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Art as a Mathform

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  1. Art as a Mathform The Intersection of Antipodal Worlds http://www.mcescher.com

  2. Game Plan • Introduction • Artists doing Math • Mathematicians doing Art Lily Pads by Laurent Davidson StabiloMobileAluminum and Steel 21.5” high 41” wide 22” deep http://www.highlands-gallery.com/Laurent_Davidson2.cfm

  3. And so Begins our Quest…

  4. Definitions • Disclaimer: • I am NOT an artist http://www.kenleap.com/

  5. Definitions • Disclaimer: • I am NOT an artist. • I don’t like art. http://www.kenleap.com/

  6. Definitions • Disclaimer: • I am NOT an artist. • I don’t like art. • I am a Mathematician. • I love Math and try to find it in all things. http://www.kenleap.com/

  7. Math & Art Differences How would a mathematician describe art? • Boring • Too abstract • Doesn’t make any sense • All artists are weirdos The Moon-Woman Jackson Pollock 1942 http://www.ibiblio.org/wm/paint/auth/pollock/pollock.moon-woman.jpg

  8. Math & Art Differences How would a mathematician describe art? • Boring • Too abstract • Doesn’t make any sense • All artists are weirdos How would an artist describe math? • Boring • Too abstract • Doesn’t make any sense • All mathematicians are weirdos

  9. Math & Art Similarities How would a mathematician describe math? • Abstract representation of our world • Makes sense to “most” people • Means different things to different people • Experience joy of creation in making something that has never been made before • The results are beautiful

  10. Math & Art Similarities How would a mathematician describe math? • Abstract representation of our world • Makes sense to “most” people • Means different things to different people • Experience joy of creation in making something that has never been made before • The results are beautiful How would an artist describe art? • Abstract representation of our world • Makes sense to “most” people • Means different things to different people • Experience joy of creation in making something that has never been made before • The results are beautiful

  11. Artists Doing Math • The Golden Ratio • Perspective (Projective Geometry) • Impossible Art • Space-Filling (Tilings)

  12. The Golden Ratio • Discovered by Pythagoreans in 5th century B.C. • The Golden Ratioby Mario Livio

  13. The Golden Ratio • Discovered by Pythagoreans in 5th century B.C. • The Golden Ratioby Mario Livio

  14. The Golden Ratio • Discovered by Pythagoreans in 5th century B.C. • The Golden Ratioby Mario Livio a b

  15. The Golden Ratio • Discovered by Pythagoreans in 5th century B.C. • The Golden Ratioby Mario Livio c b

  16. The Golden Ratio • Discovered by Pythagoreans in 5th century B.C. • The Golden Ratioby Mario Livio c d

  17. The Golden Ratio • Discovered by Pythagoreans in 5th century B.C. • The Golden Ratioby Mario Livio e d

  18. The Golden Ratio • Euclid’s Elements (300 B.C.) • The Extreme and Mean Ratio: A C B

  19. The Golden Ratio • Euclid’s Elements (300 B.C.) • The Extreme and Mean Ratio: A C B x 1

  20. The Golden Ratio • Euclid’s Elements (300 B.C.) • The Extreme and Mean Ratio: A C B x 1

  21. The Golden Ratio Simplify: Solve using Quadratic Formula: The Golden Ratio:

  22. The Golden Ratio Simplify: Solve using Quadratic Formula: The Golden Ratio:

  23. Golden Ratio in Nature • The Golden Ratio can be found in nature via Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … • The ratios of successive Fibonaccis head towards F • Formula for the nth Fibonacci number: • Logarithmic Spirals • Ram’s horns, elephant tusks, seashells, whirlpools, hurricanes, galaxies… • Peregrine Falcon

