Meeting Alhambra, Granada 2003Meeting Alhambra, Granada 2003

“Whirled White Web”

Art and Math in Snow

Carlo H. Séquin

Brent Collins, Steve Reinmuth

Dan Schwalbe, Stan Wagon

Brent CollinsBrent Collins

“Hyperbolic Hexagon II”

Brent Collins: Stacked SaddlesBrent Collins: Stacked Saddles

Scherk’s 2nd Minimal SurfaceScherk’s 2nd Minimal Surface

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)

““Hyperbolic Hexagon” by B. CollinsHyperbolic Hexagon” by B. Collins

6 saddles in a ring

6 holes passing through symmetry plane at ±45º

= “wound up” 6-story Scherk tower

Discussion: What if … we added more stories ?

or introduced a twist before closing the ring ?

Closing the LoopClosing the Loop

straight

or

twisted

Brent Collins’ Prototyping ProcessBrent Collins’ Prototyping Process

Armature for the "Hyperbolic Heptagon"

Mockup for the "Saddle Trefoil"

Time-consuming ! (1-3 weeks)

““Sculpture Generator I”, GUI Sculpture Generator I”, GUI

V-artV-art

VirtualGlassScherkTowerwithMonkeySaddles

(Radiance 40 hours)

Jane Yen

Collins’ Fabrication ProcessCollins’ Fabrication Process

Example: “Vox Solis”

Layered laminated main shapeWood master pattern

for sculpture

Slices through “Minimal Trefoil”Slices through “Minimal Trefoil”

50% 10%23%30%

45% 5%20%27%

35% 2%15%25%

One thick slicethru sculpture,from which Brent can cut boards and assemble a rough shape.

Traces represent: top and bottom,as well as cuts at 1/4, 1/2, 3/4of one board.

Profiled Slice through “Heptoroid”Profiled Slice through “Heptoroid”

Emergence of the “Heptoroid” (1)Emergence of the “Heptoroid” (1)

Assembly of the precut boards

Emergence of the “Heptoroid” (2)Emergence of the “Heptoroid” (2)

Forming a continuous smooth edge

Emergence of the “Heptoroid” (3)Emergence of the “Heptoroid” (3)

Smoothing the whole surface

The Finished The Finished “Heptoroid”“Heptoroid”

at Fermi Lab Art Gallery (1998).

Various “Scherk-Collins” SculpturesVarious “Scherk-Collins” Sculptures

Hyper-Sculpture: “Family of 12 Trefoils”Hyper-Sculpture: “Family of 12 Trefoils”

W=2

W=1

B=1 B=2 B=3 B=4

““Cohesion”Cohesion”

SIGGRAPH’2003 Art Gallery

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

Leader of Team “USA – Minnesota”

Stan WagonStan Wagon

Skier – Mountaineer – Snow Sculptor

Dan SchwalbeDan Schwalbe

Software Engineer – Snow Sculptor

Helaman FergusonHelaman Ferguson

Umbilic Torus Costa Surface

Breckenridge, 1999Breckenridge, 1999

Helaman Ferguson: “Invisible Handshake”

Breckenridge, CO, 1999Breckenridge, CO, 1999

Helaman Ferguson: “Invisible Handshake”

Robert LonghurstRobert Longhurst

Moebius Band Enneper Surface

Breckenridge, Breckenridge, 20002000

Robert Longhurst:

“Rhapsody in White”

2nd Place

Breckenridge,Breckenridge,20012001

Robert Longhurst:

“White Narcissus”

Batsheba GrossmanBatsheba Grossman

Antipot Antichron

Breckenridge, Breckenridge, 20022002

Bathsheba Grossman:

“A Twist in Time”

Honorable Mention

“Expressive Impact”

Snow-Sculpting, Breckenridge, 2003Snow-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 ReinmuthSteve Reinmuth

Collins’ Initial Suggestion: Minimal TrefoilCollins’ Initial Suggestion: Minimal Trefoil

Stan’s Objection: “Too simple …– we know we can do this.

