1.
Piero della Francesca
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Piero della Francesca was an Italian painter of the Early Renaissance. As testified by Giorgio Vasari in his Lives of the Most Excellent Painters, Sculptors, nowadays Piero della Francesca is chiefly appreciated for his art. His painting is characterized by its humanism, its use of geometric forms. His most famous work is the cycle of frescoes The History of the True Cross in the church of San Francesco in the Tuscan town of Arezzo. He was most probably apprenticed to the local painter Antonio di Giovanni dAnghiari, because in documents about payments it is noted that he was working with Antonio in 1432 and May 1438. Besides, he took notice of the work of some of the Sienese artists active in San Sepolcro during his youth. In 1439 Piero received, together with Domenico Veneziano, payments for his work on frescoes for the church of SantEgidio in Florence, in Florence he must have met leading masters like Fra Angelico, Luca della Robbia, Donatello and Brunelleschi. The classicism of Masaccios frescoes and his figures in the Santa Maria del Carmine were for him an important source of inspiration. Dating of Pieros undocumented work is difficult because his style does not seem to have developed over the years, in 1442 he was listed as eligible for the City Council of San Sepolcro. Three years later, he received the commission for the Madonna della Misericordia altarpiece for the church of the Misericordia in Sansepolcro, in 1449 he executed several frescoes in the Castello Estense and the church of SantAndrea of Ferrara, also lost. His influence was strong in the later Ferrarese allegorical works of Cosimo Tura. Two years later he was in Rimini, working for the condottiero Sigismondo Pandolfo Malatesta, in this sojourn he executed in 1451 the famous fresco of St. Sigismund and Sigismondo Pandolfo Malatesta in the Tempio Malatestiano, as well as Sigismondos portrait. Thereafter Piero was active in Ancona, Pesaro and Bologna, in 1454 he signed a contract for the Polyptych of Saint Augustine in the church of SantAgostino in Sansepolcro. The central panel of this polyptic is lost and the four panels of the wings, a few years later, summoned by Pope Nicholas V, he moved to Rome, here he executed frescoes in the Basilica di Santa Maria Maggiore, of which only fragments remain. Two years later he was again in the Papal capital, for frescoes in Vatican Palace which have also been destroyed, the Baptism of Christ, in The National Gallery in London, was executed around 1460 for the high altar of the church of the Priory of S. Other notable works of Piero della Francescas maturity are the frescoes of the Resurrection of Christ in Sansepolcro, in 1452, Piero della Francesca was called to Arezzo to replace Bicci di Lorenzo in painting the frescoes of the basilica of San Francesco. The work was finished before 1466, probably between 1452 and 1456, the cycle of frescoes, depicting the Legend of the True Cross, is generally considered among his masterworks and those of Renaissance painting in general. The story in these frescoes derives from medieval sources as to how timber relics of the True Cross came to be found

2.
Arithmetic
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Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place

3.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers

4.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

5.
Perspective (graphical)
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Perspective in the graphic arts is an approximate representation, on a flat surface, of an image as it is seen by the eye. If viewed from the spot as the windowpane was painted. Each painted object in the scene is thus a flat, scaled down version of the object on the side of the window. All perspective drawings assume the viewer is a distance away from the drawing. Objects are scaled relative to that viewer, an object is often not scaled evenly, a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening, Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewers eye, represents objects infinitely far away and they have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to the Earths horizon, any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a vanishing point, usually directly opposite the viewers eye. All lines parallel with the line of sight recede to the horizon towards this vanishing point. This is the standard receding railroad tracks phenomenon, a two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of lines that are at an angle relative to the plane of the drawing. Perspectives consisting of parallel lines are observed most often when drawing architecture. In contrast, natural scenes often do not have any sets of parallel lines, the only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles. Chinese artists made use of perspective from the first or second century until the 18th century. It is not certain how they came to use the technique, some authorities suggest that the Chinese acquired the technique from India, oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga. This was detailed within Aristotles Poetics as skenographia, using flat panels on a stage to give the illusion of depth, the philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage, Euclids Optics introduced a mathematical theory of perspective, but there is some debate over the extent to which Euclids perspective coincides with the modern mathematical definition

6.
Mathematics and art
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty, Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts, Mathematics and art have a long historical relationship. Artists have used mathematics since the 5th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1, persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, another Italian painter, Piero della Francesca, developed Euclids ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I, in modern times, the graphic artist M. C. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jaali pierced stone screens, and widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as perspective, the analysis of symmetry, and mathematical objects such as polyhedra. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching, mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Computer art often makes use of including the Mandelbrot set. Polykleitos the elder was a Greek sculptor from the school of Argos, and his works and statues consisted mainly of bronze and were of athletes. While his sculptures may not be as famous as those of Phidias, Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Next, he takes the length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, the influence of the Canon of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, many sculptors following Polykleitoss prescription. While none of Polykleitoss original works survive, Roman copies demonstrate his ideal of physical perfection, some scholars argue that Pythagorean thought influenced the Canon of Polykleitos. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects, in the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, the Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts

7.
De pictura
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De pictura is a treatise written by the Italian architect and art theorist Leon Battista Alberti. The first version, written in vernacular Italian in 1435 under the title Della pittura, was for a general audience, the Latin version, the De pictura of 1439–41, was more technical and intended for scholars. The work is the first in a trilogy of treatises on the Major arts which had a circulation during the Renaissance. Alberti was a member of Florentine family exiled in the 14th century, here he knew contemporary art innovators such as Filippo Brunelleschi, Donatello and Masaccio, with whom he shared an interest for Renaissance humanism and classical art. The treatise contained an analysis of all the techniques and painting theories known at the time, De pictura also includes the first description of linear geometric perspective around 1416, Alberti credited the discovery to Brunelleschi, and dedicated the 1435 edition to him. He placed emphasis on the ability to depict the interactions between the figures by gesture and expression. De pictura relied heavily on references to art in classical literature, De pictura influenced the work of artists including Donatello, Ghiberti, Botticelli, and Ghirlandaio. His treatment of perspective was the most influential of his recommendations, having an effect on Leonardo da Vinci. Alberti made at least 29 uses of Pliny the Elders Natural History, deriving his key themes of simplicity, for example, Alberti advised artists to use colour with restraint, and to paint in the effect of gold rather than using actual gold in their paintings. Gold did indeed vanish from Italian paintings of the part of the 15th century. Artists however found their own ways to paint with restraint, rather than following Albertis actual instructions directly, similarly, he encouraged artists to add black when modelling shapes, rather than only adding white as Cennino Cennini had advised in his c.1390 Il Libro dellArte. This advice had the effect of making Italian renaissance paintings more sombre, Alberti was here perhaps following Plinys description of the dark varnish used by Apelles. 1435/6, Della Pittura 1439–41, De pictura Spencer, J On Painting, grayson, Cecil On Painting and On Sculpture, The Latin texts of De Pictura and De Statua. A New Translation and Critical Edition

8.
Leon Battista Alberti
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Leon Battista Alberti was an Italian humanist author, artist, architect, poet, priest, linguist, philosopher and cryptographer, he epitomised the Renaissance Man. Although Alberti is known mostly for being an artist, he was also a mathematician of many sorts, Albertis life was described in Giorgio Vasaris Lives of the Most Excellent Painters, Sculptors, and Architects. Leon Battista Alberti was born in 1404 in Genoa and his mother is unknown, and his father was a wealthy Florentine who had been exiled from his own city, allowed to return in 1428. Alberti was sent to boarding school in Padua, then studied Law at Bologna and he lived for a time in Florence, then travelled to Rome in 1431 where he took holy orders and entered the service of the papal court. During this time he studied the ancient ruins, which excited his interest in architecture, Alberti was gifted in many ways. He was tall, strong and an athlete who could ride the wildest horse. He distinguished himself as a writer while he was still a child at school, in 1435, he began his first major written work, Della pittura, which was inspired by the burgeoning pictorial art in Florence in the early 15th century. In this work he analyses the nature of painting and explores the elements of perspective, composition, in 1447 he became the architectural advisor to Pope Nicholas V and was involved with several projects at the Vatican. His first major commission was in 1446 for the facade of the Rucellai Palace in Florence. This was followed in 1450 by a commission from Sigismondo Malatesta to transform the Gothic church of San Francesco in Rimini into a memorial chapel, the Tempio Malatestiano. In 1452, he completed De re aedificatoria, a treatise on architecture, using as its basis the work of Vitruvius, the work was not published until 1485. It was followed in 1464 by his less influential work, De statua, Albertis only known sculpture is a self-portrait medallion, sometimes attributed to Pisanello. Alberti was employed to design two churches in Mantua, San Sebastiano, which was never completed, and for which Albertis intention can only be speculated, and the Basilica of SantAndrea. The design for the church was completed in 1471, a year before Albertis death. As an artist, Alberti distinguished himself from the ordinary craftsman and he was a humanist, and part of the rapidly expanding entourage of intellectuals and artisans supported by the courts of the princes and lords of the time. Alberti, as a member of family and as part of the Roman curia, had special status. He was a welcomed guest at the Este court in Ferrara, the Duke of Urbino was a shrewd military commander, who generously spent money on the patronage of art. Alberti planned to dedicate his treatise on architecture to his friend, among Albertis smaller studies, pioneering in their field, were a treatise in cryptography, De componendis cifris, and the first Italian grammar

