1. Mathematics and Art: Making Beautiful Music Together Based a presentation by D.N. Seppala-Holtzman St. Joseph’s College

2. Math & Art: the Connection Many people think that mathematics and art are poles apart, the first cold and precise, the second emotional and imprecisely defined. In fact, the two come together more as a collaboration than as a collision.

3. Math & Art: Common Themes Proportions Patterns Perspective Projections Impossible Objects Infinity and Limits

4. The Divine Proportion The Divine Proportion, better known as the Golden Ratio, is usually denoted by the Greek letter Phi: .  is defined to be the ratio obtained by dividing a line segment into two unequal pieces such that the entire segment is to the longer piece as the longer piece is to the shorter.

5. A Line Segment in Golden Ratio

6. The Golden Quadratic III  is equal to the quotient a/b and it can be shown that  is equal to: (1+√5)/2

7. Properties of  o  is irrational o Its reciprocal, 1/ , is one less than  o Its square,  2, is one more than  o There’s even more, but we won’t get into that. o just think of these as strange but true facts

8. Constructing  Begin with a 2 by 2 square. Connect the midpoint of one side of the square to a corner. Rotate this line segment until it provides an extension of the side of the square which was bisected. The result is called a Golden Rectangle. The ratio of its width to its height is .

9. Constructing  A B C AB=AC

10. Properties of a Golden Rectangle If one chops off the largest possible square from a Golden Rectangle, one gets a smaller Golden Rectangle. If one constructs a square on the longer side of a Golden Rectangle, one gets a larger Golden Rectangle. Both constructions can go on forever.

11. The Golden Spiral In this infinite process of chopping off squares to get smaller and smaller Golden Rectangles, if one were to connect alternate, non-adjacent vertices of the squares, one gets a Golden Spiral.

12. The Golden Spiral

13. The Golden Spiral II

14. The Golden Triangle o An isosceles triangle with two base angles of 72 degrees and an apex angle of 36 degrees is called a Golden Triangle. o The ratio of the legs to the base is . o The regular pentagon with its diagonals is simply filled with golden ratios and triangles.

15. The Golden Triangle

16. A Close Relative: Ratio of Sides to Base is 1 to Φ

17. Golden Spirals From Triangles As with the Golden Rectangle, Golden Triangles can be cut to produce an infinite, nested set of Golden Triangles. One does this by repeatedly bisecting one of the base angles. Also, as in the case of the Golden Rectangle, a Golden Spiral results.

18. Chopping Golden Triangles

19. Spirals from Triangles

20.  In Nature o There are physical reasons that  and all things golden frequently appear in nature. o Golden Spirals are common in many plants and a few animals, as well.

21. Sunflowers

22. Pinecones

23. Pineapples

24. The Chambered Nautilus

25. A Golden Solar System?

26.  In Art & Architecture o For centuries, people seem to have found  to have a natural, nearly universal, aesthetic appeal. o Indeed, it has had near religious significance to some. o Occurrences of  abound in art and architecture throughout the ages.

27. The Pyramids of Giza

28. The Pyramids and 

29. The Pyramids were laid out in a Golden Spiral

30. The Parthenon

31. The Parthenon II

32. Cathedral of Chartres

33. Cathedral of Notre Dame

34. Michelangelo’s David

35. Michelangelo’s Holy Family

36. Rafael’s The Crucifixion

37. Da Vinci’s Mona Lisa

38. Mona Lisa II

39. Da Vinci’s Study of Facial Proportions

40. Da Vinci’s St. Jerome

41. Da Vinci’s Study of Human Proportions

42. Rembrandt’s Self Portrait

43. Seurat’s Bathers

44. Turner’s Norham Castle at Sunrise

45. Mondriaan’s Broadway Boogie- Woogie

46. Dali’s The Sacrament of the Last Supper

47. Literally an (Almost) Golden Rectangle

48. Patterns Another subject common to art and mathematics is patterns. These usually take the form of a tiling or tessellation of the plane. Many artists have been fascinated by tilings, perhaps none more than M.C. Escher.

49. Patterns & Other Mathematical Objects In addition to tilings, other mathematical connections with art include fractals, infinity and impossible objects. Real fractals are infinitely self-similar objects with a fractional dimension. Quasi-fractals approximate real ones.

50. Fractals Some art is actually created by mathematics. Fractals and related objects are infinitely complex pictures created by mathematical formulae.

51. The Koch Snowflake (real fractal)

52. The Mandelbrot Set (Quasi)

53. Blow-up 1

54. Blow-up 2

55. Blow-up 3

56. Blow-up 4

57. Blow-up 5

58. Blow-up 6

59. Blow-up 7

60. Fractals Occur in Nature (the coastline)

61. Another Quasi-Fractal

62. Yet Another Quasi-Fractal

63. And Another Quasi-Fractal

64. Tessellations There are many ways to tile the plane. One can use identical tiles, each being a regular polygon: triangles, squares and hexagons. Regular tilings beget new ones by making identical substitutions on corresponding edges.

65. Regular Tilings

66. New Tiling From Old

67. Maurits Cornelis Escher (1898-1972) Escher is nearly every mathematician’s favorite artist. Although, he himself, knew very little formal mathematics, he seemed fascinated by many of the same things which traditionally interest mathematicians: tilings, geometry,impossible objects and infinity. Indeed, several famous mathematicians have sought him out.

68. M.C. Escher A visit to the Alhambra in Granada (Spain) in 1922 made a major impression on the young Escher. He found the tilings fascinating.

69. The Alhambra

70. An Escher Tiling

71. Escher’s Butterflies

72. Escher’s Lizards

73. Escher’s Sky & Water

74. M.C. Escher Escher produced many, many different types of tilings. He was also fascinated by impossible objects, self reference and infinity.

75. Escher’s Hands

76. Escher’s Circle Limit

77. Escher’s Waterfall

78. Escher’s Ascending & Descending

79. Escher’s Belvedere

80. Escher’s Impossible Box

81. Penrose’s Impossible Triangle

82. Roger Penrose Roger Penrose is a mathematical physicist at Oxford University. His interests are many and they include cosmology (he is an expert on black holes), mathematics and the nature of comprehension. He is the author of The Emperor’s New Mind.

83. Penrose Tiles In 1974, Penrose solved a difficult outstanding problem in mathematics that had to do with producing tilings of the plane that had 5-fold symmetry and were non-periodic. There are two roughly equivalent forms: the kite and dart model and the dual rhombus model.

84. Dual Rhombus Model

85. Kite and Dart Model

86. Kites & Darts II

87. Kites & Darts III

88. Kite & Dart Tilings

89. Rhombus Tiling

90. Rhombus Tiling II

91. Rhombus Tiling III

92. Penrose Tilings There are infinitely many ways to tile the plane with kites and darts. None of these are periodic. Every finite region in any kite-dart tiling sits somewhere inside every other infinite tiling. In every kite-dart tiling of the plane, the ratio of kites to darts is .

93. Luca Pacioli (1445-1514) Pacioli was a Franciscan monk and a mathematician. He published De Divina Proportione in which he called Φ the Divine Proportion. Pacioli: “Without mathematics, there is no art.”

94. Jacopo de Barbari’s Pacioli

95. In Conclusion Although one might argue that Pacioli somewhat overstated his case when he said that “without mathematics, there is no art,” it should, nevertheless, be quite clear that art and mathematics are intimately intertwined.