SIAM Conf. on Math for Industry, Oct. 10, 2009SIAM Conf. on Math for Industry, Oct. 10, 2009

Modeling Knots for Aesthetics and Simulations

Carlo H. Séquin

U.C. Berkeley

Modeling, Analysis, Design …

Knots in Clothing Knots in Clothing

Knotted Appliances Knotted Appliances

Garden hose Power cable

Intricate Knots in the Realm of . . .Intricate Knots in the Realm of . . .

Boats Horses

Knots in ArtKnots in Art

Macrame Sculpture

Knotted PlantsKnotted Plants

Kelp Lianas

Knotted Building Blocks of LifeKnotted Building Blocks of Life

Knotted DNAModel of the most complex knotted protein (MIT 2006)

Mathematicians’ KnotsMathematicians’ Knots

Closed, non-self-intersecting curves in 3D space

0 3 4 6

Tabulated by their crossing-number :

= The minimal number of crossings visible after any deformation and projection

unknot

Various UnknotsVarious Unknots

3D Hilbert Curve (3D Hilbert Curve (SSééquin 2006quin 2006))

Pax Mundi II (Pax Mundi II (20072007))

Brent Collins, Steve Reinmuth, Carlo Séquin

The Simplest Real Knot: The TrefoilThe Simplest Real Knot: The Trefoil

José de Rivera, Construction #35

M. C. Escher, Knots (1965)

Complex, Symmetrical KnotsComplex, Symmetrical Knots

Tight “Braided” KnotsTight “Braided” Knots

Composite Knots Composite Knots

Knots can be “opened” at their periphery and then connected to each other.

Links and Linked KnotsLinks and Linked Knots

A link: comprises a set of loops

– possibly knotted and tangled together.

Two Linked Tori: Link 2Two Linked Tori: Link 22211

John Robinson, John Robinson, Bonds of Friendship (1979)Bonds of Friendship (1979)

Borromean Rings: Link 6Borromean Rings: Link 63322

John Robinson

Tetra Trefoil TanglesTetra Trefoil Tangles

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

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

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

A Loose Tangle of TrefoilsA Loose Tangle of Trefoils

Dodecahedral Pentafoil ClusterDodecahedral Pentafoil Cluster

Realization: Extrude Hone - ProMetalRealization: Extrude Hone - ProMetal

Metal sintering and infiltration process

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)

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” Breckenridge, 2005” Breckenridge, 2005

Knotty ProblemKnotty Problem

How many crossings

does this “Not-Divided” Knot have ?

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

Knot ClassificationKnot Classification

What kind of knot is this ?

Can you just look it up in the knot tables ?

How do you find a projection that yields the minimum number of crossings ?

There is still no completely safe method to assure that two knots are the same.

Project: “Beauty of Knots” Project: “Beauty of Knots”

Find maximal symmetry in 3D for simple knots.

Knot 41 and Knot 61

Computer Representation of KnotsComputer Representation of Knots

Spline representation via its control polygon.

String of piecewise-linear line segments.

But . . .

Is the Control Polygon Representative?Is the Control Polygon Representative?

A Problem:

You may construct a nice knotted control polygon,

and then find that the spline curve it defines

is not knotted at all !

Unknot With Knotted Control-PolygonUnknot With Knotted Control-Polygon

Composite of two cubic Bézier curves

Highly Knotted Control-PolygonsHighly Knotted Control-Polygons

Use the previous configuration as a building block.

Cut open lower left joint between the 2 Bézier segments.

Small changes will keep the control polygons knotted.

Assemble several such constructs in a cyclic compound.

Highly Knotted Control-PolygonsHighly Knotted Control-Polygons

The Result:

Control polygon has 12 crossings.

Compound Bézier curve is still the unknot!

An Intriguing Question:An Intriguing Question:

Can an un-knotted control polygon

produce a knotted spline curve ?

First guess: Probably NOT

Variation-diminishing property of Bézier curvesimplies that a spline cannot “wiggle”

more than its control polygon.

Cubic BCubic Béézier and Its Control Polygonzier and Its Control Polygon

Cubic Bézier curve

Region where curve is “outside” of control polygon

Two “entangled” curves

With “non-entangled” control polygons

Convex hull of control polygon

Two “Entangled” Bezier Segments “in 3D”Two “Entangled” Bezier Segments “in 3D”

NOTE: The 2 control polygons are NOT entangled!

The Building BlockThe Building Block

Two “entangled” curves

With “non-entangled” control polygons

Combining 4 such Entangled UnitsCombining 4 such Entangled Units

Use several units …

Control Polygons Are NOT Entangled …Control Polygons Are NOT Entangled …

Use several units …

Can Be Reduced to the ChordsCan Be Reduced to the Chords

This Is NOT a Knot !This Is NOT a Knot !

But This Is a Knot ! But This Is a Knot !

Knot 72

The ProblemThe Problem

When can we use the control polygon to make reliable predictions about the curve ?

Thus we have a true spline knotwhose control polygon is the unknot !

Tubular NeighborhoodsTubular Neighborhoods

A tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.

(Wikipedia)

( Tom Peters et al.)

Ambient IsotopyAmbient Isotopy

If both the curve and its control polygon lie in the same tubular neighborhood,they have the same topological surroundingsand thus have the same knotted-ness.

( Tom Peters et al.)

subdivided control polygon

A “Safe” Tubular NeighborhoodA “Safe” Tubular Neighborhood

A tube of uniform diameter equal to the minimum separation of any two branches

More Efficient Neighborhoods?More Efficient Neighborhoods?

Tube diameter is determined by tightest bottleneck

Inefficient!

Make tube diameter variable along the knot curve(s)

Difficult!

Another “Neighborhood”Another “Neighborhood”

The notion of the “control ribbon”:

control polygon

spline curve

controlribbon

A ruled surface, that connects points with equal parameter values on the spline and on the control polygon

Knots and Their Control RibbonsKnots and Their Control Ribbons

K31: “Trefoil” and K940: “Chinese Button Knot”

Crucial Test on Control RibbonCrucial Test on Control Ribbon

Any self-intersections ?

Does a Line Pass thru Control Ribbon?Does a Line Pass thru Control Ribbon?

Look at the “crossings”formed by close approaches betweenquery line (green)and the edges ofthe control ribbon.

If the two “crossings” have the same sign,line stabs the ribbon.

Current FocusCurrent Focus

Find out how this can be done most efficiently:

Find the occurrences of all “close approaches”

Determine the signs of the relevant “crossings”

ConclusionConclusion

Knots appear in many domains, in many different forms, and with highly varying degrees of complexity.

CAD tools have only tangentially addressed efficient modeling and analysis of knotted structures.

Suitable abstractions of knots, coupled with some topological guarantees, offer promise for computationally efficient solutions.

The “quest” has only just begun!

AcknowledgementsAcknowledgements

Thanks to Tom Peters for many fruitful discussions!

This work is being supported in part by the Center for Hybrid and Embedded Software Systems (CHESS) at UC Berkeley, which receives support from the National Science Foundation (NSF award #CCR-0225610 (ITR)).

Q U E S T I O N S ?Q U E S T I O N S ?

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