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

2. Game Plan 1)Introduction 2)Artists doing Math 3)Mathematicians doing Art http://www.highlands-gallery.com/Laurent_Davidson2.cfm Lily Pads by Laurent Davidson StabiloMobile Aluminum and Steel 21.5” high 41” wide 22” deep

4. And so Begins our Quest…

5. Definitions Disclaimer: 1)I am NOT an artist http://www.kenleap.com/

6. Definitions Disclaimer: 1)I am NOT an artist. 2)I don’t like art. http://www.kenleap.com/

7. Definitions Disclaimer: 1)I am NOT an artist. 2)I don’t like art. 3)I am a Mathematician. 4)I love Math and try to find it in all things. http://www.kenleap.com/

8. 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

9. 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

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

11. 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

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

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

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

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

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

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

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

19. The Golden Ratio Euclid’s Elements (300 B.C.) The Extreme and Mean Ratio: ABC

20. The Golden Ratio Euclid’s Elements (300 B.C.) The Extreme and Mean Ratio: ABC x1

21. The Golden Ratio Euclid’s Elements (300 B.C.) The Extreme and Mean Ratio: ABC x1

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

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

24. 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  Formula for the n th Fibonacci number: Logarithmic Spirals Ram’s horns, elephant tusks, seashells, whirlpools, hurricanes, galaxies… Peregrine Falcon Golden Ratio in Nature

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

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 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 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 http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

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 http://library.thinkquest.org/27890/applications6.html

35. 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/

36. 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/

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

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

39. 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 Horizon Vanishing point Vanishing point

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

41. Man: dpdp HmHm d hmhm Column: dpdp HcHc d hchc  

42. Man: dpdp HmHm d hmhm Column: dpdp HcHc d hchc  

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

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

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

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

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

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

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

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

51. Impossible Art Roger Penrose 1950s  Impossible Triangle  Tribar  Tribox  M.C. Escher (1898-1972)  Waterfall  Belvedere http://www.mcescher.com/

52. Impossible Art Roger Penrose 1950s  Impossible Triangle  Tribar  Tribox  M.C. Escher (1898-1972)  Waterfall  Belvedere  Cube With Ribbons http://www.mathacademy.com/pr/minitext/escher/index.asp

53. Impossible Art  Escher For Real http://www.cs.technion.ac.il/~gershon/EscherForReal/

54. Impossible Art  Escher For Real http://www.cs.technion.ac.il/~gershon/EscherForReal/

55. Impossible Art  Escher For Real http://www.cs.technion.ac.il/~gershon/EscherForReal/

56. Impossible Art  Escher For Real http://www.cs.technion.ac.il/~gershon/EscherForReal/

57. Impossible Art  Escher For Real http://www.cs.technion.ac.il/~gershon/EscherForReal/

58. Impossible Art  Escher For Real http://www.cs.technion.ac.il/~gershon/EscherForReal/

59. Impossible Art  Escher For Real http://www.cs.technion.ac.il/~gershon/EscherForReal/

60. Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II http://www.mcescher.com/

61. Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III http://www.mcescher.com/

62. Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III  Penrose Tiling http://goldennumber.net/penrose.htm

63. Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III  Limits  Circle Limit III http://www.mcescher.com/

64. Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III  Limits  Circle Limit III  Circle Limit IV http://www.mcescher.com/

65. Mathematicians Doing Art Larry Frazier Triple Bocote Blush http://www.highlands-gallery.com/Larry_Frazier2.cfm

66. Mathematicians Doing Art Larry Frazier http://www.highlands-gallery.com/Larry_Frazier2.cfm

67. Mathematicians Doing Art Helaman Ferguson Umbilic Torus NC http://www.angelo.edu/dept/mathematics/gallery.htm

68. Mathematicians Doing Art Ken Leap ConfluenceSalter’s Lune http://www.kenleap.com/

69. Mathematicians Doing Art Harriet Brisson Magic Cube http://www.harrietbrisson.com

70. Movie Math www.pixar.com

71. “Let no one who is not a mathematician read my works.” -Leonardo da Vinci http://www.georgehart.com/virtual-polyhedra/leonardo.html

72. Sources Hofstadter, Douglas R. Godel, Escher, Bach: An Eternal Golden Braid. Random House, New York 1979. Maor, Eli. To Infinity and Beyond: A Cultural History of the Infinite. Princeton University Press, New Jersey 1991. Livio, Mario. The Golden Ratio. Random House, New York 2002. Peterson, Ivars. Fragments of Infinity: A Kaleidoscope of Math and Art. John Wiley & Sons, Inc. New York 2001. http://www.mcescher.com/

73. Your Moment of Zen