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The art of math

A VizMath presentation featuring videos by Neil Currie on the golden ratio and by Rostom Kouyoumdjian on drawing with one point perspective. Illustrations of the use of math in art through the ages.

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The art of math

  1. 1. cyeager
  2. 2. The Art of Math Compiled by Carol Yeager with original works by Neil Currie and RostomKouyoumdjian
  3. 3. • Most images and copy are taken from Wikipedia and resource listings are contained within the Wikipedia descriptions.•• Text is available under the Creative Commons Attribution- ShareAlike License; additional terms may apply. See Terms of Use for details. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
  4. 4. Mathematics and artFrom Wikipedia, the free encyclopediaMathematics 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.
  5. 5. Pyramid of KufuIf we divide the slant height of the pyramid by half its base length, we geta ratio of 1.619, less than 1% from the golden ratio. This would alsoindicate that half the cross-section of the Khufu’s pyramid is in fact aKepler’s triangle. Debate has broken out between prominentpyramidologists, including Temple Bell, Michael Rice, and JohnTaylor, over whether the presence of the golden ratio in the pyramids isdue to design or chance.
  6. 6. 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 (3-4-5Triangle), and 1:4/πHerodotus related in one passage that the Egyptian priests told him that the dimensions of the GreatPyramid were so chosen that the area of a square whose side was the height of the great pyramidequaled the area of the triangle.[7]
  7. 7. Great Mosque of KairouanThe geometric technique of construction of the golden section seems to have determined the majordecisions of the spatial organisation. The golden section appears repeatedly in some part of thebuilding measurements. It is found in the overall proportion of the plan and in the dimensioning of theprayer space, the court and the minaret. The existence of the golden section in some parts of Kairouanmosque indicates that the elements designed and generated with this principle may have been realisedat the same period.[13]” Because of urban constraints, the mosques 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.
  8. 8. Polykleitos Polykleitos gives us a mathematical approach towards sculpturing thehuman 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.
  9. 9. RenaissanceThe 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 artistsalike 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 thesefactors, Renaissance artists became some of the best applied mathematicians of their times.
  10. 10. Pierodella FrancescaPiero della Francesca (c.1415-1492), an early Renaissance artist fromItaly, exemplified this new shift in Renaissance thinking. Though chieflyappreciated for his art, he was an expert mathematician and geometerand authored many books on solid geometry and the emerging field ofperspective
  11. 11. .DaVinciWoodcut from De DivinaProportione illustrating the golden ratio as applied to the human face.Leonardo da Vinci (1452–1519) was an Italianscientist, mathematician, engineer, inventor, anatomist, painter, sculptor, and architect. Leonardo has often been describedas the archetype of the Renaissance man.[37][38]Renowned primarily as a painter, Leonardo incorporated many mathematical concepts into his artwork despite neverhaving received any formal mathematical training. It was not until the 1490s that he trained under Luca Pacioli andprepared a series of drawings for De DivinaProportione. Leonardo studied PaciolisSumma, from which he copied tables ofproportions and multiplication tables.[39]Notably in Mona Lisa and The Last Supper, Leonardo’s work incorporated the concept of linear perspective. By making allof the lines in the painting converge on a single, invisible point on the horizon, a flat painting can appear to have depth. Increating the vanishing point, Leonardo creates the illusion that the painting is an extension of the room itself. [40]
  12. 12. 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 hismathematically inspired work.[49] Escher’s interest in tessellations, polyhedrons, shaping ofspace, and self-reference manifested itself in his work throughout his career.[50] In theAlhambra Sketch, Escher showed that art can be created with polygons or regular shapes suchas triangles, squares, and hexagons.
  13. 13. Salvador DalíSalvador 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 ofa hypercube.
  14. 14. FractalsMain article: Fractal artThe processing power of modern computers allows mathematicians andnon-mathematicians to visualise complex mathematical objects such asthe Mandelbrot set. In the modern industry of computeranimation, fractals play a key role in modelling mountains, fire, trees andother natural objects.
  15. 15. Platonic solids in artThe Platonic solids and other polyhedra are a recurring theme in Western art.Examples include:A marble mosaic featuring the small stellated dodecahedron, attributed to PaoloUccello, in the floor of the San Marco Basilica in Venice.[18]Leonardo da Vincis outstanding diagrams of regular polyhedra drawn asillustrations for Luca Paciolis book The Divine Proportion.[18]A glass rhombicuboctahedron in Jacopo de Barbaris portrait of Pacioli, painted in1495.[18]A truncated polyhedron (and various other mathematical objects) which feature inAlbrecht Dürers engraving Melancholia I.[18]Salvador Dalís painting The Last Supper in which Christ and his disciples arepictured inside a giant dodecahedron.
  16. 16. Neil Currie, of Manchester, UK discussesThe Golden Ratio in math and artistic pursuits
  17. 17. Immediately following is a film demonstrating one point perspective drawing in an originalcomposition by the Armenian artist living in Beirut … RostomKouyoumdjian
  18. 18. Thank you for spending some timewith us in an overview of the Art of Mathematics. My special thanks to Neil Currie and RostomKouyoumdjian who accepted the challenge to show the use of maths in their artistic pursuits
  19. 19. cyeager