SIGGRAPH 2004, OZONESIGGRAPH 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”Whirled White Web”

3D-Print

Day 1: The “Monolith”Day 1: The “Monolith”

Cut away prisms …

End of Day 2End of Day 2

The Torus

Day 3, pm: Flanges, Holes Day 3, pm: Flanges, Holes

Day 4: Geometry RefinementDay 4: Geometry Refinement

““House Cleaning”House Cleaning”

Memories of 2003Memories of 2003

12:40 pm -- 4212:40 pm -- 42° F° F

12:41 pm -- 4212:41 pm -- 42° F° F

The WinnersThe Winners

1st: Canada – B.C., 2nd: USA – Minnesota, 3rd: USA – Breckenridge

“… sacred geometry … very intricate … very 21st century !”

““WWW” Wins Silver MedalWWW” Wins Silver Medal

What Are We Going To Do For 2004 ?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

Sphere Eversion is Possible !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

Sphere Eversion ProcessSphere 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.

Sphere Eversion ProcessSphere 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.

Optimal Sphere EversionOptimal 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”

The Simplest Polyhedral ModelThe 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 SculptureShape Adaption for Snow Sculpture

Make Surface “Transparent”Make Surface “Transparent”

Realize surface as a grid.

Draw a mesh of smooth lines onto the surface …

Gridded Models for TransparencyGridded Models for Transparency

3D-Print from Zcorp SLIDE virtual model

““Turning a Snowball Inside-Out”Turning a Snowball Inside-Out”

Carlo H. Séquin, Alex Kozlowski, John Sullivan

Dan Schwalbe, Stan Wagon

The Final ModelThe Final Model

Morin’s Surface EversionMorin’s Surface Eversion

The Half-way PointThe Half-way Point

Finish the ProcessFinish the Process

Computer ProjectionsComputer Projections

Horizontal Slices and ProjectionsHorizontal Slices and Projections

Practice Block (Stan Wagon)Practice Block (Stan Wagon)

First NightFirst Night

Working Out Plan BWorking Out Plan B

Working on the GridWorking on the Grid

Day 1Day 1Day 1Day 1

Day 1Day 1

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

End of End of Day 1Day 1

Day 2Day 2

A Template for the “ear”A Template for the “ear”

Day 2Day 2

Day2Day2

End of End of Day 2Day 2

Day 3Day 3

Defining the GridDefining the Grid

Carving the GridCarving the Grid

Cleaning Out the Cross-TunnelCleaning Out the Cross-Tunnel

Day3Day3

End of Day 3End of Day 3

Day 4Day 4

Day 4Day 4

Day 5Day 5 End of End of Day 4Day 4

Day 5Day 5

Day 5Day 5

Day 5Day 5

““Paradigm Shift” (British Columbia)Paradigm Shift” (British Columbia)

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

““Winter Oasis” (Canada – Ontario)Winter Oasis” (Canada – Ontario)

CeremonyCeremonyHonorable Mention:“Most Ambitious Design”

Celebration Celebration

Day 6Day 6

Questions ?Questions ?

““Knot Divided”Knot Divided”

Accepted, August 5, 2004 for competition in January 2005