9.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost

10.
Biblioteca Palatina, Parma
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The Biblioteca Palatina or Palatina Library was established in 1761 in the city of Parma by Philip Bourbon, Duke of Parma. It is one of the institutions located in the Palazzo della Pilotta complex in the center of Parma. The Palatina Library was named after Apollus Palatinus, the first librarian was the Theatine priest Paolo Maria Paciaudi, who was assigned as Antiquario e Bibliotecario. The goal was to form a library as part of a project by Duke Filippos prime minister. Paciaudi failed to acquire the collections of Cardinal Domenico Passionei in Roma and of the Pertusati family of Milan and he catalogued his purchases under six main classes, Theology, Nomology, Philosophy, History, Philology, and Liberal and Mechanic Arts. The books required the importation of Louis Antoine Laferté, a book binder. The collection was kept in a gallery refurbished for the purpose by the architect, Ennemond Alexandre Petitot. In 1771, both Du Tillot and Paciaudi fell out of favor, and the library fell under the supervision of the Benedictine Andrea Mazza, however, Paciaudi was recalled from 1778 till his death in 1785 to his former office. Paciaudi was replaced by the polymath cleric Ireneo Affò, he presided over expansion into the Galleria dellIncoronata, when Affò died in 1797, he was replaced by former Jesuit priest Matteo Luigi Canonici, until 1805. In 1804, the Napoleonic administration of the Duchy named Angelo Pezzana as director, Pezzana catalogued the books under five classes, Theology, Jurisprudence, Science & arts, Belle-Lettere, and History. The ceiling of the Sala Dante was frescoed by Francesco Scaramuzza, the next librarians included Federico Odorici and Luigi Rossi. The library was shelled in 1944, today the Librarys collection contains more than 708,000 printed works, about 6620 manuscripts, and 3042 incunabula,52,470 stamps. There is a musical section of 93,000 books. The music section was established in 1889, the electronic catalogue of the Palatina was started in 1994. The library holds mediaeval manuscripts, among them the biblical manuscripts 360 and 361. Official Website Biblioteca Palatina at the Consortium of European Research Libraries

11.
Parma
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Parma listen is a city in the northern Italian region of Emilia-Romagna famous for its prosciutto, cheese, architecture, music and surrounding countryside. It is home to the University of Parma, one of the oldest universities in the world, Parma is divided into two parts by the stream of the same name. The district on the far side of the river is Oltretorrente, Parmas Etruscan name was adapted by Romans to describe the round shield called Parma. The Italian poet Attilio Bertolucci wrote, As a capital city it had to have a river, as a little capital it received a stream, which is often dry. Parma was already an area in the Bronze Age. In the current position of the city rose a terramare, the terramare were ancient villages built of wood on piles according to a defined scheme and squared form, constructed on dry land and generally in proximity to the rivers. During this age the first necropolis were constructed, diodorus Siculus reported that the Romans had changed their rectangular shields for round ones, imitating the Etruscans. Whether the Etruscan encampment was so named because it was round, like a shield, the Roman colony was founded in 183 BC, together with Mutina,2,000 families were settled. Parma had an importance as a road hub over the Via Aemilia. It had a forum, in what is today the central Garibaldi Square, in 44 BC, the city was destroyed, and Augustus rebuilt it. During the Roman Empire, it gained the title of Julia for its loyalty to the imperial house, the city was subsequently sacked by Attila, and later given by the Germanic king Odoacer to his followers. During the Gothic War, however, Totila destroyed it and it was then part of the Byzantine Exarchate of Ravenna and, from 569, of the Lombard Kingdom of Italy. Under Frankish rule, Parma became the capital of a county, like most northern Italian cities, it was nominally a part of the Holy Roman Empire created by Charlemagne, but locally ruled by its bishops, the first being Guibodus. In the subsequent struggles between the Papacy and the Empire, Parma was usually a member of the Imperial party, two of its bishops became antipopes, Càdalo, founder of the cathedral, as Honorius II, and Guibert, as Clement III. An almost independent commune was created around 1140, a treaty between Parma and Piacenza of 1149 is the earliest document of a comune headed by consuls, the struggle between Guelphs and Ghibellines was a feature of Parma too. In 1213, her podestà was the Guelph Rambertino Buvalelli, then, after a long stance alongside the emperors, the Papist families of the city gained control in 1248. The city was besieged in 1247–48 by Emperor Frederick II, who was crushed in the battle that ensued. Parma fell under the control of Milan in 1341, after a short-lived period of independence under the Terzi family, the Sforza imposed their rule through their associated families of Pallavicino, Rossi, Sanvitale and Da Correggio

12.
Biblioteca Ambrosiana
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The Biblioteca Ambrosiana is a historic library in Milan, Italy, also housing the Pinacoteca Ambrosiana, the Ambrosian art gallery. Named after Ambrose, the saint of Milan, it was founded by Cardinal Federico Borromeo, whose agents scoured Western Europe and even Greece and Syria for books. To house the cardinals 15,000 manuscripts and twice that many printed books, construction began in 1603 under designs and direction of Lelio Buzzi and Francesco Maria Richini. When its first reading room, the Sala Fredericiana, opened to the public on 8 December 1609 it was, after the Bodleian Library in Oxford, a printing press was attached to the library, and a school for instruction in the classical languages. Constant acquisitions, soon augmented by bequests, required enlargement of the space, Borromeo intended an academy and a collection of pictures, for which a new building was initiated in 1611–18 to house the Cardinals paintings and drawings, the nucleus of the Pinacoteca. Cardinal Borromeo gave his collection of paintings and drawings to the library, shortly after the cardinals death, his library acquired twelve manuscripts of Leonardo da Vinci, including the Codex Atlanticus. Prized manuscripts, including the Leonardo codices, were requisitioned by the French during the Napoleonic occupation, on 15 October 1816 the Romantic poet Lord Byron visited the library. He was delighted by the letters between Lucrezia Borgia and Pietro Bembo and claimed to have managed to steal a lock of her hair held on display. The novelist Mary Shelley visited the library on 14 September 1840 but was disappointed by the tight security occasioned by the recent attempted theft of some of the relics of Petrarch housed there. Among Christian and Islamic Arabic manuscripts are treatises on medicine, a unique 11th-century diwan of poets, the library has a college of Doctors, similar to the scriptors of the Vatican Library. The building was damaged in World War II, with the loss of the archives of opera libretti of La Scala, artwork at the Pinacoteca Ambrosiana includes da Vincis Portrait of a Musician, Caravaggios Basket of Fruit, and Raphaels cartoon of The School of Athens. S

