SIGGRAPH 2004 OZONE Turning a Snowball Inside Out
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: Com. Squared Systems, Eagan, MN Stan Wagon: Macalester College, St. Paul, MN John M. Sullivan, Tech. University, Berlin
“Whirled White Web” 3 D-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.
Shape Adaption for Snow Sculpture Restructured Morin surface to fit block size: (10’ x 12’)
Make Surface “Transparent” u Realize surface as a grid. u Draw a mesh of smooth lines onto the surface …
Gridded Models for Transparency 3 D-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
Day 1
Shovels, Drills, Pick-axes. . .
End of Day 1
Day 2 A Template for the “ear”
Day 2
Day 2
End of Day 2
Day 3
Defining the Grid
Carving the Grid
Cleaning Out the Cross-Tunnel
Day 3
End of Day 3
Day 4
Day 4
Day 5 End of Day 4
Day 5
Day 5
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
- Slides: 60