Graphics Lunch, Feb. 5, 2009Graphics Lunch, Feb. 5, 2009

Carlo H. Séquin

Graphics Lunch, Feb. 5, 2009Graphics Lunch, Feb. 5, 2009

NaughtyKnotty Sculptures

Carlo H. Séquin

U.C. Berkeley

Knotty problems in knot theory

NOT This:NOT This:

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

Knots as constructive sculptural building blocks.

Math-Art ConnectionMath-Art Connection

When does a mathematical model

become a piece of art ?

Previous explorations:• Minimal surfaces• Hyperbolic tessellations• 4-dimensional polytopes

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 Tangle (2)Tetra Trefoil Tangle (2)

Complex linking -- two different views

Tetra Trefoil TangleTetra Trefoil Tangle

Complex linking (two views)

Octahedral Trefoil TangleOctahedral Trefoil Tangle

Octahedral Trefoil Tangle (1)Octahedral Trefoil Tangle (1)

Simplest linking

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;explore several different methods… -->

Make aesthetically pleasing artifacts!

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

PART BPART BWays to Make Complicated KnotsWays to Make Complicated Knots

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

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).

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

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)

Splitting Moebius BandsSplitting Moebius Bands

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

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 ?

A More General QuestionA More General Question

Take any knot made from an n-sided prismatic cord.

Split that cord lengthwise into n strands.

Cut the bundle of strands at one point and reconnect,after giving the bundle of n strands a twistequivalent of t strand-spacings (where n, t are mutually prime).

How complex is the resulting knot ?

PART CPART C

Space-filling KnotsSpace-filling Knots

Can we pack knots so tightly

that they fill all of 3D space ?

Ian Stewart, Mathematical Recreations,

Scientific American, Nov. 1995

4 (convoluted) Trefoils Make a Cube4 (convoluted) Trefoils Make a Cube

Cubes stack up to fill space

A Simpler, More Elegant SolutionA Simpler, More Elegant Solution

Three congruent

interlocking trefoils

make a hexagonal

prismatic block.

Isohedral Trefoil-Knot Tile of 3D SpaceIsohedral Trefoil-Knot Tile of 3D Space

What We Would Really Like ...What We Would Really Like ...

Stacking cubes or prisms in 3D space …

is a “cheap” way to fill space with knots!

Neighboring knots should mutually link,

so that the “fabric of space holds together.”

Toroidal Tile (Linking Unknots)Toroidal Tile (Linking Unknots)

Assembly of 5 Tiles

The Basic Tile

Extensible Linkage of Toroidal TilesExtensible Linkage of Toroidal Tiles

The Use of Figure-8 KnotsThe Use of Figure-8 Knots

Figure-8 knot can also have 4 lobes

sticking out in tetrahedral directions.

Figure-8 Knots in Diamond Lattice CellFigure-8 Knots in Diamond Lattice Cell

A Denser Lattice of Figure-8 KnotsA Denser Lattice of Figure-8 Knots

Dense Figure-8 Knot LatticeDense Figure-8 Knot Lattice

Model made with selective laser sintering.

My Conceptual 3D-CAD ToolsMy Conceptual 3D-CAD Tools

PART DPART DRibbed Sculptures Ribbed Sculptures

Suspended in Generating KnotsSuspended in Generating Knots

Example: Charles O. Perry’s Solstice

In Tampa Florida

Work with James Hamlin

Emulation of Perry’s Emulation of Perry’s SolsticeSolstice

Solstice: A ribbed surface in a (3,2) torus knot

Generating PrincipleGenerating Principle

Line swept along a (2,1) torus knot generates a Moebius band.

Hyperbolic triangle swept along a (3,2) torus knot yields Solstice

A Variation of the Generating KnotA Variation of the Generating Knot

Original (3,2) torus knot Modified (4,3) knot

A Variation of the Generating KnotA Variation of the Generating Knot

Original (3,2) torus knot Modified (2,3) knot

… and of the angle offset of the ribs around major circle

ConclusionsConclusions

Knots are mathematically intriguing and they are also inspiring artistic elements.

They can be used as building blocks for sophisticated aesthetic assemblies.

They can be extended recursively to form much more complicated knots.

They can be split lengthwise to make interesting knots and tangles.

They can be used to tile 3D space.

They can be used as the generating frameworkfor large-scale sculpture.

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

it is:

“KNOT-ART”

ChineseChineseButton KnotButton Knot

(Knot 9(Knot 94040))

Bronze, Dec. 2007Bronze, Dec. 2007

Carlo SCarlo Sééquinquin

cast & patina bycast & patina bySteve ReinmuthSteve Reinmuth

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

Carlo SCarlo Sééquinquin

2nd Prize, AMS Exhibit 2009