13.
Luca Pacioli
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Fra Luca Bartolomeo de Pacioli was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and a seminal contributor to the field now known as accounting. He is referred to as The Father of Accounting and Bookkeeping in Europe and he was also called Luca di Borgo after his birthplace, Borgo Sansepolcro, Tuscany. Luca Pacioli was born between 1446 and 1448 in Sansepolcro where he received an abbaco education and this was education in the vernacular rather than Latin and focused on the knowledge required of merchants. His father was Bartolomeo Pacioli, however Luca Pacioli was said to have lived with the Befolci family as a child in his birth town Sansepolcro. He moved to Venice around 1464, where he continued his own education while working as a tutor to the three sons of a merchant and it was during this period that he wrote his first book, a treatise on arithmetic for the boys he was tutoring. Between 1472 and 1475, he became a Franciscan friar, in 1475, he started teaching in Perugia, first as a private teacher, from 1477 holding the first chair in mathematics. He wrote a textbook in the vernacular for his students. He continued to work as a tutor of mathematics and was, in fact. In 1494, his first book to be printed, Summa de arithmetica, proportioni et proportionalita, was published in Venice. In 1497, he accepted an invitation from Duke Ludovico Sforza to work in Milan, there he met, taught mathematics to, collaborated and lived with Leonardo da Vinci. In 1499, Pacioli and Leonardo were forced to flee Milan when Louis XII of France seized the city and their paths appear to have finally separated around 1506. Pacioli died at about the age of 70 in 1517, most likely in Sansepolcro where it is thought that he had spent much of his final years, the manuscript was written between December 1477 and 29 April 1478. It contains 16 sections on merchant arithmetic, such as barter, exchange, profit, mixing metals, one part of 25 pages is missing from the chapter on algebra. A modern transcription has been published by Calzoni and Cavazzoni along with a translation of the chapter on partitioning problems. Proportioni et proportionalita, a textbook for use in the schools of Northern Italy and it was a synthesis of the mathematical knowledge of his time and contained the first printed work on algebra written in the vernacular. It is also notable for including the first published description of the method that Venetian merchants used during the Italian Renaissance. The system he published included most of the cycle as we know it today. He described the use of journals and ledgers, and warned that a person should not go to sleep at night until the debits equaled the credits

14.
De divina proportione
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De divina proportione is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da Vinci, composed around 1498 in Milan and first printed in 1509. Its subject was mathematical proportions and their applications to geometry, visual art through perspective, the clarity of the written material and Leonardos excellent diagrams helped the book to achieve an impact beyond mathematical circles, popularizing contemporary geometric concepts and images. The book consists of three manuscripts, which Pacioli worked on between 1496 and 1498. The first part, Compendio divina proportione, studies the golden ratio from a perspective and explores its applications to various arts. The second part, Trattato dellarchitettura, discusses the ideas of Vitruvius on the application of mathematics to architecture in twenty chapters, the text compares the proportions of the human body to those of artificial structures, with examples from classical Greco-Roman architecture. The third part, Libellus in tres partiales divisus, is mainly an Italian translation of Piero della Francescas Latin writings On Five Regular Solids and mathematical examples. In 1550 Giorgio Vasari wrote a biography of della Francesca, in which he accused Pacioli of plagiarism and claimed that he stole della Francescas work on perspective, on arithmetic, Leonardo drew the illustrations of the regular solids while he lived with and took mathematics lessons from Pacioli. Leonardos drawings are probably the first illustrations of skeletonic solids which allowed an easy distinction between front and back, Pacioli produced three manuscripts of the treatise by different scribes. He gave the first copy with a dedication to the Duke of Milan, Ludovico il Moro, a second copy was donated to Galeazzo da Sanseverino and now rests at the Biblioteca Ambrosiana in Milan. The third, which has gone missing, was given to Pier Soderini, on 1 June 1509 the first printed edition was published in Venice by Paganino Paganini, it has since been reprinted several times. The book was displayed as part of an exhibition in Milan between October 2005 and October 2006 together with the Codex Atlanticus, the M logo used by the Metropolitan Museum of Art in New York is adapted from one in De divina proportione. Full text of original edition Full text of 1509 edition Title page of a reprint in Vienna,1889 A video featuring a 1509 edition on display at Stevens Institute of Technology

15.
Leonardo da Vinci
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He has been variously called the father of palaeontology, ichnology, and architecture, and is widely considered one of the greatest painters of all time. Sometimes credited with the inventions of the parachute, helicopter and tank, many historians and scholars regard Leonardo as the prime exemplar of the Universal Genius or Renaissance Man, an individual of unquenchable curiosity and feverishly inventive imagination. Much of his working life was spent in the service of Ludovico il Moro in Milan. He later worked in Rome, Bologna and Venice, and he spent his last years in France at the home awarded to him by Francis I of France, Leonardo was, and is, renowned primarily as a painter. Among his works, the Mona Lisa is the most famous and most parodied portrait, Leonardos drawing of the Vitruvian Man is also regarded as a cultural icon, being reproduced on items as varied as the euro coin, textbooks, and T-shirts. Perhaps fifteen of his paintings have survived, Leonardo is revered for his technological ingenuity. He conceptualised flying machines, a type of armoured fighting vehicle, concentrated power, an adding machine. Some of his inventions, however, such as an automated bobbin winder. A number of Leonardos most practical inventions are nowadays displayed as working models at the Museum of Vinci. He made substantial discoveries in anatomy, civil engineering, geology, optics, and hydrodynamics, today, Leonardo is widely considered one of the most diversely talented individuals ever to have lived. Leonardo was born on 15 April 1452 at the hour of the night in the Tuscan hill town of Vinci. He was the son of the wealthy Messer Piero Fruosino di Antonio da Vinci, a Florentine legal notary, and Caterina. Leonardo had no surname in the modern sense – da Vinci simply meaning of Vinci, his birth name was Lionardo di ser Piero da Vinci, meaning Leonardo. The inclusion of the title ser indicated that Leonardos father was a gentleman, little is known about Leonardos early life. He spent his first five years in the hamlet of Anchiano in the home of his mother and his father had married a sixteen-year-old girl named Albiera Amadori, who loved Leonardo but died young in 1465 without children. When Leonardo was sixteen, his father married again to twenty-year-old Francesca Lanfredini, pieros legitimate heirs were born from his third wife Margherita di Guglielmo and his fourth and final wife, Lucrezia Cortigiani. Leonardo received an education in Latin, geometry and mathematics. In later life, Leonardo recorded only two childhood incidents, one, which he regarded as an omen, was when a kite dropped from the sky and hovered over his cradle, its tail feathers brushing his face

16.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

17.
Flagellation of Christ (Piero della Francesca)
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The Flagellation of Christ is a painting by Piero della Francesca in the Galleria Nazionale delle Marche in Urbino, Italy. Called by one writer an enigmatic little painting, the composition is complex and unusual, kenneth Clark placed The Flagellation in his personal list of the best ten paintings, calling it the greatest small painting in the world. The theme of the picture is the Flagellation of Christ by the Romans during his Passion, the biblical event takes place in an open gallery in the middle distance, while three figures in the foreground on the right-hand side apparently pay no attention to the event unfolding behind them. The painting is signed under the seated emperor OPVS PETRI DE BVRGO SCI SEPVLCRI – the work of Piero of Borgo Santo Sepolcro, the portrait of the bearded man at the front is considered unusually intense for Pieros time. Much of the debate surrounding the work concerns the identities or significance of the three men at the front. Depending on the interpretation of the subject of the painting, they may represent contemporary figures or people related to the passion of Christ, or they may even have multiple identities. The latter is suggested with respect to the sitting man on the left, who is in one sense certainly Pontius Pilate. Originally the painting had a frame on which the Latin phrase Convenerunt in Unum, taken from Psalm 2 and this text is cited in Book of Acts 4,26 and related to Pilate, Herod and the Jews. Both advisers were held responsible for Oddantonios death due to their unpopular government, Oddantonios death would be compared, in its innocence, to that of Christ. The painting would then have been commissioned by Federico da Montefeltro, moreover, Oddantonios corpse was buried in an unnamed grave. A painting dedicated to the memory of Duke Oddantonio and to his rehabilitation would thus have been a case of betrayal to the citizens of Urbino, another traditional view considers the picture a dynastic celebration commissioned by Duke Federico da Montefeltro, Oddantonios successor and half-brother. The three men would simply be his predecessors, however, since Duke Guidobaldo was a son of Federico born in 1472, this information has to be erroneous. Instead, the rightmost figure may represent Oddantonios and Federicos father Guidantonio, according to this other old-fashioned view, the figure in the middle would represent an angel, flanked by the Latin and the Greek Churches, whose division created strife in the whole of Christendom. The young man would be Bonconte II da Montefeltro, who died of plague in 1458, in this way, the sufferings of Christ are paired both to those of the Byzantines and of Bonconte. The figure on the left watching would be sultan Murad II, the three men on the right are identified as, from left, Cardinal Bessarion, Thomas Palaiologos and Niccolò III dEste, host of the council of Mantua after its move to his lordship of Ferrara. Piero della Francesca painted the Flagellation some 20 years after the fall of Constantinople, again, the man in the far left would be the Byzantine Emperor. Another explanation of the painting is offered by Marilyn Aronberg Lavin in Piero della Francesca, the interior scene represents Pontius Pilate showing Herod with his back turned, because the scene closely resembles numerous other depictions of the flagellation that Piero would have known. Ottavio is dressed in the garb of an astrologer, even down to his forked beard

