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

My Background: Geometry ! u Descriptive Geometry – love since high school

Descriptive Geometry

40 Years of Geometry and Design CCD TV Camera Soda Hall RISC 1 Computer Chip Octa-Gear (Cyberbuild)

More Recent Creations

Frank Smullin (1943 – 1983) u Tubular sculptures; u Apple II program for u calculating intersections.

Frank Smullin: u “ 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 Block u 4 tetrahedral links... u like a carbon atom... u can be assembled into a diamond-lattice leads to the “Granny-Knot-Lattice” 

Granny Knot Lattice (1981)

The Strands in the G.K.L.

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

The Process: ( For Scherk-Collins Toroids ) Inspirational Model Generative Paradigm Computer Program Many New Models Insight, Analysis Math, Geometry Selection, Design

Brent Collins: Hyperbolic Hexagon

Scherk’s 2nd Minimal Surface 2 planes: the central core 4 planes: bi-ped saddles 4-way saddles = “Scherk tower”

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

V-art (1999) Virtual Glass Scherk Tower with Monkey Saddles (Radiance 40 hours) Jane Yen

Closing the Loop straight or twisted “Scherk Tower”“Scherk-Collins Toroids”

Sculpture Generator 1, GUI

Shapes from Sculpture Generator 1

Some of the Parameters in “SG1”

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

2003: “Whirled White Web”

Brent Collins and David Lynn

Inauguration Sutardja Dai Hall 2/27/09

Details of Internal Representation u Boundary Representations u Meshes of small triangles defining surface

Base Geometry: One “Scherk Story” u Taylored hyperbolas, hugging a circle u Hyperbolic Slices  Triangle Strips

The Basic Saddle Element with surface normals u precomputed -- then warped into toroid

Shape Generation: u by stacking this basic hyperbolic element, u twisting that stack along z-axis, u bending (warping) it into an arch or loop.

Knot Representations u Knot tables ! u A particular realization of an individual knot is just a closed space curve in 3D space. u It can be represented as a sequence of vertices: V0 (x,y,z); V1 (x,y,z) … u Connected with a poly-line for visualization.

A Simple Tool to Display Knots u B-Splines with their corresponding control-polygons

Knot Representation u Control Polygon of Trefoil Knot: Then just drag this text file onto “KnotView-3D.exe”

Turning Knots into Sculptures u Define a cross-section and sweep it along the given 3D knot curve.

Brent Collins’ Pax Mundi 1997: wood, 30”diam. 2006: Commission from H&R Block, Kansas City to make a 70”diameter version in bronze. My task: to define the master geometry. CAD tools played important role.

How to Model Pax Mundi... u Already addressed that question in 1998: u Pax Mundi could not be done with Sculpture Generator I u Needed a more general program ! u Used the Berkeley SLIDE environment. u First: Needed to find the basic paradigm   

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

4-period “Gabo Curve” Same construction as for as for 2-period curve

Pax Mundi Revisited u Can be seen as: Amplitude modulated, 4-period Gabo curve

SLIDE-GUI for “Pax Mundi” Shapes Good combination of interactive 3D graphics and parameterizable procedural constructs.

2-period Gabo Sculpture Tennis ball – or baseball – seam used as sweep curve.

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

Via Globi 5 (Virtual Wood) Wilmin Martono

Modularity of Gabo Sweep Generator u Sweep Curve Generator: l Gabo Curves as B-splines u Cross Section Fine Tuner: l Paramererized shapes u Sweep / Twist Controller

Sweep / Twist Control u How do we orient, move, scale, morph... the cross section along the sweep path ? Natural orientation with Frenet frame Torsion Minimization: Azimuth: tangential / normal 900° of twist added.

Extension: Free-form Curve on a Sphere Spherical Spline Path Editor (Jane Yen) Smooth interpolating curve through sparse data points

Many Different Viae Globi Models

Paradigm Extension: Sweep Path is no longer confined to a sphere! Music of the Spheres (Brent Collins)

Allows Knotted Sweep Paths Chinese Button Knot

Really Free-form 3D Space Curves Figure-8 knot

The Process: Example: Pax Mundi Wood Pax Mundi Sweep curve on a sphere Via Globi Framework In Slide Bronze Pax Mundi Inspirational Model Generative Paradigm Computer Program Many New Models Insight, Analysis Math, Geometry Selection, Design