2. SIGGRAPH 2004, OZONE Turning a Snowball Inside Out: Mathematical Visualization at the 12-foot Scale Alex Kozlowski & Carlo H. Séquin: U.C. Berkeley Dan Schwalbe: ComSquared Systems, Eagan, MN Stan Wagon: Macalester College, St. Paul, MN John M. Sullivan, Tech. University, Berlin

3. “Whirled White Web” 3D-Print

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

5. End of Day 2 The Torus

6. Day 3, pm: Flanges, Holes

7. Day 4: Geometry Refinement

8. “House Cleaning”

9. Memories of 2003

10. 12:40 pm -- 42° F

11. 12:41 pm -- 42° F

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

13. “WWW” Wins Silver Medal

14. What Are We Going To Do For 2004 ? “Turning a Snowball Inside Out” Making a Model of the Half-way Point of the Sphere Eversion Process

15. Sphere Eversion is Possible ! u First proven by Steve Smale around 1960 from complex topological arguments. u But he could not say HOW it can be done … ! l Surface may pass through itself, l but no ripping, puncturing, creasing allowed, e.g., this is not an acceptable solution: PINCH

16. Sphere Eversion Process u A few years later Bernard Morin, a blind mathematician, figured out how to do it. u In his honor, the half-way point, where half each of the inside and outside of the sphere shell can be seen, is called the Morin surface.

17. Sphere Eversion Process u You need a rather contorted move to achieve the desired goal. u Bernard Morin figured out one such path. u Charles Pugh made models from chicken wire. u Nelson Max made a first computer simulation.

18. Optimal Sphere Eversion u In the 1990’s John Sullivan found the most efficient way (using the least surface bending) to accomplish this eversion, and made a beautiful movie of it. From: John Sullivan: “The Optiverse”

19. The Simplest Polyhedral Model Partial cardboard model based on cuboctahedron eversion by Apéry & Denner.

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

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

22. Gridded Models for Transparency 3D-Print from Zcorp SLIDE virtual model

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

24. The Final Model

25. Morin’s Surface Eversion

26. The Half-way Point

27. Finish the Process

28. Computer Projections

29. Horizontal Slices and Projections

30. Practice Block (Stan Wagon)

31. First Night

32. Working Out Plan B

33. Working on the Grid

34. Day 1

36. Shovels, Drills, Pick-axes...

37. End of Day 1

38. Day 2 A Template for the “ear”

39. Day 2

40. Day2

41. End of Day 2

42. Day 3

43. Defining the Grid

44. Carving the Grid

45. Cleaning Out the Cross-Tunnel

46. Day3

47. End of Day 3

48. Day 4

50. Day 5 End of Day 4

51. Day 5

54. “Paradigm Shift” (British Columbia)

55. “Year of the Dragon” (USA – Tennessee)

56. “Winter Oasis” (Canada – Ontario)

57. Ceremony Honorable Mention: “Most Ambitious Design”

58. Celebration

59. Day 6

60. Questions ?

61. “Knot Divided” Accepted, August 5, 2004 for competition in January 2005