18.
The History of the True Cross
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The History of the True Cross or The Legend of the True Cross is a sequence of frescoes painted by Piero della Francesca in the Basilica of San Francesco in Arezzo. It is his largest work, and generally considered one of his finest, and this work demonstrates Piero’s advanced knowledge of perspective and colour, his geometric orderliness and skill in pictorial construction. The main episodes depicted are, Death of Adam, the Queen of Sheba in Adoration of the Wood and The Meeting of Solomon and the Queen of Sheba. According to the legend, the Queen of Sheba worshiped the beams made from the tree, and informed Solomon that the Saviour would hang from that tree and this caused Solomon to hew it down and bury it, until it was found by the Romans. Constantines Dream Emperor Constantine the Great, before the battle of Milvian Bridge, is awakened by an angel who shows him the cross in heaven, with the cross on his shield, he slew the enemy, and later converted to Christianity. Discovery and Proof of the True Cross, helena, Constantines mother, finds the cross in Jerusalem. It was not easy to get information and when the queen had called them and demanded them the place where our Lord Jesus Christ had been crucified, then commanded she to burn them all or cast them into a dry pit for seven days and there torment them with hunger. The Jew is shown in one fresco being pulled from the pit by a rope, whereupon he confessed that Jesus was his lord, the proof of the cross was that it was used to resurrect a dead man. The cross played a role in battles during the war between the Eastern Roman Empire and the Sassanid Empire, the cycle ends with a depiction of the Annunciation, not strictly part of the Legend of the True Cross but probably included by Piero for its universal meaning. Dating of the frescoes is uncertain, but they are believed to date from after 1447 and it would have been finished around 1466. Most of the choir was painted in the early- to mid-1450s, although the design of the frescoes is evidently Pieros, he seems to have delegated parts of the painting to assistants, as was usual. The hand of Giovanni da Piamonte, in particular, can be recognised in some of the frescoes, the Dictionary of Art, Grove Belton, R. & Kersten, B. Vision and Visions in Piero della Francesca’s Legend of the True Cross, translated into English prose by Lucius Hudson Holt. All images in series and interactive model Institute for Advanced Studies web site, the Legend of the True Cross fresco cycle on YouTube, here and here. Legend of the True Cross fresco cycle at art-threads

19.
The Baptism of Christ (Piero della Francesca)
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The Baptism of Christ is a painting by the Italian Renaissance master Piero della Francesca, finished around 1448–50. It is housed in the National Gallery, London, the panel was commissioned presumably some time about 1440 by the Camaldolese Monastery of Sansepolcro in Tuscany, originally part of a triptych. Its dating to Piero della Francescas early career is evidenced by the relationship with the light painting of his master. It portrays Christ being baptised by John, his head surmounted by a representing the Holy Spirit. Christ, Johns hand, the bird and the form an axis which divides the painting in two symmetrical parts. A second division is created by the tree on the left, behind John, a man in white briefs, his feet already in the water, is struggling to get out of his undershirt. The three angels on the left wear different clothes and, in a break from traditional iconography, are not supporting Christs garments and this could be an allusion to the contemporary Council of Florence, whose goal was the unification of the Western and Eastern Churches. The Camaldolese monk and theologian, Saint Ambrose Traversari, who had been Prior General of the Camaladolese congregation, had been a supporter of the union. Such symbolism is also suggested by the presence, behind the neophyte on the right, of figures dressed in an oriental fashion. Piero della Francesca was renowned in his times as an authority on perspective and geometry, his attention to the theme is shown by Johns arm and leg, milan, Mondadori Arte. more reading at Great Works of Western Art Page at artonline. it

20.
Madonna del Parto
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A Madonna del Parto is an iconic depiction of the Virgin Mary shown as pregnant, which was developed in Italy, mainly in Tuscany in the 14th century. Examples include works by Taddeo Gaddi, Bernardo Daddi and Nardo di Cione, the Madonna was portrayed standing, alone, often with a closed book on her belly, an allusion to the Incarnate Word. The works were associated with the devotions of pregnant women, praying for a safe delivery, here the Virgin wears the Girdle of Thomas, a belt of knotted cloth cord that was a relic held in Prato Cathedral, which many depictions wear. The most famous work showing this subject is a painting by the Italian Renaissance master Piero della Francesca. It is housed in the Museo della Madonna del Parto of Monterchi, Tuscany, Piero della Francesca finished it in seven days, using first-rate colors, including a large extent of blu oltremare obtained by lapis lazuli imported from Afghanistan by the Republic of Venice. The fresco was at one time located in Santa Maria di Momentana, the work was attributed to Piero della Francesca only in 1889. Its dating has been the subject of debate, ranging from 1450 to 1475, the 16th century artist and writer Giorgio Vasari wrote that it was completed in 1459, when Piero della Francesca was in Sansepolcro for his mothers death. The fresco also plays an important role in Richard Hayers novel Visus, in Andrei Tarkovskys film Nostalghia, Piero della Francescas Madonna has neither books nor royal attributes as in most predecessors of the image, nor does she wear the girdle. She is portrayed with a hand against her side to support her prominent belly and she is flanked by two angels, who are holding open the curtains of a pavilion decorated with pomegranates, a symbol of Christs Passion. The upper part of the painting is lost, the two angels are specular, as they were executed by the artist using with the same perforated cartoons. The theological symbolism behind the representation is complex, maurizio Calvesi has suggested that the tent represents the Ark of the Covenant. Mary would be thus the new Ark of Alliance in her role as Mother of Christ, for other scholars the tent is a symbol of the Catholic Church and the Madonna would symbolize the tabernacle, as she is portrayed containing Jesus body. Cassidy, Brendan, A Relic, Some Pictures and the Mothers of Florence in the Late Fourteenth Century, Gesta, Vol.30,2, pp. 91-99, The University of Chicago Press on behalf of the International Center of Medieval Art, JSTOR Longhi, Roberto. Gli affreschi di Piero della Francesca ad Arezzo e Monterchi, luogo teologico mariano, Piero della Francesca, Madonna del parto, ein Kunstwerk zwischen Politik und Devotion

21.
The Resurrection (Piero della Francesca)
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The Resurrection is a fresco painting by the Italian Renaissance master Piero della Francesca, painted in the 1460s in the Palazzo della Residenza in the town of Sansepolcro, Tuscany, Italy. The secular and spiritual meanings of the painting were always intimately intertwined, pieros Christ is also present on the towns Coat of Arms. His stern, impassive figure, depicted in an iconic and abstract fixity, rises over four sleeping soldiers and his figure in the communes council hall both protects the judge and purifies the judged according to Marilyn Aronberg Lavin. Andrew Graham-Dixon notes that apart from the wound, Christs body is as perfectly sculpted, but there are touches of intense humanity about him too, the unidealised, almost coarse-featured face, and those three folds of skin that wrinkle at his belly as he raises his left leg. Piero emphasises his twofold nature, as man and God. The guard holding the lance is depicted sitting in an impossible pose. Piero probably left them out not to break the balance of the composition, according to tradition and by comparison with the woodcut illustrating Vasaris Lives of the Painters, the sleeping soldier in brown armor on Christs right is a self-portrait of Piero. The contact between the head and the pole of the banner carried by Christ is supposed to represent his contact with the divinity. The composition is unusual in that it contains two vanishing points, one is in the center of the sarcophagus, because the faces of the guards are seen from below, and the other is in Jesuss face. The top of the forms a boundary between the two points of view, and the steepness of the hills prevents the transition between the two points of view from being too jarring. Sansepolcro was spared damage during World War 2 when British artillery officer Tony Clarke defied orders. Clarke had read Huxleys 1925 essay describing the Resurrection, which states, It stands there before us in entire and actual splendour and it was later ascertained that the Germans had already retreated from the area — the bombardment had not been necessary. The town, along with its famous painting, survived, when the events of the episode eventually became clear, Clarke was lauded as a local hero and to this day a street in Sansepolcro bears his name. Il Cristo Risorto di Piero immagini rare e desuete, amintore Fanfani e Giulio Gambassi, Il Cristo Risorto di Piero della Francesca, Una battaglia per l’arte. Firenze, Grafica European Center of Fine Arts

22.
Polyptych of Perugia
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The Polyptych of Perugia is a complex of paintings by the Italian Renaissance master Piero della Francesca, finished around 1470. It is housed in the Galleria Nazionale dellUmbria, Perugia, Italy, the work was executed for the new Franciscan convent of SantAntonio da Padova in Perugia, most likely in the years following his sojourn in Rome. It portrays the Virgin enthroned with the Child in the part, flanked by several saints, Anthony of Padua and John the Baptist on the left, Francis. In the cusp is the Annunciation, the upper part of the predella shows the saints Clare and Lucy, while in the lower part are miracles stories of the main Franciscan saints. More innovative and typical of the style is the Annunciation, set in a bright cloister

