3. Miles of Tiles The Exhibit

4. Miles of Tiles The Exhibit

5. Miles of Tiles The Exhibit

6. Miles of Tiles The Exhibit

7. Miles of Tiles The Exhibit

8. The concept of repeated shapes touches all disciplines.

9. ART

10. M. C. Escher

19. ART

20. SCIENCE Geology

21. SCIENCE Geology Sodium Chloride

22. SCIENCE Geology Sodium Chloride

23. SCIENCE Geology Sodium Chloride

24. SCIENCE Geology Galena

25. SCIENCE Geology Galena

26. SCIENCE Geology Calcite

27. SCIENCE Geology Calcite

28. SCIENCE Biology

29. SCIENCE Biology Influenza

30. SCIENCE Biology Influenza

31. SCIENCE Biology HIV

32. SCIENCE Biology HIV

33. SCIENCE Biology T4 bacteriophage

34. SCIENCE Biology Ingest Nutrients Respiration Excretion

35. SCIENCE Biology Alive

36. SCIENCE Biology It is the virus’ shape (structure) that lets it replicate

37. SCIENCE Biology

38. SCIENCE Biology Only 4 repeated nucleotides make an entire human being

39. SCIENCE Architecture

40. SCIENCE Architecture

41. SCIENCE Architecture

42. SCIENCE Architecture

43. SCIENCE Nature

44. SCIENCE Nature

45. SCIENCE Nature

46. Math Geometry

47. Math Geometry

48. Math Geometry

49. Math Geometry

50. Math Geometry

51. Math Geometry + Nature

52. Math Geometry + Nature

53. Math Geometry + Nature

54. Math Geometry + Nature Why do bees use hexagons? Is there something special about them?

55. Math Geometry + Nature Why do bees use hexagons? Why not build using some other shape?

56. Math Geometry + Nature Why not build using some other shape? +

57. Math Geometry + Nature Why not build using some other shape? + Some shapes don’t tessellate!

58. Math Geometry + Nature Some shapes don’t tessellate! Tessellate? What’s that?

59. Math Geometry + Nature Tessellated [tes-uh-ley-ted] adjective 1.of, pertaining to, or like a mosaic. 2.arranged in or having the appearance of a mosaic

60. Math Geometry + Nature Some shapes don’t tessellate! + The bees would have wasted space in the hive

61. Math Geometry + Nature + The bees would have wasted space in the hive

62. Math Geometry + Nature But many shapes do tessellate!

63. Math Geometry + Nature But many shapes do tessellate! Squares seem to be an obvious choice

64. Math Geometry + Nature They are simple and don’t waste hive space Squares seem to be an obvious choice

65. Math Geometry + Nature So why do bees choose the more complex shape of a hexagon?

66. Math Geometry + Nature So why do bees choose the more complex shape of a hexagon? For the answer we’ll need to use some math

67. Math Geometry + Nature For the answer we’ll need to use some math

68. Math Geometry + Nature Let’s see what happens when the bees create their hive using different shapes.

69. Math Geometry + Nature We’ll start with the seemingly simplest design; a square.

70. Math Geometry + Nature Let’s assume each side of the square is three units in length. 3 3 3 3

71. Math Geometry + Nature This gives us a square with a perimeter of 12. 3 3 3 3 3 + 3 + 3 + 3 = 12 P = 12

72. Math Geometry + Nature The area of this square is 9. 3 3 3 3 3 x 3 = 9 P = 12 A = 9

73. Math Geometry + Nature Now let’s do the same calculations for a hexagon also with perimeter of 12.

74. Math Geometry + Nature 2 2 P = 12 2 2 2 2 Now let’s do the same calculations for a hexagon also with perimeter of 12.

75. Math Geometry + Nature 2 2 P = 12 A = ? 2 2 2 2 Now let’s calculate the area for this hexagon.

76. Math Geometry + Nature 2 2 2 2 2 2 Divide the hexagon into triangles 2 P = 12 A = ?

77. Math Geometry + Nature 2 2 2 2 2 2 Each is an equilateral triangle P = 12 A = ? 2

78. Math Geometry + Nature 2 2 2 2 2 2 Calculate area of one equilateral triangle 22 P = 12 A = ? 2

79. Math Geometry + Nature 2 2 2 2 2 2 Area = ½ b x h 22 P = 12 A = ? 2

80. Math Geometry + Nature 2 2 2 2 2 2 A = ½ b x h 22 P = 12 A = ? 2

81. Math Geometry + Nature 2 2 2 2 2 2 A = ½ b x h 22 P = 12 A = ? 11

82. Math Geometry + Nature 2 2 2 2 2 2 A = ½ b x h 2 P = 12 A = ? 1 h

83. Math Geometry + Nature 2 2 2 2 2 2 A = ½ b x h 2 P = 12 A = ? 1 h Determine the value of “h”

84. Math Geometry + Nature Word of the Day apothem - a perpendicular from the center of a regular polygon to one of its sides.

