BRIDGES 2004, Winfield KS

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BRIDGES 2004, Winfield KS

• 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
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Stan Wagon, Macalester College, St. Paul, MN

• Leader of Team USA Minnesota since 1999

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Monkey Saddle Trefoil

• from Sculpture Generator I

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Whirled White Web

• 3D-Print

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ACCEPTED !

• Now how do we get this design into a
  10x10x12 block of snow ?

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Construction Drawings

• Top View Side View
  Axial View

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Day 1
Removing lots of snow
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Day 1 The Monolith

• Cut away prisms

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Day 2 Making a Torus

• Mark center, circles
  Bulls-eye !

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Chipping away
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Making a Torus
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End of Day 2

• The Torus

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Day 3, am Drawing Flanges
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Day 3, pm Flanges, Holes
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End of Day 3 Proper Topology
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Day 4 Geometry Refinement
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End of Day 4 Desired Geometry
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Day 5, am Surface Refinement
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House Cleaning
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Whirled White Web
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Memories of 2003
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1240 pm -- 42 F
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1241 pm -- 42 F
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124001
Photo StRomain
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3 pm
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(No Transcript)
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The Winners

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

sacred geometry very intricate very 21st
century !
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WWW Wins Silver Medal
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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

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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 ?

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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 !

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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
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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 ! )

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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

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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.

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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.

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Optimal Sphere Eversion

• In the 1990s 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
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The Simplest Polyhedral Model

• Partial cardboard model based on
  cuboctahedron eversion by Apery .

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Shape Adaption for Snow Sculpture

• Restructured Morin surface to fit block size
  (10 x 10 x 12)

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Make Surface Transparent

• Realize surface as a grid.
• Draw a mesh of smooth lines onto the surface

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Gridded Models for Transparency

• 3D-Print from Zcorp

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Turning a Snowball Inside-Out
Carlo H. Séquin, Alex Kozlowski, John
Sullivan Dan Schwalbe, Stan Wagon
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The Final Model
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Morins Surface Eversion
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The Half-way Point
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Finish the Process
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Computer Projections
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Horizontal Slices and Projections
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Practice Block (Stan Wagon)
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First Night
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Working Out Plan B
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Working on the Grid
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Day 1
Day 1
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Day 1
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Shovels, Drills, Pick-axes ...
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End of Day 1
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Day 2
Day 2
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Day 2
A Template for the ear
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Day 2
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Day 2
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Day2
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End of Day 2
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Day 3
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Defining the Grid
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Carving the Grid
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Cleaning Out the Cross-Tunnel
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Day3
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End of Day 3
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Day 4
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Day 4
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Day 5
End of Day 4
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Day 5
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Day 5
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Day 5
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Day 5
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Paradigm Shift (British Columbia)
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Year of the Dragon (USA Tennessee)
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Winter Oasis (Canada Ontario)
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Ceremony
Honorable MentionMost Ambitious Design
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Celebration
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Toasting...
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Day 6
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Questions ?