23.
Brera Madonna
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The Brera Madonna is a painting by the Italian Renaissance master Piero della Francesca, executed in 1472-1474. It is housed in the Pinacoteca di Brera of Milan, where it was deposited by Napoleon, the work, of a type known as a sacra conversazione, was commissioned by Federico III da Montefeltro, Duke of Urbino, to celebrate the birth of Federicos son, Guidobaldo. According to other sources, it would celebrate his conquest of several castles in the Maremma, some sources suggest that the work was commissioned to celebrate the birth of Federicos son, Guidobaldo, who was born in 1472. The work represents a sacred conversation, with the Virgin enthroned, on the right low corner, kneeling and wearing his armor, the patron of arts, duke and condottiero Federico da Montefeltro. At the center, hanging by a thread from the shell is an egg, emblem alike of Marys fecundity. The Child wears a necklace of deep red coral beads, a color which alludes to blood, a symbol of life and death, coral was also used for teething, and often worn by babies. The saints at the left of the Madonna are generally identified as John the Baptist, Bernardino of Siena and Jerome, on the right would be Francis, Peter Martyr and Andrew. In the last figure, the Italian historian Ricci has identified a portrait of Luca Pacioli, the presence of John the Baptist would be explained as he was the patron saint of Federicos wife, while St. Jerome was the protector of Humanists. Francis, finally, would be present as the painting was thought for the Franciscan church of San Donato degli Osservanti. The apse ends with a shell semi-dome from which an egg is hanging. The shell was a symbol of the new Venus, Mary, according to another hypothesis, the egg would be a pearl, and the shell would refer to the miracle of the virginal conception. The egg is considered a symbol of the Creation and, in particular, to Guidobaldos birth. According to Italian art historian Carlo Ludovico Ragghianti, the work has been cut down on both sides, as shown by the portions of entablatures barely visible in the upper corners

24.
Madonna di Senigallia
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The Madonna di Senigallia is a painting by the Italian Renaissance master Piero della Francesca, finished around 1474. It is housed in the Galleria Nazionale delle Marche, in the Ducal Palace of Urbino, from its small scale the painting was intended for private devotion. It was noticed for the first time in 1822 in the church of the Observant Franciscan convent of Santa Maria delle Grazie just outside Senigallia, Senigallia was wrested from Sigismondo Malatesta by Federico Montefeltro, both men were patrons of Piero. The commission was likely from or on behalf of Giovanni Della Rovere, betrothed in 1474 to Giovanna Montefeltro, at which time Federico made Giovanni, following its rediscovery the painting was taken to the Ducal Palace, Urbino. The light, which enters from the window on the left, is a symbol of the Virgins conception. The staring, thoughtful immobility of all the characters would be also an allusion to the latter

25.
Algorithm
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In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391, English adopted the French term, but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu and it begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as, Algorism is the art by which at present we use those Indian figures, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. An informal definition could be a set of rules that precisely defines a sequence of operations, which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. An enumerably infinite set is one whose elements can be put into one-to-one correspondence with the integers, the concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case

26.
Catenary
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In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The curve has a U-like shape, superficially similar in appearance to a parabola, the curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid, chainette, or, particularly in the materials sciences, mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the curve, the catenoid, is a minimal surface. The mathematical properties of the curve were first studied by Robert Hooke in the 1670s. Catenaries and related curves are used in architecture and engineering, in the design of bridges and arches, in the offshore oil and gas industry, catenary refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. The word catenary is derived from the Latin word catēna, which means chain and it appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are and it is often said that Galileo thought the curve of a hanging chain was parabolic. That the curve followed by a chain is not a parabola was proven by Joachim Jungius, some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon. David Gregory wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the differential equation. Euler proved in 1744 that the catenary is the curve which, nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796. Catenary arches are used in the construction of kilns. To create the desired curve, the shape of a chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material. The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an catenary and it is close to a more general curve called a flattened catenary, with equation y = A cosh, which is a catenary if AB =1. While a catenary is the shape for a freestanding arch of constant thickness. According to the U. S. National Historic Landmark nomination for the arch and its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form. In free-hanging chains, the force exerted is uniform with respect to length of the chain, the same is true of a simple suspension bridge or catenary bridge, where the roadway follows the cable. A stressed ribbon bridge is a sophisticated structure with the same catenary shape

27.
Fractal
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A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry, if the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge, Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set, Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals

28.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio

29.
Hyperboloid structure
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Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov, the worlds first hyperboloid tower is located in Polibino, Dankovsky District, Lipetsk Oblast, Russia. Hyperbolic structures have a negative Gaussian curvature, meaning they curve inward rather than outward or being straight. As doubly ruled surfaces, they can be made with a lattice of beams, hence are easier to build than curved surfaces that do not have a ruling. With cooling towers, a structure is preferred. At the bottom, the widening of the tower provides an area for installation of fill to promote thin film evaporative cooling of the circulated water. In the 1880s, Shukhov began to work on the problem of the design of systems to use a minimum of materials, time. His calculations were most likely derived from mathematician Pafnuty Chebyshevs work on the theory of best approximations of functions, Shukhovs mathematical explorations of efficient roof structures led to his invention of a new system that was innovative both structurally and spatially. The steel gridshells of the pavilions of the 1896 All-Russian Industrial. Two pavilions of this type were built for the Nizhni Novgorod exposition, one oval in plan, the roofs of these pavilions were doubly curved gridshells formed entirely of a lattice of straight angle-iron and flat iron bars. Shukhov himself called them azhurnaia bashnia, the patent of this system, for which Shukhov applied in 1895, was awarded in 1899. Shukhov also turned his attention to the development of an efficient and his solution was inspired by observing the action of a woven basket holding up a heavy weight. Again, it took the form of a curved surface constructed of a light network of straight iron bars. Over the next twenty years, he designed and built close to two hundred of these towers, no two alike, most with heights in the range of 12m to 68m. At least as early as 1911, Shukhov began experimenting with the concept of forming a tower out of stacked sections of hyperboloids. Stacking the sections permitted the form of the tower to taper more at the top, increasing the number of sections would increase the tapering of the overall form, to the point that it began to resemble a cone. By 1918 Shukhov had developed this concept into the design of a nine-section stacked hyperboloid radio transmission tower for Moscow, Shukhov designed a 350m tower, which would have surpassed the Eiffel Tower in height by 50m, while using less than a quarter of the amount of material. In July 1919, Lenin decreed that the tower should be built to a height of 150m, construction of the smaller tower with six stacked hyperboloids began within a few months, and Shukhov Tower was completed by March 1922

30.
Minimal surface
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In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature, the term minimal surface is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a solution, forming a soap film. However the term is used for general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas, Minimal surfaces can be defined in several equivalent ways in R3. Local least area definition, A surface M ⊂ R3 is minimal if, note that this property is local, there might exist other surfaces that minimize area better with the same global boundary. Variational definition, A surface M ⊂ R3 is minimal if and this definition makes minimal surfaces a 2-dimensional analogue to geodesics. Note that spherical bubbles are not minimal surfaces as per this definition, while they minimize total area subject to a constraint on internal volume, mean curvature definition, A surface M ⊂ R3 is minimal if and only if its mean curvature vanishes identically. A direct implication of this definition is that point on the surface is a saddle point with equal. This definition ties minimal surfaces to harmonic functions and potential theory, harmonic definition, If X =, M → R3 is an isometric immersion of a Riemann surface into 3-space, then X is said to be minimal whenever xi is a harmonic function on M for each i. A direct implication of this definition and the principle for harmonic functions is that there are no compact complete minimal surfaces in R3. This definition uses that the curvature is half of the trace of the shape operator. If the projected Gauss map obeys the Cauchy–Riemann equations then either the trace vanishes or every point of M is umbilic, mean curvature flow definition, Minimal surfaces are the critical points for the mean curvature flow. The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3, Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z of least area stretched across a given closed contour. He derived the Euler–Lagrange equation for the solution d d x + d d y =0 He did not succeed in finding any solution beyond the plane. By expanding Lagranges equation to z y y −2 z x z y z x y + z x x =0 Gaspard Monge, while these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface, progress had been fairly slow until the middle of the century, when the Björling problem was solved using complex methods. The first golden age of minimal surfaces began, schwarz found the solution of the Plateau problem for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 using complex methods

