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Breckenridge, CO, January 2004

Explore the intricate process of turning snow into art using mathematical concepts in Breckenridge. Learn about the Sphere Eversion Process, construction, and geometry refinement for snow sculptures.

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Breckenridge, CO, January 2004

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  1. Breckenridge, CO, January 2004 art & math in snow Carlo H. Séquin U.C. Berkeley

  2. Memories of 2003

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

  4. Monkey Saddle Trefoil from Sculpture Generator I

  5. “Whirled White Web” 3D-Print FDM

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

  7. Construction Drawings Top View Side View Axial View

  8. Plan of Attack • Cut away 2 large triangular prisms; • Make a perfect donut with a small hole(using a semicircular template); • Mark edges of flanges on torus surface (using marked-up template); • Drill, carve, refine … (using maquette for reference).

  9. Day 1 Removing lot’s of snow …

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

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

  12. Chipping away …

  13. Making a Torus

  14. Tools and Techniques ... Use of template Need for a sun shield

  15. End of Day 2 The Torus

  16. Day 3, am: Drawing Flanges

  17. Day 3, pm: Flanges, Holes

  18. End of Day 3: Proper Topology

  19. Day 4: Geometry Refinement

  20. End of Day 4: Desired Geometry

  21. Day 5, am: Surface Refinement

  22. “House Cleaning”

  23. “Whirled White Web”

  24. 12:40 pm -- 42° F

  25. 12:41 pm -- 42° F

  26. 12:50 pm

  27. 12:40:01 Photo: StRomain

  28. 3 pm

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

  30. “WWW” Wins Silver Medal

  31. 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

  32. Sphere Eversion ... – what is that ?? How is this supposed to work ??? Can you really turn a beach ball inside outwithout cutting a hole into it ?

  33. A Simpler Task: Circle Reversal Make chain go in clock-wise direction ! Of course, this is easy, if we can flip through the 3rd dimension !

  34. Circle Eversion in the Plane Only allowed to push the string within plane. Self-intersections are allowed ! Not possible without forming cuspswith infinitely sharp curvature ! PINCHIS NOTALLOWED

  35. Back to Sphere Eversion Can it be done … ? -- without going through the 4th dimension, -- without any cuts, tears, sharp creases…( Of course we must allow the surface to pass through itself ! )

  36. Sphere Eversion is Possible ! • 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: PINCH

  37. 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.

  38. Sphere Eversion Process • But there are more contorted paths that can 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.

  39. 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”

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

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

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

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

  44. Our Submission: 7/1/03

  45. The Final Model

  46. A Visualization Sequence Sphere inversion in a dozen easy steps… Color surface: green on one side, red on the other. Make it partially gridded, so we can see inside… … but only from the green side, make red opaque -- like a one-way mirror: -- can see through from the green side, -- but visibility is blocked from red side.

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