2. Meeting Alhambra, Granada 2003 “Whirled White Web” Art and Math in Snow Carlo H. Séquin Brent Collins, Steve Reinmuth Dan Schwalbe, Stan Wagon

3. Brent Collins “Hyperbolic Hexagon II”

4. Brent Collins: Stacked Saddles

5. Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles (monkey saddle)

6. “Hyperbolic Hexagon” by B. Collins u 6 saddles in a ring u 6 holes passing through symmetry plane at ±45º u = “wound up” 6-story Scherk tower u Discussion: What if … l we added more stories ? l or introduced a twist before closing the ring ?

7. Closing the Loop straight or twisted

8. Brent Collins’ Prototyping Process Armature for the "Hyperbolic Heptagon" Mockup for the "Saddle Trefoil" Time-consuming ! (1-3 weeks)

9. “Sculpture Generator I”, GUI

10. V-art Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

11. Collins’ Fabrication Process Example: “Vox Solis” Layered laminated main shape Wood master pattern for sculpture

12. Slices through “Minimal Trefoil” 50%10%23%30% 45%5%20%27% 35%2%15%25%

13. u One thick slice thru sculpture, from which Brent can cut boards and assemble a rough shape. u Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board. Profiled Slice through “Heptoroid”

14. Emergence of the “Heptoroid” (1) Assembly of the precut boards

15. Emergence of the “Heptoroid” (2) Forming a continuous smooth edge

16. Emergence of the “Heptoroid” (3) Smoothing the whole surface

17. The Finished “Heptoroid” u at Fermi Lab Art Gallery (1998).

18. Various “Scherk-Collins” Sculptures

19. Hyper-Sculpture: “Family of 12 Trefoils” W=2 W=1 B=1 B=2 B=3 B=4

20. “Cohesion” SIGGRAPH’2003 Art Gallery

21. Stan Wagon, Macalester College, St. Paul, MN u Leader of Team “USA – Minnesota”

22. Stan Wagon u Skier – Mountaineer – Snow Sculptor

23. Dan Schwalbe u Software Engineer – Snow Sculptor

24. Helaman Ferguson Umbilic Torus Costa Surface

25. Breckenridge, 1999 Helaman Ferguson: “Invisible Handshake”

26. Breckenridge, CO, 1999 Helaman Ferguson: “Invisible Handshake”

27. Robert Longhurst Moebius Band Enneper Surface

28. Breckenridge, 2000 Robert Longhurst: “Rhapsody in White” 2 nd Place

29. Breckenridge, 2001 Robert Longhurst: “White Narcissus”

30. Batsheba Grossman Antipot Antichron

31. Breckenridge, 2002 Bathsheba Grossman: “A Twist in Time” Honorable Mention “Expressive Impact”

32. Snow-Sculpting, Breckenridge, 2003 Brent Collins and Carlo Séquin are invited to join the team and to provide a design. Other Team Members: Stan Wagon, Dan Schwalbe, Steve Reinmuth (= Team “Minnesota”)

33. Steve Reinmuth

34. Collins’ Initial Suggestion: Minimal Trefoil Stan’s Objection: “Too simple … – we know we can do this. No risk – no fun !”

35. Monkey Saddle Trefoil from Sculpture Generator I

36. Maquettes 3D-Print FDM

37. Name, Story u “Snow Flower, Winter Rose, Winter Whirl, Wild White Whirl, Webbed Wild Whirl, Whirled Wild Web …” u finally the perfect homonym: “Whirled White Web” u Like this global network, the ridges of our sculpture span the outer perimeters of the whole “globe,” and at the same time come close together in the central hole. It illustrates how the WWW can link together people from all over the world.

38. ACCEPTED ! Now – how do we get this design into a 10’x10’x12’ block of snow ?

39. Construction Drawings Top View Side View Axial View

40. Plan of Attack u Cut away 2 large triangular prisms; u Hang plastic template (12’x18’) over slab; u Mark projections of holes, flanges; u Drill, carve, refine … (using maquette for reference) u Try this first on a smaller practice block!

41. 1:1 Templates Magnified from computer prints with the classical grid method

42. Practice Block (1) Template transfer Reference holes

43. Practice Block (2) Free-hand carving  Irregular ribs

44. Practice Block (3)  Need new plan !

46. Removing lots of snow … Day 1

47. Day 1: The “Monolith” Cut away prisms …

48. Day 2: Making a Torus Mark center, circles … Bull’s-eye !

50. Chipping away …

52. Tools, Templates

53. Making a Torus Use of template Need for a sun shield

54. End of Day 2 The Torus

55. Day 3, am: Drawing Flanges

56. Day 3, pm: Flanges, Holes

57. End of Day 3: Proper Topology

58. Day 4: Geometry Refinement

60. End of Day 4: Desired Geometry

61. Day 5, am: Surface Refinement

62. “House Cleaning”

63. “Whirled White Web”

65. Official Team Photo

66. 12:40 pm -- 42° F

67. 12:41 pm -- 42° F

68. 12:40:01 Photo: StRomain

69. 12:50 pm

70. 3 pm

72. The Winners 1 st : Canada – B.C., 2 nd : USA – Minnesota, 3 rd : USA – Breckenridge “… sacred geometry … very intricate … very 21 st century !”

73. “WWW” Wins Silver Medal

74. British Columbia: “Winter Comes”

76. Day 2 Day 3

77. Breckenridge: “A Fishing Tail”

78. Mexico City: “Capilla Posa”

79. China: “The Love of Mother”

80. Quebec: “Rève’Olution”

81. Manitoba: “Birth of a Nation”

82. USA – New York: “94 Hour Photo”

84. What Are We Going To Do For 2004 ? “Turning a Snowball Inside Out”

85. Sphere Eversion u ~ 1960, the blind mathematician B. Morin, (born 1931) conceived of a way how a sphere can be turned inside-out: l Surface may pass through itself, l but no ripping, puncturing, creasing allowed, e.g., this is not an acceptable solution: PINCH

86. Morin Surface u But there are more contorted paths that can achieve the desired goal. u The Morin surface is the half-way point of one such path: John Sullivan: “The Optiverse”

87. Simplest Model Partial cardboard model based on the simplest polyhedral sphere (= cuboctahedron) eversion.

88. Gridded Models for Transparency 3D-Print from ZcorpSLIDE virtual model

89. Shape Adaption for Snow Sculpture Restructured Morin surface to fit block size: (10’ x 10’ x 12’)

90. Make Surface “Transparent” u Realize surface as a grid. u Draw a mesh of smooth lines onto the surface …

91. “Turning a Snowball Inside-Out” Carlo H. Séquin, Alex Kozlowski, John Sullivan Dan Schwalbe, Stan Wagon

92. Our Submission: 7/1/03 QUESTIONS ? -- DISCUSSION ?