31.
Paraboloid
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In geometry, a paraboloid is a quadric surface that has one axis of symmetry and no center of symmetry. The term paraboloid is derived from parabola, which refers to a section that has the same property of symmetry. In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation z = x 2 a 2 + y 2 b 2. Where a and b are constants that dictate the level of curvature in the xz, in this position, the elliptic paraboloid opens upward. A hyperbolic paraboloid is a ruled surface shaped like a saddle. In a suitable system, a hyperbolic paraboloid can be represented by the equation z = y 2 b 2 − x 2 a 2. In this position, the hyperbolic paraboloid opens down along the x-axis, obviously both the paraboloids contain a lot of parabolas. But there are differences, too, an elliptic paraboloid contains ellipses. With a = b an elliptic paraboloid is a paraboloid of revolution and this shape is also called a circular paraboloid. This also works the way around, a parallel beam of light incident on the paraboloid parallel to its axis is concentrated at the focal point. This applies also for other waves, hence parabolic antennas, for a geometrical proof, click here. The hyperbolic paraboloid is a ruled surface, it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, hence the hyperbolic paraboloid is a conoid. This property makes easy to realize a hyperbolic paraboloid with concrete, the widely sold fried snack food Pringles potato crisps resemble a truncated hyperbolic paraboloid. The distinctive shape of these allows them to be stacked in sturdy tubular containers. Examples in architecture St. a point, if the plane is a tangent plane, remark, an elliptic paraboloid is projectively equivalent to a sphere. Remarks, A hyperbolic paraboloid is a surface, but not developable. The Gauss curvature at any point is negative, hence it is a saddle surface

32.
Camera lucida
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A camera lucida is an optical device used as a drawing aid by artists. The camera lucida performs an optical superimposition of the subject being viewed upon the surface upon which the artist is drawing, the artist sees both scene and drawing surface simultaneously, as in a photographic double exposure. This allows the artist to duplicate key points of the scene on the drawing surface, the camera lucida was patented in 1807 by William Hyde Wollaston. The basic optics were described 200 years earlier by Johannes Kepler in his Dioptrice, by the 19th century, Kepler’s description had fallen into oblivion, so Wollaston’s claim was never challenged. The term camera lucida is Wollastons, while on honeymoon in Italy in 1833, the photographic pioneer William Fox Talbot used a camera lucida as a sketching aid. He later wrote that it was a disappointment with his efforts which encouraged him to seek a means to cause these natural images to imprint themselves durably. In 2001, artist David Hockneys book Secret Knowledge, Rediscovering the Lost Techniques of the Old Masters was met with controversy and their evidence is based largely on the characteristics of the paintings by great artists of later centuries, such as Ingres, Van Eyck, and Caravaggio. The camera lucida is still available today through art-supply channels but is not well known or widely used and it has enjoyed a resurgence recently through a number of Kickstarter campaigns. The name camera lucida is obviously intended to recall the much older drawing aid, there is no optical similarity between the devices. The camera lucida is a light, portable device that not require special lighting conditions. No image is projected by the camera lucida, in the simplest form of camera lucida, the artist looks down at the drawing surface through a half-silvered mirror tilted at 45 degrees. This superimposes a direct view of the surface beneath. This design produces an image which is right-left reversed when turned the right way up. Also, light is lost in the imperfect reflection, Wollastons design used a prism with four optical faces to produce two successive reflections, thus producing an image that is not inverted or reversed. Angles ABC and ADC are 67. 5° and BCD is 135°, hence, the reflections occur through total internal reflection, so very little light is lost. It is not possible to see straight through the prism, so it is necessary to look at the edge to see the paper. The instrument often includes a weak negative lens, creating an image of the scene at about the same distance as the drawing surface. If white paper is used with the camera lucida, the superimposition of the paper with the scene tends to wash out the scene, when working with a camera lucida, it is often beneficial to use black paper and to draw with a white pencil

33.
Camera obscura
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The surroundings of the projected image have to be relatively dark for the image to be clear, so many historical camera obscura experiments were performed in dark rooms. The term camera obscura also refers to constructions or devices that use of the principle within a box. Camerae obscurae with a lens in the opening have been used since the second half of the 16th century, before the term camera obscura was first used in 1604, many other expressions were used including cubiculum obscurum, cubiculum tenebricosum, conclave obscurum and locus obscurus. Rays of light travel in straight lines and change when they are reflected and partly absorbed by an object, retaining information about the color, lit objects reflect rays of light in all directions. The human eye itself works much like a camera obscura with an opening, a biconvex lens, a camera obscura device consists of a box, tent or room with a small hole in one side. Light from a scene passes through the hole and strikes a surface inside, where the scene is reproduced, inverted and reversed. The image can be projected onto paper, and can then be traced to produce an accurate representation. In order to produce a reasonably clear projected image, the aperture has to be about 1/100th the distance to the screen, many camerae obscurae use a lens rather than a pinhole because it allows a larger aperture, giving a usable brightness while maintaining focus. As the pinhole is made smaller, the image gets sharper, with too small a pinhole, however, the sharpness worsens, due to diffraction. Using mirrors, as in an 18th-century overhead version, it is possible to project a right-side-up image, another more portable type is a box with an angled mirror projecting onto tracing paper placed on the glass top, the image being upright as viewed from the back. There are theories that occurrences of camera obscura effects inspired paleolithic cave paintings and it is also suggested that camera obscura projections could have played a role in Neolithic structures. Perforated gnomons projecting an image of the sun were described in the Chinese Zhoubi Suanjing writings. The location of the circle can be measured to tell the time of day. In Arab and European cultures its invention was later attributed to Egyptian astronomer. Some ancient sightings of gods and spirits, especially in worship, are thought to possibly have been conjured up by means of camera obscura projections. In these writings it is explained how the image in a collecting-point or treasure house is inverted by an intersecting point that collected the light. Light coming from the foot of a person would partly be hidden below. Rays from the head would partly be hidden above and partly form the part of the image

34.
Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry

35.
Proportion (architecture)
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Proportion is a central principle of architectural theory and an important connection between mathematics and art. It is the effect of the relationships of the various objects and spaces that make up a structure to one another. These relationships are governed by multiples of a standard unit of length known as a module. Proportion in architecture was discussed by Vitruvius, Alberti, Andrea Palladio, Architecture in Roman antiquity was rarely documented except in the writings of Vitruvius treatise De Architectura. Vitruvius served as an engineer under Julius Caesar during the first Gallic Wars, the treatise was dedicated to Emperor Augustus. Moreover, Vitruvius identified the Six Principles of Design as order, arrangement, proportion, symmetry, propriety, among the six principles, proportion interrelates and supports all the other factors in geometrical forms and arithmetical ratios. The word symmetria, usually translated to symmetry in modern renderings, in ancient times meant something more closely related to mathematical harmony, Vitruvius tried to describe his theory in the makeup of the human body, which he referred to as the perfect or golden ratio. The principles of measurement units digit, foot, and cubit also came from the dimensions of a Vitruvian Man, more specifically, Vitruvius used the total height of 6 feet of a person, and each part of the body takes up a different ratio. For example, the face is about 1/10 of the height. Vitruvius used these ratios to prove that the composition of classical orders mimicked the human body, in classical architecture, the module was established as the radius of the lower shaft of a classical column, with proportions expressed as a fraction or multiple of that module. In his Le Modulor, Le Corbusier presented a system of proportions which took the golden ratio, history of architecture Mathematics and architecture Mathematics and art P. H. Scholfield. The Theory of Proportion in Architecture

