Knotting Mathematics and ArtKnotting Mathematics and Art University of Southern Florida, Nov.3, 2007University of Southern Florida, Nov.3, 2007

NaughtyKnotty Sculptures

Carlo H. Séquin

U.C. Berkeley

Knotty problems in knot theory

Sculptures Made from Knots (1)Sculptures Made from Knots (1)

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 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.

Tetrahedron, Octahedron, ... done !

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 Knots (2)Sculptures Made from Knots (2)

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

I. Bottom-up assembly of knots

II. Top-down mesh infilling

III. Longitudinal knot splitting

Make aesthetically pleasing artifacts

For this conference I have been looking for sculptureswhere the whole piece is just a single knot and

which also involve some “interesting” knots.

OutlineOutline

I. Bottom-up assembly of knots

II. Top-down mesh infilling

III. 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:

Perhaps knotting together crossing strands . . .

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

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

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 assembly of knots

II. Top-down mesh infilling

III. Longitudinal knot splitting

Recursive Figure-8 KnotRecursive Figure-8 Knot

Recursion stepMark crossings over/under to form alternating knot

Result after 2 more recursion steps

Recursive Figure-8 KnotRecursive Figure-8 Knot

Scale stroke-width proportional to recursive reduction

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

Robert Fathauer: “Recursive Trefoil Knot”

Trefoil Recursion

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 assembly of knots

II. Top-down mesh infilling

III. Longitudinal knot splitting

A Split TrefoilA Split Trefoil

To open: Rotate around z-axis

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

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

Another Split TrefoilAnother Split Trefoil

How much “wiggle room” is there ?

Trefoil “Harmonica”Trefoil “Harmonica”

An Iterated Trefoil-Path of TrefoilsAn Iterated Trefoil-Path of Trefoils

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 ?

ConclusionsConclusions

Knots are mathematically intriguing and they are inspiring artistic elements.

They can be used as building blocks for sophisticated constellations.

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

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

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

it is:

“KNOT-ART”