490 likes | 508 Views
Breckenridge, CO, January 2004. art & math in snow. Carlo H. Séquin U.C. Berkeley. Memories of 2003. Stan Wagon, Macalester College, St. Paul, MN. Leader of Team “USA – Minnesota”. Monkey Saddle Trefoil. from Sculpture Generator I. “Whirled White Web”.
E N D
Breckenridge, CO, January 2004 art & math in snow Carlo H. Séquin U.C. Berkeley
Stan Wagon, Macalester College, St. Paul, MN • Leader of Team “USA – Minnesota”
Monkey Saddle Trefoil from Sculpture Generator I
“Whirled White Web” 3D-Print FDM
ACCEPTED ! Now – how do we get this design into a 10’x10’x12’ block of snow ?
Construction Drawings Top View Side View Axial View
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).
Day 1 Removing lot’s of snow …
Day 1: The “Monolith” Cut away prisms …
Day 2: Making a Torus Mark center, circles … Bull’s-eye !
Tools and Techniques ... Use of template Need for a sun shield
End of Day 2 The Torus
12:40:01 Photo: StRomain
The Winners 1st: Canada – B.C., 2nd: USA – Minnesota, 3rd: USA – Breckenridge “… sacred geometry … very intricate … very 21st century !”
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 ... – what is that ?? How is this supposed to work ??? Can you really turn a beach ball inside outwithout cutting a hole into it ?
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 !
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
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 ! )
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 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 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.
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”
Gridded Models for Transparency 3D-Print from Zcorp SLIDE virtual model
Shape Adaption for Snow Sculpture Restructured Morin surface to fit block size: (10’ x 10’ x 12’)
Make Surface “Transparent” • Realize surface as a grid. • Draw a mesh of smooth lines onto the surface …
“Turning a Snowball Inside-Out” Carlo H. Séquin, Alex Kozlowski, John Sullivan Dan Schwalbe, Stan Wagon
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.