36.
Body proportions
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While there is significant variation in anatomical proportions between people, there are many references to body proportions that are intended to be canonical, either in art, measurement, or medicine. In measurement, body proportions are used to relate two or more measurements based on the body. A cubit, for instance, is supposed to be six palms, a span is taken to be 9 inches and was previously considered as half a cubit. While convenient, these ratios may not reflect the variation of the individuals using them. Similarly, in art, body proportions are the study of relation of human or animal parts to each other. These ratios are used in depictions of the figure. It is important in drawing to draw the human figure in proportion. In modern figure drawing, the unit of measurement is the head. This unit of measurement is reasonably standard, and has long used by artists to establish the proportions of the human figure. Ancient Egyptian art used a canon of proportion based on the fist, measured across the knuckles, with 18 fists from the ground to the hairline on the forehead. This was already established by the Narmer Palette from about the 31st century BC, the proportions used in figure drawing are, An average person, is generally 7-and-a-half heads tall. An ideal figure, used when aiming for an impression of nobility or grace, is drawn at 8 heads tall, a heroic figure, used in the heroic for the depiction of gods and superheroes, is eight-and-a-half heads tall. Most of the length comes from a bigger chest and longer legs. A study using Polish participants by Sorokowski found 5% longer legs than a used as a reference was considered most attractive. The study concluded this preference might stem from the influence of leggy runway models, the Sorokowski study was criticized for using a picture of the same person with digitally altered leg lengths which Marco Bertamini felt were unrealistic. Another study using British and American participants, found mid-ranging leg-to-body ratios to be most ideal, the Swami et al. study was criticized for using a picture of the same person with digitally altered leg lengths which Marco Bertamini felt were unrealistic. Bertamini also criticized the Swami study for only changing the leg length while keeping the arm length constant, bertaminis own study which used stick figures mirrored Swamis study, however, by finding a preference for leggier women. Another common measurement of related to leg-to-body ratio is sitting-height ratio, sitting height ratio is the ratio of the head plus spine length to total height which is highly correlated to leg-to-body ratio

37.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left

38.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules

39.
Wallpaper group
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A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, There are 17 possible distinct groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the space groups. Wallpaper groups categorize patterns by their symmetries, subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group. Consider the following examples, Examples A and B have the same group, it is called p4m in the IUC notation. Example C has a different wallpaper group, called p4g or 4*2, a complete list of all seventeen possible wallpaper groups can be found below. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance, think of shifting a set of vertical stripes horizontally by one stripe. Strictly speaking, a true symmetry only exists in patterns that repeat exactly, a set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end disappears and a new stripe is added at the other end. In practice, however, classification is applied to finite patterns, sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry, the types of transformations that are relevant here are called Euclidean plane isometries. This type of symmetry is called a translation, Examples A and C are similar, except that the smallest possible shifts are in diagonal directions. If we turn example B clockwise by 90°, around the centre of one of the squares, Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can also flip example B across a horizontal axis that runs across the middle of the image, example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A. However, example C is different and it only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a line, we do not get the same pattern back. This is part of the reason that the group of A and B is different from the wallpaper group of C. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891, the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done

40.
Algorithmic art
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Algorithmic art, also known as algorithm art, is art, mostly visual art, of which the design is generated by an algorithm. Algorithmic artists are sometimes called algorists, algorithmic art, also known as computer-generated art, is a subset of generative art and is related to systems art. Fractal art is an example of algorithmic art, the final output is typically displayed on a computer monitor, printed with a raster-type printer, or drawn using a plotter. Variability can be introduced by using pseudo-random numbers, there is no consensus as to whether the product of an algorithm that operates on an existing image can still be considered computer-generated art, as opposed to computer-assisted art. Some of the earliest known examples of computer-generated algorithmic art were created by Georg Nees, Frieder Nake, A. Michael Noll, Manfred Mohr and these artworks were executed by a plotter controlled by a computer, and were therefore computer-generated art but not digital art. The act of creation lay in writing the program, which specified the sequence of actions to be performed by the plotter and her early work with copier and telematic art focused on the differences between the human hand and the algorithm. Aside from the work of Roman Verostko and his fellow algorists. These are important here because they use a different means of execution, whereas the earliest algorithmic art was drawn by a plotter, fractal art simply creates an image in computer memory, it is therefore digital art. The native form of an artwork is an image stored on a computer –this is also true of very nearly all equation art. However, in a stricter sense fractal art is not considered algorithmic art, from one point of view, for a work of art to be considered algorithmic art, its creation must include a process based on an algorithm devised by the artist. This input may be mathematical, computational, or generative in nature, inasmuch as algorithms tend to be deterministic, meaning that their repeated execution would always result in the production of identical artworks, some external factor is usually introduced. This can either be a number generator of some sort. By this definition, fractals made by a program are not art. However, defined differently, algorithmic art can be seen to include fractal art, the artist Kerry Mitchell stated in his 1999 Fractal Art Manifesto, Fractal Art is not. Computer Art, in the sense that the computer does all the work. The work is executed on a computer, but only at the direction of the artist, turn a computer on and leave it alone for an hour. When you come back, no art will have been generated, algorist is a term used for digital artists who create algorithmic art. Algorists formally began correspondence and establishing their identity as artists following a panel titled Art, the co-founders were Roman Verostko and Jean-Pierre Hébert. Fractal art consists of varieties of computer-generated fractals with colouring chosen to give an attractive effect, especially in the western world, it is not drawn or painted by hand

41.
Anamorphosis
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Anamorphosis is a distorted projection or perspective requiring the viewer to use special devices or occupy a specific vantage point to reconstitute the image. The word anamorphosis is derived from the Greek prefix ana‑, meaning back or again, an optical anamorphism is the visualization of a mathematical operation called an affine transformation. There are two types of anamorphosis, perspective and mirror. More-complex anamorphoses can be devised using distorted lenses, mirrors, or other optical transformations, examples of perspectival anamorphosis date to the early Renaissance. Examples of mirror anamorphosis were first seen in the late Renaissance, the deformed image is painted on a plane surface surrounding the mirror. By looking into the mirror, a viewer can see the image undeformed, leonardos Eye is the earliest known definitive example of perspective anamorphosis in modern times. The prehistoric cave paintings at Lascaux may also use this technique, Hans Holbein the Younger is well known for incorporating an oblique anamorphic transformation into his painting The Ambassadors. In this artwork, a distorted shape lies diagonally across the bottom of the frame, viewing this from an acute angle transforms it into the plastic image of a human skull, a symbolic memento mori. During the seventeenth century, Baroque trompe loeil murals often used anamorphism to combine actual architectural elements with illusory painted elements, when a visitor views the art work from a specific location, the architecture blends with the decorative painting. The dome and vault of the Church of St. Ignazio in Rome, painted by Andrea Pozzo, due to neighboring monks complaining about blocked light, Pozzo was commissioned to paint the ceiling to look like the inside of a dome, instead of building a real dome. As the ceiling is flat, there is one spot where the illusion is perfect. Mirror anamorphosis emerged early in the 17th century in Italy and China and it remains uncertain whether Jesuit missionaries imported or exported the technique. Anamorphosis could be used to conceal images for privacy or personal safety, a secret portrait of Bonnie Prince Charlie is painted in a distorted manner on a tray and can only be recognized when a polished cylinder is placed in the correct position. To possess such an image would have seen as treason in the aftermath of the 1746 Battle of Culloden. In the eighteenth and nineteenth centuries, anamorphic images had come to be used more as childrens games than fine art, in the twentieth century, some artists wanted to renew the technique of anamorphosis. Marcel Duchamp was interested in anamorphosis, and some of his installations are visual paraphrases of anamorphoses, Jan Dibbets conceptual works, the so-called perspective corrections are examples of linear anamorphoses. In the late century, mirror anamorphosis was revived as childrens toys. Beginning in 1967, Dutch artist Jan Dibbets based a series of photographic work titled Perspective Corrections on the distortion of reality through perspective anamorphosis

42.
Computer art
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Computer art is any art in which computers play a role in 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 traditional disciplines are now integrating digital technologies and, as a result, for instance, an artist may combine traditional painting with algorithm art and other digital techniques. As a result, defining computer art by its end product can thus be difficult, Computer art is by its nature evolutionary since changes in technology and software directly affect what is possible. Notable artists in this vein include Manfred Mohr, Ronald Davis, Harold Cohen, Joseph Nechvatal, George Grie, Olga Kisseleva, John Lansdown, and Jean-Pierre Hébert. On the title page of the magazine Computers and Automation, January 1963, Edmund Berkeley published a picture by Efraim Arazi from 1962 and this picture inspired him to initiate the first Computer Art Contest in 1963. The annual contest was a key point in the development of art up to the year 1973. The precursor of computer art dates back to 1956-1958, with the generation of what is probably the first image of a human being on a computer screen, a pin-up girl at a SAGE air defense installation. Desmond Paul Henry invented the Henry Drawing Machine in 1960, his work was shown at the Reid Gallery in London in 1962, many artists tentatively began to explore the emerging computing technology for use as a creative tool. In the summer of 1962, A. Michael Noll programmed a computer at Bell Telephone Laboratories in Murray Hill. His later computer-generated patterns simulated paintings by Piet Mondrian and Bridget Riley, Noll also used the patterns to investigate aesthetic preferences in the mid-1960s. The Stuttgart exhibit featured work by Georg Nees, the New York exhibit featured works by Bela Julesz, a third exhibition was put up in November 1965 at Galerie Wendelin Niedlich in Stuttgart, Germany, showing works by Frieder Nake and Georg Nees. Analogue computer art by Maughan Mason along with computer art by Noll were exhibited at the AFIPS Fall Joint Computer Conference in Las Vegas toward the end of 1965. In 1968, the Institute of Contemporary Arts in London hosted one of the most influential exhibitions of computer art called Cybernetic Serendipity. The exhibition included many of whom often regarded as the first digital artists, Nam June Paik, Frieder Nake, Leslie Mezei, Georg Nees, A. Michael Noll, John Whitney, one year later, the Computer Arts Society was founded, also in London. At the time of the opening of Cybernetic Serendipity, in August 1968 and it took up the European artists movement of New Tendencies that had led to three exhibitions in Zagreb of concrete, kinetic, and constructive art as well as op art and conceptual art. New Tendencies changed its name to Tendencies and continued with more symposia, exhibitions, a competition, katherine Nash and Richard Williams published Computer Program for Artists, ART1 in 1970. Xerox Corporation’s Palo Alto Research Center designed the first Graphical User Interface in the 1970s, the first Macintosh computer is released in 1984, since then the GUI became popular

