Leonardo Meeting, San Francisco, May 12, 2008Leonardo Meeting, San Francisco, May 12, 2008

NaughtyKnotty Sculptures

Carlo H. Séquin

EECS, Computer Science

University of CA, Berkeley

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-Art ConnectionMath-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.

Hamiltonian Cycles on 4D Cross Polytope

Sculptures Made from KnotsSculptures Made from Knots

2004 - 2007:Knots 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: Extrude Hone - ProMetalRealization: Extrude Hone - ProMetal

Metal sintering and infiltration process

Sculptures Made from KnotsSculptures Made from Knots

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

Make aesthetically pleasing artifacts

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

Many Different Ways . . .Many Different Ways . . .

I. Bottom-up knot construction

II. Fusing simple knots together

III. Top-down mesh infilling

IV. Longitudinal knot splitting

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

A plane-filling Peano curve

Do This In 3 D !

““Hilbert” Curve in 3DHilbert” Curve in 3D

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

Recursive 9-Crossing KnotRecursive 9-Crossing Knot

Is this really a 81-crossing knot ?

9 crossings

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

33rdrd Generation of Trefoil-Sierpinsky Generation of Trefoil-Sierpinsky

From Paintings to SculpturesFrom Paintings to Sculptures

Do something like this in 3D !

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

INTERMEZZO:INTERMEZZO:

Homage toHomage toFrank Smullin (1943 – 1983)Frank Smullin (1943 – 1983)

Frank Smullin (1943 – 1983) Frank Smullin (1943 – 1983)

Tubular sculptures;

Apple II program for

calculating intersections.

Frank Smullin (Nashville, 1981):Frank Smullin (Nashville, 1981):

“ 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

Four tetrahedral links, like a carbon atom ...

can be assembled into diamond-lattice ...

... leads to the “Granny-Knot-Lattice”

Smullin: “TetraGranny”

Strands in the Granny-Knot-LatticeStrands in the Granny-Knot-Lattice

Granny-Knot-Lattice (SGranny-Knot-Lattice (Séquin, 1981)quin, 1981)

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

!

A Nice Symmetrical Starting KnotA Nice Symmetrical Starting Knot

Granny Knot with cross-connected ends

4-fold symmetric Knot 819

(3,4) Torus Knot

Recursion StepRecursion Step

Placement of the 8 substitution knots

Establishing ConnectivityEstablishing Connectivity

Grow knots until they almost touch

Work in Progress ...Work in Progress ...

Connectors added to close the knot

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

Recursion on a 7-crossing KnotRecursion on a 7-crossing Knot

Robert Fathauer, Bridges Conference, 2007

...

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

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

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

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

““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”