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SIGGRAPH 2004, OZONE

Explore the intricate process of mathematical visualization with large-scale 3D models, detailing the sphere eversion process, artworks by Alex Kozlowski, Carlo H. Séquin, and more. Learn about the innovative techniques used for optimal sphere eversion and shape adaptation for snow sculptures. Follow the journey from initial models to the final representation of Morin's surface eversion. Witness the transformation of a simple snowball into a complex mathematical wonder.

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SIGGRAPH 2004, OZONE

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  1. 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

  2. “Whirled White Web” 3D-Print

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

  4. End of Day 2 The Torus

  5. Day 3, pm: Flanges, Holes

  6. Day 4: Geometry Refinement

  7. “House Cleaning”

  8. Memories of 2003

  9. 12:40 pm -- 42° F

  10. 12:41 pm -- 42° F

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

  12. “WWW” Wins Silver Medal

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

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

  15. Sphere Eversion Process • A few years later Bernard Morin, a blind mathematician, figured out how to do it. • 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.

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

  17. Optimal Sphere Eversion • 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”

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

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

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

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

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

  23. The Final Model

  24. Morin’s Surface Eversion

  25. The Half-way Point

  26. Finish the Process

  27. Computer Projections

  28. Horizontal Slices and Projections

  29. Practice Block (Stan Wagon)

  30. First Night

  31. Working Out Plan B

  32. Working on the Grid

  33. Day 1 Day 1

  34. Day 1

  35. Shovels, Drills, Pick-axes ...

  36. End of Day 1

  37. Day 2 A Template for the “ear”

  38. Day 2

  39. Day2

  40. End of Day 2

  41. Day 3

  42. Defining the Grid

  43. Carving the Grid

  44. Cleaning Out the Cross-Tunnel

  45. Day3

  46. End of Day 3

  47. Day 4

  48. Day 4

  49. Day 5 End of Day 4

  50. Day 5

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