G4G9 G4G9 The Beauty of Knots EECS Computer Science Division EECS Computer Science Division University of California, Berkeley University of California, Berkeley Carlo H. Séquin
Dec 21, 2015
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”