No risk – no fun !”

Monkey Saddle TrefoilMonkey Saddle Trefoil

from Sculpture Generator I

MaquettesMaquettes

3D-Print FDM

Name, StoryName, Story

“Snow Flower, Winter Rose, Winter Whirl, Wild White Whirl, Webbed Wild Whirl, Whirled Wild Web …”

finally the perfect homonym:

“Whirled White Web”

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 !ACCEPTED !

Now – how do we get this design into a 10’x10’x12’ block of snow ?

Construction DrawingsConstruction Drawings

Top View Side View Axial View

Plan of AttackPlan of Attack

Cut away 2 large triangular prisms;

Hang plastic template (12’x18’) over slab;

Mark projections of holes, flanges;

Drill, carve, refine …(using maquette for reference)

Try this first on a smaller practice block!

1:1 Templates1:1 Templates

Magnified from computer prints with the classical grid method

Practice Block (1)Practice Block (1)

Template transfer Reference holes

Practice Block (2)Practice Block (2)

Free-hand carving Irregular ribs

Practice Block (3)Practice Block (3)

Need new plan !

Removing lots of snow …

Day 1Day 1

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

Cut away prisms …

Day 2: Making a TorusDay 2: Making a Torus

Mark center, circles … Bull’s-eye !

Chipping away …Chipping away …

Tools, TemplatesTools, Templates

Making a TorusMaking a Torus

Use of template Need for a sun shield

End of Day 2End of Day 2

The Torus

Day 3, am: Drawing FlangesDay 3, am: Drawing Flanges

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

End of Day 3: Proper TopologyEnd of Day 3: Proper Topology

Day 4: Geometry RefinementDay 4: Geometry Refinement

End of Day 4: Desired GeometryEnd of Day 4: Desired Geometry

Day 5, am: Surface RefinementDay 5, am: Surface Refinement

““House Cleaning”House Cleaning”

““Whirled White Web”Whirled White Web”

Official Team PhotoOfficial Team Photo

12:40 pm -- 4212:40 pm -- 42° F° F

12:41 pm -- 4212:41 pm -- 42° F° F

12:40:0112:40:01

Photo:StRomain

12:50 pm12:50 pm

3 pm3 pm

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

British Columbia: “Winter Comes” British Columbia: “Winter Comes”

Day 2 Day 3

Breckenridge: “A Fishing Tail”Breckenridge: “A Fishing Tail”

Mexico City: “Capilla Posa”Mexico City: “Capilla Posa”

China: “The Love of Mother”China: “The Love of Mother”

Quebec: “RQuebec: “Rèève’Olution”ve’Olution”

Manitoba: “Birth of a Nation”Manitoba: “Birth of a Nation”

USA – New York: “94 Hour Photo”USA – New York: “94 Hour Photo”

What Are We Going To Do For 2004 ?What Are We Going To Do For 2004 ?

“Turning a Snowball Inside Out”

Sphere EversionSphere Eversion

~ 1960, the blind mathematician B. Morin, (born 1931) conceived of a way how a sphere can be turned inside-out:

Surface may pass through itself,

but no ripping, puncturing, creasing allowed,e.g., this is not an acceptable solution:

PINCH

Morin SurfaceMorin Surface But there are more contorted paths

that can achieve the desired goal.

The Morin surface is the half-way point of one such path:

John Sullivan: “The Optiverse”

Simplest ModelSimplest Model

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

Gridded Models for TransparencyGridded Models for Transparency

3D-Print from Zcorp SLIDE virtual model

Shape Adaption for Snow SculptureShape Adaption for Snow Sculpture

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

Make Surface “Transparent”Make Surface “Transparent”

Realize surface as a grid.

Draw a mesh of smooth lines onto the surface …

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

Carlo H. Séquin, Alex Kozlowski, John Sullivan

Dan Schwalbe, Stan Wagon

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