1. Mathematics and Art: Making Beautiful Music Together 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 Quadratic Equation The definition of  leads to the following equation, if the line is divided into segments of lengths a and b:

7. The Golden Quadratic II Cross multiplication yields:

8. The Golden Quadratic III Setting  equal to the quotient a/b and manipulating this equation shows that  satisfies the quadratic equation:

9. The Golden Quadratic IV Applying the quadratic formula to this simple equation and taking  to be the positive solution yields:

10. Properties of   is irrational Its reciprocal, 1/ , is one less than  Its square,  2, is one more than 

11.  Is an Infinite Square Root

12. Φ is an Infinite Continued Fraction

13. 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 .

14. Constructing  A B C AB=AC

15. 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.

16. 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.

17. The Golden Spiral

18. The Golden Spiral II

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

20. The Golden Triangle

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

22. 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.

23. Chopping Golden Triangles

24. Spirals from Triangles

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

26. Sunflowers

27. Pinecones

28. Pineapples

29. The Chambered Nautilus

30. Angel Fish

31. Tiger

32. Human Face I

33. Human Face II

34. Le Corbusier’s Man

35. A Golden Solar System?

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

37. The Pyramids of Giza

38. The Pyramids and 

39. The Pyramids were laid out in a Golden Spiral

40. The Parthenon

41. The Parthenon II

42. The Parthenon III

43. Cathedral of Chartres

44. Cathedral of Notre Dame

45. Michelangelo’s David

46. Michelangelo’s Holy Family

47. Rafael’s The Crucifixion

48. Da Vinci’s Mona Lisa

49. Mona Lisa II

50. Da Vinci’s Study of Facial Proportions

51. Da Vinci’s St. Jerome

52. Da Vinci’s The Annunciation

53. Da Vinci’s Study of Human Proportions

54. Rembrandt’s Self Portrait

55. Seurat’s Parade

56. Seurat’s Bathers

57. Turner’s Norham Castle at Sunrise

58. Mondriaan’s Broadway Boogie- Woogie

59. Hopper’s Early Sunday Morning

60. Dali’s The Sacrament of the Last Supper

61. Literally an (Almost) Golden Rectangle

62. 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.

63. 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.

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

65. The Koch Snowflake (real fractal)

66. The Mandelbrot Set (Quasi)

67. Blow-up 1

68. Blow-up 2

69. Blow-up 3

70. Blow-up 4

71. Blow-up 5

72. Blow-up 6

73. Blow-up 7

74. Fractals Occur in Nature (the coastline)

75. Another Quasi-Fractal

76. Yet Another Quasi-Fractal

77. And Another Quasi-Fractal

78. 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.

79. Regular Tilings

80. New Tiling From Old

81. 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.

82. 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.

83. The Alhambra

84. An Escher Tiling

85. Escher’s Butterflies

86. Escher’s Lizards

87. Escher’s Sky & Water

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

89. Escher’s Hands

90. Escher’s Circle Limit

91. Escher’s Waterfall

92. Escher’s Ascending & Descending

93. Escher’s Belvedere

94. Escher’s Impossible Box

95. Penrose’s Impossible Triangle

96. 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.

97. 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.

98. Dual Rhombus Model

99. Kite and Dart Model

100. Kites & Darts II

101. Kites & Darts III

102. Kite & Dart Tilings

103. Rhombus Tiling

104. Rhombus Tiling II

105. Rhombus Tiling III

106. 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 .

107. 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.”

108. Jacopo de Barbari’s Pacioli

109. 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.