G4G9G4G9

The Beauty of Knots

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Carlo H. Séquin

Classical Knot TablesClassical Knot Tables

Flat (2.5D), uninspiring, lack of symmetry …

Trefoil KnotTrefoil Knot

Figure-8 KnotFigure-8 KnotBronze, Dec. 2007Bronze, Dec. 2007

Carlo SCarlo Sééquinquin

2nd Prize, AMS Exhibit 2009

““The Beauty of Knots”The Beauty of Knots”

Undergraduate research group in 2009

What is the most symmetrical configuration?

What is the most 3-dimensional configuration?

Make aesthetically pleasing artifacts!

Some Results Emphasizing SymmetrySome Results Emphasizing Symmetry

Knot 77 Knot 74

Knot TransformationsKnot Transformations

Bring out special, desirable qualities: A graceful “evenly-spaced” curve:

Minimize electrostatic repulsion potential on a flexible wire.

Tightest configuration: Pull tight a rope of fixed diameter without self-intersections.

The least “wiggly” curve: Minimize the arc-length integral of curvature squared.

The most 3D-filling configuration:Wrap knot around a sphere or a cylinder;Turn configuration inside out (point inversion);Play with wires, alu-foil, pipe cleaners!

Knot 5Knot 522

Knot 6Knot 611

““Signature Knot” for G4G9Signature Knot” for G4G9

Has to be a 9-crossing knot … -- but which one ?

““Signature Knot” for G4G9Signature Knot” for G4G9

… a 9-crossing knot:

Knot 940

the same Knot !

It has 3-fold symmetry!

Knot 9Knot 94040: “Chinese Button Knot”: “Chinese Button Knot”

It has interesting 3D properties !

Knot 9Knot 94040: Chinese Button Knot: Chinese Button Knot

Knot 9Knot 94040: Chinese Button Knot: Chinese Button Knot

ChineseChineseButton KnotButton Knot

(Knot 9(Knot 94040))

Bronze, Dec. 2007Bronze, Dec. 2007

Carlo SCarlo Sééquinquin

cast & patina bycast & patina bySteve ReinmuthSteve Reinmuth

Knot 9Knot 94040 in Ribbon Form in Ribbon Form

Will be the subject of some hands-on “constructivist activities” on Fri./Sat. pm.

From Simple Knots to Complicated KnotsFrom Simple Knots to Complicated Knots

“Hilbert Cube 512” – looks complicated … but it is not; -- just a simple, unknotted loop!

Generating Complicated KnotsGenerating Complicated Knots

Is there a procedure to make knots of arbitrary complexity…?

Perhaps by fusing simple knots together…

Perhaps by applying recursive techniques…

Start with: 2.5D - Celtic Knots

2.5D Celtic Knots – Basic Step2.5D Celtic Knots – Basic Step

Celtic Knot – Denser ConfigurationCeltic Knot – Denser Configuration

Celtic Knot – Second IterationCeltic Knot – Second Iteration

Another Approach: Knot-FusionAnother Approach: Knot-Fusion

Combine 3 trefoils into a 12-crossing knot

Sierpinski Trefoil KnotSierpinski Trefoil Knot

Close-up of Sierpinski Trefoil KnotClose-up of Sierpinski Trefoil Knot

33rdrd Generation of Sierpinski Knot Generation of Sierpinski Knot

Another Approach: Mesh-InfillingAnother Approach: Mesh-Infilling

Robert Fathauer, Bridges Conference, 2007

...

Map “the whole thing” into all meshes of similar shape

2.5D Recursive (Fractal) Knot2.5D Recursive (Fractal) Knot

Robert Fathauer: “Recursive Trefoil Knot”

Trefoil Recursion3 views step

Recursive Figure-8 Knot Recursive Figure-8 Knot (4 crossings)(4 crossings)

Recursion stepMark crossings over/under, form alternating knot

Result after 2 more recursion steps

Recursive Figure-8 KnotRecursive Figure-8 Knot

Scale the stroke-width proportional to recursive reduction

From 2D Drawings to 3D SculptureFrom 2D Drawings to 3D Sculpture

Too flat ! Switch plane orientations

Recursive Figure-8 Knot 3DRecursive Figure-8 Knot 3D

Maquette emerging from FDM machine

Recursive Recursive Figure-8 KnotFigure-8 Knot

9 loop iterations

Is It Math ?Is It Math ?Is It Art ?Is It Art ?

it is:

“KNOT-ART”