43.
Fourth dimension in art
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New possibilities opened up by the concept of four-dimensional space helped inspire many modern artists in the first half of the twentieth century. Early Cubists, Surrealists, Futurists, and abstract artists took ideas from higher-dimensional mathematics, french mathematician Maurice Princet was known as le mathématicien du cubisme. Picassos Portrait of Daniel-Henry Kahnweiler in 1910 was an important work for the artist, early cubist Max Weber wrote an article entitled In The Fourth Dimension from a Plastic Point of View, for Alfred Stieglitzs July 1910 issue of Camera Work. Another influence on the School of Paris was that of Jean Metzinger and Albert Gleizes, in 1936 in Paris, Charles Tamkó Sirató published his Manifeste Dimensioniste, which described how the Dimensionist tendency has led to, Literature leaving the line and entering the plane. Painting leaving the plane and entering space, sculpture stepping out of closed, immobile forms. …The artistic conquest of space, which to date has been completely art-free. The manifesto was signed by prominent modern artists worldwide. In 1953, the surrealist Salvador Dalí proclaimed his intention to paint an explosive, nuclear and he said that, This picture will be the great metaphysical work of my summer. Completed the next year, Crucifixion depicts Jesus Christ upon the net of a hypercube, the unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares. The Metropolitan Museum of Art describes the painting as a new interpretation of an oft-depicted subject, christs spiritual triumph over corporeal harm. Some of Piet Mondrians abstractions and his practice of Neoplasticism are said to be rooted in his view of a utopian universe, the fourth dimension has been the subject of numerous fictional stories. De Stijl Five-dimensional space Four-dimensional space Duration Philosophy of space and time Octacube Clair, spirits, Art, and the Fourth Dimension. Dalí, Salvador, Gómez de la Serna, Ramón, Maurice Princet, Le Mathématicien du Cubisme. Overview of The Fourth Dimension And Non-Euclidean Geometry In Modern Art, traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n dimensions. Art in the Fourth Dimension, Giving Form to Form – The Abstract Paintings of Piet Mondrian, spaces of Utopia, An Electronic Journal, 23–35. Einstein, Picasso, space, time, and beauty that causes havoc, making Music Modern, New York in the 1920s. Shadows of Reality, The Fourth Dimension in Relativity, Cubism, in The Fourth Dimension from a Plastic Point of View. The Fourth Dimension And Non-Euclidean Geometry In Modern Art, duchamp in Context, Science and Technology in the Large Glass and Related Works

44.
Fractal art
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Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid-1980s onwards and it is a genre of computer art and digital art which are part of new media art. The Julia set and Mandelbrot sets can be considered as icons of fractal art, Fractal art is rarely drawn or painted by hand. In some cases, other programs are used to further modify the images produced. Non-fractal imagery may also be integrated into the artwork and it was assumed that fractal art could not have developed without computers because of the calculative capabilities they provide. Fractals are generated by applying iterative methods to solving non-linear equations or polynomial equations, Fractals are any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size. There are many different kinds of images and can be subdivided into several groups. Fractals derived from standard geometry by using iterative transformations on a common figure like a straight line. The first fractal figures invented near the end of the 19th, IFS Strange attractors Fractal flame L-system fractals Fractals created by the iteration of complex polynomials, perhaps the most famous fractals. Newton fractals, including Nova fractals Quaternionic and hypernionic fractals Fractal terrains generated by random fractal processes Mandelbulbs are a kind of three dimensional fractal, Fractal Expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such as self-similarity for example. Perhaps the best example of fractal expressionism is found in Jackson Pollocks dripped patterns and they have been analysed and found to contain a fractal dimension which has been attributed to his technique. Fractals of all kinds have used as the basis for digital art. High resolution color graphics became available at scientific research labs in the mid-1980s. Scientific forms of art, including art, have developed separately from mainstream culture. Many fractal images are admired because of their perceived harmony and this is typically achieved by the patterns which emerge from the balance of order and chaos. Similar qualities have been described in Chinese painting and miniature trees and rockeries, Fractal rendering programs used to make fractal art include Ultra Fractal, Apophysis, Bryce and Sterling. Fractint was the first widely used fractal generating program, the first fractal image that was intended to be a work of art was probably the famous one on the cover of Scientific American, August 1985. This image showed a landscape formed from the function on the domain outside the Mandelbrot set

45.
Islamic geometric patterns
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Islamic decoration, which tends to avoid using figurative images, makes frequent use of geometric patterns which have developed over the centuries. These may constitute the entire decoration, may form a framework for floral or calligraphic embellishments, interest in Islamic geometric patterns is increasing in the West, both among craftsmen and artists including M. C. Islamic art mostly avoids figurative images to avoid becoming objects of worship, Islamic geometric patterns derived from simpler designs used in earlier cultures, Greek, Roman, and Sasanian. They are one of three forms of Islamic decoration, the others being the based on curving and branching plant forms. Geometric designs and arabesques are forms of Islamic interlace patterns, david Wade states that Much of the art of Islam, whether in architecture, ceramics, textiles or books, is the art of decoration – which is to say, of transformation. Wade argues that the aim is to transfigure, turning mosques into lightness and pattern and she argues that beauty, whether in poetry or in the visual arts, was enjoyed for its own sake, without commitment to religious or moral criteria. Many Islamic designs are built on squares and circles, typically repeated, overlapped and interlaced to form intricate, a recurring motif is the 8-pointed star, often seen in Islamic tilework, it is made of two squares, one rotated 45 degrees with respect to the other. The fourth basic shape is the polygon, including pentagons and octagons, all of these can be combined and reworked to form complicated patterns with a variety of symmetries including reflections and rotations. Such patterns can be seen as mathematical tessellations, which can extend indefinitely and they are constructed on grids that require only ruler and compass to draw. Artist and educator Roman Verostko argues that such constructions are in effect algorithms, the circle symbolizes unity and diversity in nature, and many Islamic patterns are drawn starting with a circle. On this basis is constructed a star surrounded by six smaller irregular hexagons to form a tessellating star pattern. This forms the basic design which is outlined in white on the wall of the mosque and that design, however, is overlaid with an intersecting tracery in blue around tiles of other colours, forming an elaborate pattern that partially conceals the original and underlying design. A similar design forms the logo of the Mohammed Ali Research Center and he observed that many different combinations of polygons can be used as long as the residual spaces between the polygons are reasonably symmetrical. For example, a grid of octagons in contact has squares as the residual spaces, every octagon is the basis for an 8-point star, as seen at Akbars tomb, Sikandra. Hankin considered the skill of the Arabian artists in discovering suitable combinations of polygons and he further records that if a star occurs in a corner, exactly one quarter of it should be shown, if along an edge, exactly one half of it. The mathematical properties of the tile and stucco patterns of the Alhambra palace in Granada. Some authors have claimed on dubious grounds to have found most or all of the 17 wallpaper groups there, the earliest geometrical forms in Islamic art were occasional isolated geometric shapes such as 8-pointed stars and lozenges containing squares. These date from 836 in the Great Mosque of Kairouan, Tunisia, the next development, marking the middle stage of Islamic geometric pattern usage, was of 6- and 8-point stars, which appear in 879 at the Ibn Tulun Mosque, Cairo, and then became widespread