1. Solid Freeform Fabrication of Aesthetic Objects George W. Hart Stony Brook University Dept. Computer Science

2. 1.Sculpture 2.Mathematical Models: Fractals Goldberg Polyhedra 4D Polytope Projections 3.Historical Leonardo Models 4.Orderly Tangles 5.Puzzles 6.Student Projects

3. 1. Sculpture

4. 72 Pencils

5. Chronosynclastic Infundibulum

6. Just Two Cavities

7. Disk Combobulation

8. The Plastic Tableware of Damocles

9. Rainbow Bits

10. Fire and Ice

11. I’d like to make one thing perfectly clear

12. Battered Moonlight

13. Yin and Yang

14. Gazmogenesis

15. Rorschachohedron

16. Fat and Skinny

17. Whoville

18. Whoville -- detail

19. Roads Untaken

20. The Susurrus of the Sea

21. Loopy

22. Millennium Bookball

23. Twisted Rivers, Knotted Sea

25. Twisted Rivers -- reconstruction

26. Twisted Rivers – RP design

27. Tangled Reindeer

28. This End Up

29. Knot Structured

30. Nessie

31. Quintessence of Hedgehog

32. Bouquet

33. Frabjous

34. Salamanders

35. SFF Model (2.5 inch diam.)

36. Complete!

37. “People”

38. Snarl — wood model, 14 inch

39. Snarl — SFF model, 1 inch

40. Snarl — SFF model, 2 inch

41. Star Corona

42. Compass Points

44. “Cagework”

45. Aardvarks

46. Spaghetti Code

47. Rendering

48. Rendering, along 5-fold axis

49. Solid Freeform Fabrication Model

50. Spaghetti Code

51. Deep Structure

52. Gonads of the Rich and Famous

53. Gonads of the Rich and Famous -- design --

55. Untitled

60. 2. Mathematical Models

61. Fractal — Sierpinski Tetrahedron

62. Connected Structure

63. Snub Truncated Icosahedron

64. Zonohedron-61

65. 4D Polytope Projection Models

66. FDM Machine with 120-Cell Model

67. 120-Cell, before dissolving support

68. 120-Cell 120 cells, 720 faces, 1200 edges, 600 vertices. Cell-first parallel projection, (shows less than half of the edges.)

69. Five-Legged-Bee Hive

70. 120-Cell – perspective projection

71. (smallest ?) 120-Cell Wax model, made on Sanders machine

72. 120-Cell – perspective projection Selective laser sintering

73. Truncated 120-Cell

74. Truncated 120-Cell - Stereolithography

75. Zometool Truncated 120-Cell MathCamp 2000

76. Ambo 600-Cell Bridges Conference, 2001

77. Ambo 120-Cell Orthogonal projection Stereolithography Can do with Zome

78. 48 Truncated Cubes Poor FDM model

79. Goldberg Polyhedra (Fullerenes)

80. “Buckyballs” C60, form of carbon 1985 discovery 1996 Nobel Prize in chemistry By: –Richard Smalley (Rice Univ.) –Robert Curl (Rice Univ.) –Sir Harold Kroto

81. Geodesic Dome from Icosahedron

82. Dual Tessellations

83. Triangular Subdivisions Area = x 2 + xt + t 2

84. Goldberg Polyhedra (2,0) (2,1) (3,0)

85. Goldberg(2,2) Goldberg(4,1)

86. Goldberg (3,3) Goldberg(6,0)

87. Goldberg (7,3)

90. Ernst Haeckel

92. Artificial Radiolarian

93. Goldberg (2,1)

94. Goldberg (3,1)

96. Metal Goldberg (8, 3)

97. Nested Spheres

98. Nested Balls

99. Untitled (Abraham Sharp)

101. 3. Historical Models Leonardo / Pacioli

102. Sphere Model from Euclid

103. Leonardo Model

105. 4. Orderly Tangles

106. Regular Polylinks Symmetric linkages of regular polygons Alan Holden built models –Cardboard or dowels Holden wrote: –Shapes, Spaces and Symmetry,1971 –“Regular Polylinks”, 1980 –Orderly Tangles, 1983 Table of lengths 4 Triangles

107. Regular Polylinks 4 Triangles6 Squares Left and right hand forms

108. Regular Polylinks 12 Pentagons

109. Regular Polylinks 6 Pentagons - size scaled

110. Derivation from Regular Polyhedron Rotate faces Slide in or out

111. Wooden Puzzles Taiwan –Teacher Lin –Sculptor Wu Square cross sections Simple lap joint No glue Trial and error to determine length 12 Pentagons

112. Second Puzzle from Lin and Wu 10 Triangles

113. Many Analogous Puzzles Possible Each regular polylink gives a puzzle Also can combine several together: –Different ones interweaved –Same one nested Need critical dimensions to cut lengths No closed-form formulas for lengths Wrote program to: –Determine dimensions –Output templates to print, cut, assemble –Output STL files for solid freeform fabrication

114. Generates Template to Print and Cut 4 Triangles

115. Paper or Wood Models 6 Squares

116. Solid Freeform Fabrication 6 Squares

117. Square Cross Section 6 Pentagons

118. Paper or Wood Models

119. Orderly Tangle Puzzle aluminum 30 parts

120. Selective Laser Sintering (SLS)

121. Selective Laser Sintering

122. Curved Components Central Inversion 4 TrianglesInverted

123. Curved Components Central Inversion 20 TrianglesInverted

124. SLS Models — Inverted Tangles 4 triangles5 tetrahedra

125. Tangle Software Available free at http://www.georgehart.com

126. 5. Puzzles

127. Sculpture Puzzles aluminum 12 parts

128. Sculpture Puzzles SLS 3 inch

129. 30-Piece Puzzle

130. 6. Student Projects CSE391 Special Topics Algorithms for Solid Modeling

131. Student Projects