Meeting Alhambra, Granada 2003 “Whirled White Web” Art and Math in Snow Carlo H. Séquin Brent Collins, Steve Reinmuth Dan Schwalbe, Stan Wagon

Brent Collins “Hyperbolic Hexagon II”

Brent Collins: Stacked Saddles

Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles (monkey saddle)

“Hyperbolic Hexagon” by B. Collins u 6 saddles in a ring u 6 holes passing through symmetry plane at ±45º u = “wound up” 6-story Scherk tower u Discussion: What if … l we added more stories ? l or introduced a twist before closing the ring ?

Closing the Loop straight or twisted

Brent Collins’ Prototyping Process Armature for the "Hyperbolic Heptagon" Mockup for the "Saddle Trefoil" Time-consuming ! (1-3 weeks)

“Sculpture Generator I”, GUI

V-art Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

Collins’ Fabrication Process Example: “Vox Solis” Layered laminated main shape Wood master pattern for sculpture

Slices through “Minimal Trefoil” 50%10%23%30% 45%5%20%27% 35%2%15%25%

u One thick slice thru sculpture, from which Brent can cut boards and assemble a rough shape. u Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board. Profiled Slice through “Heptoroid”

Emergence of the “Heptoroid” (1) Assembly of the precut boards

Emergence of the “Heptoroid” (2) Forming a continuous smooth edge

Emergence of the “Heptoroid” (3) Smoothing the whole surface

The Finished “Heptoroid” u at Fermi Lab Art Gallery (1998).

Various “Scherk-Collins” Sculptures

Hyper-Sculpture: “Family of 12 Trefoils” W=2 W=1 B=1 B=2 B=3 B=4

“Cohesion” SIGGRAPH’2003 Art Gallery

Stan Wagon, Macalester College, St. Paul, MN u Leader of Team “USA – Minnesota”

Stan Wagon u Skier – Mountaineer – Snow Sculptor

Dan Schwalbe u Software Engineer – Snow Sculptor

Helaman Ferguson Umbilic Torus Costa Surface

Breckenridge, 1999 Helaman Ferguson: “Invisible Handshake”

Breckenridge, CO, 1999 Helaman Ferguson: “Invisible Handshake”

Robert Longhurst Moebius Band Enneper Surface

Breckenridge, 2000 Robert Longhurst: “Rhapsody in White” 2 nd Place

Breckenridge, 2001 Robert Longhurst: “White Narcissus”

Batsheba Grossman Antipot Antichron

Breckenridge, 2002 Bathsheba Grossman: “A Twist in Time” Honorable Mention “Expressive Impact”

Snow-Sculpting, Breckenridge, 2003 Brent Collins and Carlo Séquin are invited to join the team and to provide a design. Other Team Members: Stan Wagon, Dan Schwalbe, Steve Reinmuth (= Team “Minnesota”)

Steve Reinmuth

Collins’ Initial Suggestion: Minimal Trefoil Stan’s Objection: “Too simple … – we know we can do this. No risk – no fun !”

Monkey Saddle Trefoil from Sculpture Generator I

Maquettes 3D-Print FDM

Name, Story u “Snow Flower, Winter Rose, Winter Whirl, Wild White Whirl, Webbed Wild Whirl, Whirled Wild Web …” u finally the perfect homonym: “Whirled White Web” u Like this global network, the ridges of our sculpture span the outer perimeters of the whole “globe,” and at the same time come close together in the central hole. It illustrates how the WWW can link together people from all over the world.

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 u Cut away 2 large triangular prisms; u Hang plastic template (12’x18’) over slab; u Mark projections of holes, flanges; u Drill, carve, refine … (using maquette for reference) u Try this first on a smaller practice block!

1:1 Templates Magnified from computer prints with the classical grid method

Practice Block (1) Template transfer Reference holes

Practice Block (2) Free-hand carving  Irregular ribs

Practice Block (3)  Need new plan !

Removing lots of snow … Day 1

Day 1: The “Monolith” Cut away prisms …

Day 2: Making a Torus Mark center, circles … Bull’s-eye !

Chipping away …

Tools, Templates

Making a Torus Use of template Need for a sun shield

End of Day 2 The Torus

Day 3, am: Drawing Flanges

Day 3, pm: Flanges, Holes

End of Day 3: Proper Topology

Day 4: Geometry Refinement

End of Day 4: Desired Geometry

Day 5, am: Surface Refinement

“House Cleaning”

“Whirled White Web”

Official Team Photo

12:40 pm -- 42° F

12:41 pm -- 42° F

12:40:01 Photo: StRomain

12:50 pm

3 pm

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

British Columbia: “Winter Comes”

Day 2 Day 3

Breckenridge: “A Fishing Tail”

Mexico City: “Capilla Posa”

China: “The Love of Mother”

Quebec: “Rève’Olution”

Manitoba: “Birth of a Nation”

USA – New York: “94 Hour Photo”

What Are We Going To Do For 2004 ? “Turning a Snowball Inside Out”

Sphere Eversion u ~ 1960, the blind mathematician B. Morin, (born 1931) conceived of a way how a sphere can be turned inside-out: l Surface may pass through itself, l but no ripping, puncturing, creasing allowed, e.g., this is not an acceptable solution: PINCH

Morin Surface u But there are more contorted paths that can achieve the desired goal. u The Morin surface is the half-way point of one such path: John Sullivan: “The Optiverse”

Simplest Model Partial cardboard model based on the simplest polyhedral sphere (= cuboctahedron) eversion.

Gridded Models for Transparency 3D-Print from ZcorpSLIDE virtual model

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

Make Surface “Transparent” u Realize surface as a grid. u 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

Our Submission: 7/1/03 QUESTIONS ? -- DISCUSSION ?