí Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (11 May 190423 January 1989), known as Salvador Dalí ( ; ; ), was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, ...
's 1954 ''Crucifixion (Corpus Hypercubus)'' depicts the cross as a
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
, representing the divine perspective with four dimensions rather than the usual three. In Dalí's ''The Sacrament of the Last Supper'' (1955) Christ and his disciples are pictured inside a giant
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
. File:God the Geometer.jpg, God the geometer. ''Codex Vindobonensis'', c. 1220 File:Bible moralisée de Tolède - Dieu pantocrator.jpg, The creation, with the Pantocrator bearing. Bible of St Louis, c. 1220–1240 File:Kepler-solar-system-2.png, Johannes Kepler's
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
model of planetary spacing in the Solar System from ''Mysterium Cosmographicum'', 1596 File:The Ancient of Days.jpg, William Blake's ''The Ancient of Days'', 1794 File:William Blake - Newton.png, William Blake's ''Newton (Blake), Newton'', c. 1800


See also

* Mathematics and architecture * Music and mathematics * List of mathematical art software


Notes


References


External links


Bridges Organization
conference on connections between art and mathematics
Bridging the Gap Between Math and Art
– Slide Show from ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...
''
Discovering the Art of Mathematics

Mathematics and Art
– American Mathematical Society, AMS
Mathematics and Art
– Cut-the-Knot
Mathematical Imagery
– American Mathematical Society
Mathematics in Art and Architecture
– National University of Singapore

– Virtual Math Museum
When art and math collide
– Science News
Why the history of maths is also the history of art
Lynn Gamwell in ''The Guardian'' {{DEFAULTSORT:Mathematics and Art Mathematics and art, History of art Mathematics and culture, Art Visual arts Applied mathematics