  24. Golden Ratio in Art • Great Pyramid at Giza http://people.bath.ac.uk/jaj21/disprovingmyth.html

  25. Golden Ratio in Art • Great Pyramid at Giza • Parthenon http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  26. Golden Ratio in Art • Great Pyramid at Giza • Parthenon http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  27. Golden Ratio in Art • Great Pyramid at Giza • Parthenon http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  28. Golden Ratio in Art • Great Pyramid at Giza • Parthenon • Leonardo da Vinci’s Saint Jerome http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  29. Golden Ratio in Art • Great Pyramid at Giza • Parthenon • Leonardo da Vinci’s Saint Jerome http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  30. Golden Ratio in Art • Great Pyramid at Giza • Parthenon • Leonardo da Vinci’s Saint Jerome http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  31. Golden Ratio in Art • Great Pyramid at Giza • Parthenon • Leonardo da Vinci’s Saint Jerome • Michelangelo’s Holy Family http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  32. Golden Ratio in Art • Great Pyramid at Giza • Parthenon • Leonardo da Vinci’s Saint Jerome • Michelangelo’s Holy Family http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

  33. Golden Ratio in Art • Great Pyramid at Giza • Parthenon • Leonardo da Vinci’s Saint Jerome • Michelangelo’s Holy Family • Leonardo da Vinci’s Mona Lisa http://library.thinkquest.org/27890/applications6.html

  34. Golden Ratio in Art • Great Pyramid at Giza • Parthenon • Leonardo da Vinci’s Saint Jerome • Michelangelo’s Holy Family • Leonardo da Vinci’s Mona Lisa • Salvador Dali’s Sacrament of the Last Supper http://plus.maths.org/issue22/features/golden/

  35. Renaissance Art • Three of the best known Renaissance artists also made contributions to mathematics: • Piero della Francesca (ca. 1412-1492): • On Perspective in Painting • Short Book on the Five Regular Solids • Treatise on the Abacus • Leonardo da Vinci (1452-1519) • Illustrator of The Divine Proportion (Luca Pacioli) • Quadrature of the Circle (Squaring the Circle) • Areas of regions bounded by curves • Albrecht Durer (1471-1528) • Treatise on Measurement with Compass and Ruler • One of first Math books published in German • Earliest Nets of Polyhedra • Tiling of the plane http://www.intriguing.com/mp/

  36. Albrecht Durer Melencolia I http://www.ibiblio.org/wm/paint/auth/durer/

  37. Putting it in Perspective http://www.intriguing.com/mp/

  38. Putting it in Perspective • Pre-Renaissance subjects were depicted according to status in Church or social hierarchy • Represent a scene in true and objective way • Projective Geometry: what properties of an object are preserved under a projection? • Parallel lines intersect at horizon (vanishing point) • Circles become ellipses • Squares become trapezoids Vanishing point Vanishing point Horizon

  39. Putting it in Perspective • Dimensions should decrease at same rate as we move towards the horizon • Compare heights of objects • Similar Triangles preserve ratios of corresponding sides http://plus.maths.org/issue23/features/criminisi/

  40. Hm hm a d Man: dp Column: Hc hc b d dp

  41. Hm hm a d Man: dp Column: Hc hc b d dp

  42. and So we must have Cross-multiplying gives us

  43. Piero della Francesca The Flagellation www.artchive.com

  44. Piero della Francesca The Flagellation www.artchive.com

  45. Sandro Botticelli The Annunciation http://www.kap.pdx.edu/trow/winter01/perspective/persp-images.htm

  46. Impossible Art • Roger Penrose 1950s • Impossible Triangle http://mathworld.wolfram.com/PenroseTriangle.html

  47. Impossible Art • Roger Penrose 1950s • Impossible Triangle • Tribar http://icl.pku.edu.cn/yujs/MathWorld/math/t/t317.htm

  48. Impossible Art • Roger Penrose 1950 • Impossible Triangle • Tribar • Tribox http://icl.pku.edu.cn/yujs/MathWorld/math/t/t318.htm

  49. Impossible Art • Roger Penrose 1950s • Impossible Triangle • Tribar • Tribox • M.C. Escher (1898-1972) • Waterfall http://www.mathacademy.com/pr/minitext/escher/index.asp