The art of math

cyeager

The Art of Math

Compiled by Carol Yeager with original works by

Neil Currieand

Rostom Kouyoumdjian

• Most images and copy are taken from Wikipedia and resource listings are contained within the Wikipedia descriptions.

• http://en.wikipedia.org/wiki/Mathematics_and_art

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Mathematics and art

From Wikipedia, the free encyclopedia

Mathematics and art have a long historical relationship. The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid,[1] the Parthenon, the Colosseum. There are many examples of artists who have been inspired by mathematics and studied mathematics as a means of complementing their works. The Greek sculptor Polykleitos prescribed a series of mathematical proportions for carving the ideal male nude. Renaissance painters turned to mathematics and many, including Piero della Francesca, became accomplished mathematicians themselves.

Pyramid of Kufu

If we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619, less than 1% from the golden ratio. This would also indicate that half the cross-section of the Khufu’s pyramid is in fact a Kepler’s triangle. Debate has broken out between prominent pyramidologists, including Temple Bell, Michael Rice, and John Taylor, over whether the presence of the golden ratio in the pyramids is due to design or chance.

Pyramidologists, Martin Gardner, Herbert Turnbull, and David Burton contend that:Possible base:hypotenuse(b:a) ratios for the Pyramid of Khufu: 1:φ (Kepler’s Triangle), 3:5 (3-4-5 Triangle), and 1:4/πHerodotus related in one passage that the Egyptian priests told him that the dimensions of the Great Pyramid were so chosen that the area of a square whose side was the height of the great pyramid equaled the area of the triangle.[7]

Great Mosque of Kairouan

The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organisation. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period.[13] ” Because of urban constraints, the mosque's floor plan is not a perfect rectangle. Even so, for example, the division of the courtyard and prayer hall is almost a perfect golden ratio.

Polykleitos

Polykleitos gives us a mathematical approach towards sculpturing the human body. The influence of the Canon of Polykleitos is immense both in

Classical Greek, Roman, and Renaissance sculpture, with many sculptors after him following Polykleitos’ prescription. While none of Polykleitos’ original

works survive, Roman copies of his works demonstrate and embody his ideal of physical perfection and mathematical precision.

Renaissance

The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of

mathematics as a relevant subject needed to understand nature and the arts. Two major reasons drove Renaissance artists towards the pursuit of mathematics. First, painters

needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas. Second, philosophers and artists alike were convinced that mathematics was the

true essence of the physical world and that the entire universe, including the arts, could be explained in

geometric terms.[17] In light of these factors, Renaissance artists became some of the best applied mathematicians of

their times.

Piero della FrancescaPiero della Francesca (c.1415-1492), an early Renaissance artist from Italy, exemplified this new shift in Renaissance thinking. Though chiefly appreciated for his art, he was an expert mathematician and geometer and authored many books on solid geometry and the emerging field of perspective

DaVinciWoodcut from De Divina Proportione illustrating the golden ratio as applied to the human face.

Leonardo da Vinci (1452–1519) was an Italian scientist, mathematician, engineer, inventor, anatomist, painter, sculptor, and architect. Leonardo has often been described as the archetype of the Renaissance man. [37][38]

Renowned primarily as a painter, Leonardo incorporated many mathematical concepts into his artwork despite never having received any formal mathematical training. It was not until the 1490s that he trained under Luca Pacioli and prepared a series of drawings for De Divina Proportione. Leonardo studied Pacioli's Summa, from which he copied tables of proportions and multiplication tables.[39]

Notably in Mona Lisa and The Last Supper, Leonardo’s work incorporated the concept of linear perspective. By making all of the lines in the painting converge on a single, invisible point on the horizon, a flat painting can appear to have depth. In creating the vanishing point, Leonardo creates the illusion that the painting is an extension of the room itself. [40]

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M. C. EscherCircle Limit III by M.C. Escher (1959)A renowned artist born in 1898 and died in 1972, M.C. Escher was known for his mathematically inspired work.[49] Escher’s interest in tessellations, polyhedrons, shaping of space, and self-reference manifested itself in his work throughout his career.[50] In the Alhambra Sketch, Escher showed that art can be created with polygons or regular shapes such as triangles, squares, and hexagons.

Salvador Dalí

Salvador Dalí (1904–1989) incorporated mathematical themes in several of his later works. His 1954 painting Crucifixion (Corpus Hypercubus) depicts a crucified figure upon the net of a hypercube.

FractalsMain article: Fractal artThe processing power of modern computers allows mathematicians and non-mathematicians to visualise complex mathematical objects such as the Mandelbrot set. In the modern industry of computer animation, fractals play a key role in modelling mountains, fire, trees and other natural objects.

Platonic solids in art

The Platonic solids and other polyhedra are a recurring theme in Western art. Examples include:A marble mosaic featuring the small stellated dodecahedron, attributed to Paolo Uccello, in the floor of the San Marco Basilica in Venice.[18]

Leonardo da Vinci's outstanding diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's book The Divine Proportion.[18]

A glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495.[18]

A truncated polyhedron (and various other mathematical objects) which feature in Albrecht Dürer's engraving Melancholia I.[18]

Salvador Dalí's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.

Neil Currie, of Manchester, UK discusses

The Golden Ratio in math and artistic pursuits

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Immediately following isa film demonstrating one point perspective drawing

in an original composition by the Armenian artistliving in Beirut …

Rostom Kouyoumdjian

Thank you for spending some time with us in an overview of the Art of

Mathematics.

My special thanks to Neil Currie and Rostom Kouyoumdjian who accepted the challenge to show the use of maths in their artistic pursuits

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