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