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

“Whirled White Web” 3D-Print

Day 1: The “Monolith” Cut away prisms …

End of Day 2 The Torus

Day 3, pm: Flanges, Holes

Day 4: Geometry Refinement

“House Cleaning”

Memories of 2003

12:40 pm -- 42° F

12:41 pm -- 42° F

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

“WWW” Wins Silver Medal

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

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

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.

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.

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”

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

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

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

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

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

The Final Model

Morin’s Surface Eversion

The Half-way Point

Finish the Process

Computer Projections

Horizontal Slices and Projections

Practice Block (Stan Wagon)

First Night

Working Out Plan B

Working on the Grid

Day 1

Shovels, Drills, Pick-axes...

End of Day 1

Day 2 A Template for the “ear”

Day 2

Day2

End of Day 2

Day 3

Defining the Grid

Carving the Grid

Cleaning Out the Cross-Tunnel

Day3

End of Day 3

Day 4

Day 5 End of Day 4

Day 5

“Paradigm Shift” (British Columbia)

“Year of the Dragon” (USA – Tennessee)

“Winter Oasis” (Canada – Ontario)

Ceremony Honorable Mention: “Most Ambitious Design”

Celebration

Day 6

Questions ?

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