1 / 73

Art as a Mathform

Art as a Mathform. The Intersection of Ant ipodal Worlds. http://www.mcescher.com. Game Plan. Introduction Artists doing Math Mathematicians doing Art. Lily Pads by Laurent Davidson StabiloMobile Aluminum and Steel 21.5” high 41” wide 22” deep.

noble-osborne
Download Presentation

Art as a Mathform

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related