URAP, September 16, 2013 Carlo H. Séquin University of California, Berkeley The Beauty of Knots

URAP, September 16, 2013URAP, September 16, 2013

Carlo H. Séquin

University of California, Berkeley

The Beauty of Knots

My Background: Geometry ! My Background: Geometry !

Descriptive Geometry – love since high school

Descriptive GeometryDescriptive Geometry

40 Years of Geometry and Design40 Years of Geometry and Design

CCD TV Camera Soda Hall

RISC 1 Computer Chip Octa-Gear (Cyberbuild)

More Recent CreationsMore Recent Creations

Frank Smullin (1943 – 1983) Frank Smullin (1943 – 1983)

Tubular sculptures;

Apple II program for

calculating intersections.

Frank Smullin:Frank Smullin: “ The Granny knot has more artistic merits

than the square knot because it is more 3D;its ends stick out in tetrahedral fashion... ”

Square Knot Granny Knot

Granny Knot as a Building BlockGranny Knot as a Building Block

4 tetrahedral links ...

like a carbon atom ...

can be assembled intoa diamond-lattice ...

... leads to the “Granny-Knot-Lattice”

Granny Knot Lattice (1981)Granny Knot Lattice (1981)

The Strands in the G.K.L.The Strands in the G.K.L.

Capturing Geometry ProcedurallyCapturing Geometry Procedurally

Collaboration with sculptor Brent Collins: “Hyperbolic Hexagon” 1994 “Hyperbolic Hexagon II”, 1996 “Heptoroid”, 1998

The Process: (The Process: (For Scherk-Collins ToroidsFor Scherk-Collins Toroids))

InspirationalModel

GenerativeParadigm

ComputerProgram

Many NewModels

Insight,Analysis

Math,Geometry

Selection,Design

Brent Collins: Brent Collins: Hyperbolic HexagonHyperbolic Hexagon

ScherkScherk’’s 2nd Minimal Surfaces 2nd Minimal Surface

2 planes: the central core 4 planes:bi-ped saddles 4-way saddles

= “Scherk tower”

ScherkScherk’’s 2nd Minimal Surfaces 2nd Minimal Surface

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)“Scherk Tower”

V-artV-art(1999)(1999)

VirtualGlassScherkTowerwithMonkeySaddles

(Radiance 40 hours)

Jane Yen

Closing the LoopClosing the Loop

straight

or

twisted

“Scherk Tower” “Scherk-Collins Toroids”

Sculpture Generator 1Sculpture Generator 1, GUI , GUI

Shapes from Shapes from Sculpture Generator 1Sculpture Generator 1

Some of the Parameters in “SG1”Some of the Parameters in “SG1”

The Finished The Finished HeptoroidHeptoroid

at Fermi Lab Art Gallery (1998).

2003: 2003: ““Whirled White WebWhirled White Web””

Brent Collins and David LynnBrent Collins and David Lynn

Inauguration Sutardja Dai Hall 2/27/09Inauguration Sutardja Dai Hall 2/27/09

Details of Internal RepresentationDetails of Internal Representation

Boundary Representations

Meshes of small triangles defining surface

Base Geometry: One “Scherk Story”Base Geometry: One “Scherk Story”

Taylored hyperbolas, hugging a circle

Hyperbolic Slices Triangle Strips

The Basic Saddle ElementThe Basic Saddle Elementwith surface normals

precomputed -- then warped into toroid

Shape Generation:Shape Generation: by stacking this basic hyperbolic element,

twisting that stack along z-axis,

bending (warping) it into an arch or loop.

Knot RepresentationsKnot Representations

Knot tables !

A particular realization of an individual knotis just a closed space curve in 3D space.

It can be represented as a sequence of vertices: V0 (x,y,z); V1 (x,y,z) …

Connected with a poly-line for visualization.

A Simple Tool to Display KnotsA Simple Tool to Display Knots

http://www.cs.berkeley.edu/~sequin/X/Knot-View/

B-Splines with their corresponding control-polygons

Knot RepresentationKnot Representation

Control Polygon of Trefoil Knot:

10.0 -2.0 4.0-6.732 7.66 -4.0-6.732 -7.66 4.0 10.0 2.0 -4.0-3.268 9.66 4.0-3.268 -9.66 -4.0

Then just drag this text file onto “KnotView-3D.exe”

Turning Knots into SculpturesTurning Knots into Sculptures

Define a cross-section and sweep it along the given 3D knot curve.

Brent Collins’ Brent Collins’ Pax MundiPax Mundi

1997: wood, 30”diam.1997: wood, 30”diam.

2006: Commission from H&R Block, Kansas Cityto make a 70”diameter

version in bronze.

My task: to define the master geometry.

CAD tools played important role.

How to Model How to Model Pax MundiPax Mundi ... ...

Already addressed that question in 1998:

Pax Mundi could not be done with Sculpture Generator I

Needed a more general program !

Used the Berkeley SLIDE environment.

First: Needed to find the basic paradigm

Sculptures by Naum GaboSculptures by Naum Gabo

Pathway on a sphere:

Edge of surface is like seam of tennis- or base-ball;

2-period Gabo curve.

2-period “Gabo Curve”2-period “Gabo Curve”

Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used (symmetry!).

4-period “Gabo Curve”4-period “Gabo Curve”

Same construction as for as for 2-period curve

Pax MundiPax Mundi Revisited Revisited

Can be seen as:

Amplitude modulated, 4-period Gabo curve

SLIDE-GUI for “SLIDE-GUI for “Pax MundiPax Mundi” Shapes” ShapesGood combination of interactive 3D graphicsand parameterizable procedural constructs.

2-period Gabo Sculpture2-period Gabo Sculpture

Tennis ball – or baseball –

seam used as

sweep curve.

Viae Globi Viae Globi Family Family (Roads on a Sphere)(Roads on a Sphere)

2 3 4 5 periods

Via Globi 5Via Globi 5 (Virtual Wood) (Virtual Wood)

Wilmin Martono

Modularity of Modularity of Gabo Sweep GeneratorGabo Sweep Generator

Sweep Curve Generator:

Gabo Curves as B-splines

Cross Section Fine Tuner:

Paramererized shapes

Sweep / Twist Controller

Sweep / Twist ControlSweep / Twist Control

How do we orient, move, scale, morph ...the cross section along the sweep path ?

Natural orientationwith Frenet frame

Torsion Minimization:Azimuth: tangential / normal

900° of twistadded.

Extension:Extension: Free-form Curve on a Sphere Free-form Curve on a Sphere

Spherical Spline Path Editor (Jane Yen)

Smooth interpolating curve through sparse data points

Many Different Many Different Viae GlobiViae Globi ModelsModels

Paradigm Extension:Paradigm Extension: Sweep Path Sweep Path is no longer confined to a sphere!is no longer confined to a sphere!

Music of the Spheres (Brent Collins)

Allows Knotted Sweep PathsAllows Knotted Sweep Paths

Chinese Button Knot

Really Free-form 3D Space CurvesReally Free-form 3D Space Curves

Figure-8 knot

The Process: Example: The Process: Example: Pax MundiPax Mundi

WoodPax Mundi

Sweep curve on a

sphere

ViaGlobi

FrameworkIn Slide

Bronze Pax Mundi

InspirationalModel

GenerativeParadigm

ComputerProgram

Many NewModels

Insight,Analysis

Math,Geometry

Selection,Design