Leonardo Meeting, SETI Institute, Feb. 10, 2010Leonardo Meeting, SETI Institute, Feb. 10, 2010

NaughtyKnotty Sculptures

Carlo H. Séquin

U.C. Berkeley

NOT This:NOT This:

But This: But This: Sculptures Made from Knots Sculptures Made from Knots

Knots as constructive sculptural building blocks.

Technical Designs …Technical Designs …

CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994

RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000

Since 1994: Aesthetic Designs …Since 1994: Aesthetic Designs …

What is the role of the computer in:

aesthetic optimization,

the creative process ?

Collaboration with Brent CollinsCollaboration with Brent Collins

“Hyperbolic Hexagon II”

““Sculpture Generator I Sculpture Generator I ” GUI ” GUI

Math Math Art Connection Art Connection

When does a mathematical model

become a piece of art ?

Rapid Prototyping Model of the 24-CellRapid Prototyping Model of the 24-Cell

Noticethe 3-fold

permutationof colors

Made on the Z-corp machine.

3 Hamiltonian Cycles on 4D Cross Polytope

Hamiltonian Cycles on 4D Cross Polytope

PART APART A

Knots as Constructive Building BlocksKnots as Constructive Building Blocks

Tetrahedral Trefoil Tangle Tetrahedral Trefoil Tangle (FDM)(FDM)

Tetra Trefoil TanglesTetra Trefoil Tangles

Simple linking (1) -- Complex linking (2)

{over-over-under-under} {over-under-over-under}

Tetra Trefoil TangleTetra Trefoil Tangle

Complex linking (two views)

Platonic Trefoil TanglesPlatonic Trefoil Tangles

Take a Platonic polyhedron made from triangles,

Add a trefoil knot on every face,

Link with neighboring knots across shared edges.

Icosahedral Trefoil TangleIcosahedral Trefoil Tangle

Simplest linking (type 1)

Icosahedral Icosahedral Trefoil Trefoil TangleTangle(type 3)(type 3)

Doubly linked with each neighbor

Arabic IcosahedronArabic Icosahedron

Dodecahedral Pentafoil ClusterDodecahedral Pentafoil Cluster

Realization by ProMetal (Ex One Co.)Realization by ProMetal (Ex One Co.)

Metal sintering and infiltration process

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

Undergraduate research group in 2009

More recently, I have been looking for sculptureswhere the whole piece is just a single knot.

Make aesthetically pleasing artifacts!

Classical Knot TablesClassical Knot Tables

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

Knot 5.2Knot 5.2

Knot 6.1Knot 6.1

PART BPART BComputer-Generated KnotsComputer-Generated Knots

I. Bottom-up knot construction

II. Fusing simple knots together

III. Top-down mesh infilling

IV. Longitudinal knot splitting

Generate knots & increase their complexity in a structured, procedural way.

Explore several different methods…

The 2D Hilbert Curve (1891)The 2D Hilbert Curve (1891)

A plane-filling Peano curve

Do This In 3 D !

““Hilbert” Curve in 3D (1999)Hilbert” Curve in 3D (1999)

Start with Hamiltonian path on cube edges and recurse ...

Replaces an “elbow”

Jane Yen: “Jane Yen: “Hilbert Radiator PipeHilbert Radiator Pipe” ” (2000)(2000)

Flaws( from a sculptor’s . point of view ):

4 coplanar segments

Not a closed loop

Broken symmetry

Metal Sculpture at SIGGRAPH 2006Metal Sculpture at SIGGRAPH 2006

A Knot Theorist’s ViewA Knot Theorist’s View

It is still just the un-knot !

Thus our construction element should use a “more knotted thing”:

e.g. an overhand knot:

Recursion StepRecursion Step

Replace every 90° turn with a knotted elbow.

Also: Start from a True KnotAlso: Start from a True Knot

e.g., a “cubist” trefoil knot.

Recursive Cubist Trefoil KnotRecursive Cubist Trefoil Knot

A Knot Theorist’s ViewA Knot Theorist’s View

This is just a compound-knot !

It does not really lead to a complex knot !

Thus our assembly step should cause a more serious entanglement:

adjacent knots should entangle one another, or

crossing strands should be knotted together . . .

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

OutlineOutline

I. Bottom-up knot construction

II. Fusing simple knots together

III. Top-down mesh infilling

IV. Longitudinal knot splitting

Knot-FusionKnot-Fusion

Combine 3 trefoils into a 9-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

From Paintings to SculpturesFrom Paintings to Sculptures

Do something like this in 3D !

Perhaps using two knotted strands(like your shoe laces).

A “Knotty” “3D” Recursion StepA “Knotty” “3D” Recursion Step

Use the Granny knot as a replacement element where two strands cross ...

Next Recursion StepNext Recursion Step

Substitute the 8 crossings with 8 Granny-knots

One More Recursion StepOne More Recursion Step

Now use eight of these composite elements;

connect;

beautify. Too much

com

plexity

!

Too much

com

plexity

!

OutlineOutline

I. Bottom-up knot construction

II. Fusing simple knots together

III. Top-down mesh infilling

IV. Longitudinal knot splitting

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

Recursion stepMark crossings over/under to form alternating knot

Result after 2 more recursion steps

Recursive Figure-8 KnotRecursive Figure-8 Knot

Scale the stroke-width proportional to recursive reduction

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

Robert Fathauer: “Recursive Trefoil Knot”

Trefoil Recursion3 views step

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

OutlineOutline

I. Bottom-up knot construction

II. Fusing simple knots together

III. Top-down mesh infilling

IV. Longitudinal knot splitting

Splitting Moebius BandsSplitting Moebius Bands

Litho by FDM-model FDM-modelM.C.Escher thin, colored thick

A Split TrefoilA Split Trefoil

To open: Rotate one half around z-axis

Split Trefoil (side view, closed)Split Trefoil (side view, closed)

Split Trefoil (side view, open)Split Trefoil (side view, open)

Split Moebius Trefoil (SSplit Moebius Trefoil (Sééquin, 2003)quin, 2003)

““Knot DividedKnot Divided” by Team Minnesota” by Team Minnesota

Knotty ProblemKnotty Problem

How many crossings

does this Not-Divided Knot have ?

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

it is:

“KNOT-ART”

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

Carlo SCarlo Sééquinquin

ChineseChineseButton KnotButton Knot

(Knot 9(Knot 94040))

Bronze, Dec. 2007Bronze, Dec. 2007

Carlo SCarlo Sééquinquin

cast & patina bycast & patina bySteve ReinmuthSteve Reinmuth