85. Math Geometry + Nature 2 2 2 2 2 2 A = ½ b x h 2 P = 12 A = ? 1 h Determine the value of “h”

86. Math Geometry + Nature 2 2 2 2 2 2 a 2 + b 2 = c 2 2 P = 12 A = ? 1 h Determine the value of “h”

87. Math Geometry + Nature 2 2 2 2 2 2 1 2 + h 2 = 2 2 2 P = 12 A = ? 1 h Determine the value of “h”

88. Math Geometry + Nature 2 2 2 2 2 2 1 + h 2 = 4 2 P = 12 A = ? 1 h Determine the value of “h”

89. Math Geometry + Nature 2 2 2 2 2 2 h 2 = 3 2 P = 12 A = ? 1 h Determine the value of “h”

90. Math Geometry + Nature 2 2 2 2 2 2 h 2 = 3 2 P = 12 A = ? 1 h Determine the value of “h”

91. Math Geometry + Nature 2 2 2 2 2 2 h = 3 2 P = 12 A = ? 1 h Determine the value of “h”

92. Math Geometry + Nature 2 2 2 2 2 2 h ≈ 1.73 2 P = 12 A = ? 1 h ≈ 1.73 Determine the value of “h”

93. Math Geometry + Nature 2 2 2 2 2 2 A = ½ b x h P = 12 A = ? 11 h Now we can calculate the area of one equilateral triangle

94. Math Geometry + Nature 2 2 2 2 2 2 A = 1 x 1.73 P = 12 A = ? 11 h Now we can calculate the area of one equilateral triangle

95. Math Geometry + Nature 2 2 2 2 2 2 A = 1.73 P = 12 A = ? The area of one equilateral triangle is 1.73 1.73

96. Math Geometry + Nature 2 2 2 2 2 2 Now calculate the area of the hexagon P = 12 A = ? 1.73

97. Math Geometry + Nature 2 2 2 2 2 2 Now calculate the area of the hexagon P = 12 A = ? 1.73

98. Math Geometry + Nature 2 2 2 2 2 2 A=1.73 x 6 P = 12 A = ? 1.73

99. Math Geometry + Nature A = 10.38 P = 12 A = 10.38 10.38

100. Math Geometry + Nature Let’s do the same calculations one last time for a circle also with perimeter of 12. P = 12 A = ?

101. Math Geometry + Nature C = ∏ x d P = 12 A = ?

102. Math Geometry + Nature 12 = ∏ x d P = 12 A = ?

103. Math Geometry + Nature 12/∏ = d P = 12 A = ?

104. Math Geometry + Nature 3.82 ≈ d P = 12 A = ?

105. Math Geometry + Nature 3.82 ≈ d r = ½ d P = 12 A = ?

106. Math Geometry + Nature 3.82 ≈ d r = ½ 3.82 P = 12 A = ?

107. Math Geometry + Nature 3.82 ≈ d r = 1.91 P = 12 A = ?

108. Math Geometry + Nature r = 1.91 P = 12 A = ?

109. Math Geometry + Nature r = 1.91 A = ∏ r 2 P = 12 A = ?

110. Math Geometry + Nature r = 1.91 A = ∏ 1.91 2 P = 12 A = ?

111. Math Geometry + Nature r = 1.91 A = ∏ 3.65 P = 12 A = ?

112. Math Geometry + Nature r = 1.91 A = 11.47 P = 12 A = 11.47

113. Math Geometry + Nature Let’s compare what we calculated P = 12 A = 9 P = 12 A = 10.38 P = 12 A = 11.47

114. Math Geometry + Nature P = 12 A = 9 If bees built their hive using squares they would have used 12 units of wax to create the walls of each cell, and gotten 9 units of area inside.

115. Math Geometry + Nature If they build using hexagons they would use 12 units of wax to create the walls of each cell, to get 10.38 units of area inside. P = 12 A = 10.38

116. Math Geometry + Nature In other words, they get a larger interior for each cell for the same amount of wax when using hexagons. P = 12 A = 10.38

117. Math Geometry + Nature If they build using circles they would use 12 units of wax to create the walls of each cell, to get 11.47 units of area inside. P = 12 A = 11.47

118. Math Geometry + Nature A circle yields the greatest interior space for the smallest perimeter of any shape! P = 12 A = 11.47

119. Math Geometry + Nature But circles don’t tessellate! P = 12 A = 11.47 +

120. Math Geometry + Nature P = 12 A = 10.38 Hexagons yields the greatest interior space for the smallest perimeter of any shape that tessellates!

121. Math Geometry + Nature P = 12 A = 10.38 How do the bees know this?

122. Miles of Tiles Going Further Penrose tessellations

123. Miles of Tiles Going Further Penrose tessellations