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