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

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**SIAM Conf. on Math for Industry, Oct. 10, 2009**Carlo H. Séquin U.C. Berkeley • Modeling Knots for Aesthetics and Simulations Modeling, Analysis, Design …**Knotted Appliances**• Garden hose Power cable**Intricate Knots in the Realm of . . .**• Boats Horses**Knots in Art**• Macrame Sculpture**Knotted Plants**• Kelp Lianas**Knotted Building Blocks of Life**• Knotted DNA Model of the most complex knotted protein (MIT 2006)**Mathematicians’ Knots**unknot • 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**Pax Mundi II (2007)**• Brent Collins, Steve Reinmuth, Carlo Séquin**The Simplest Real Knot: The Trefoil**• José de Rivera, Construction #35 M. C. Escher, Knots (1965)**Composite Knots**• Knots can be “opened” at their periphery and then connected to each other.**Links and Linked Knots**• A link: comprises a set of loops • – possibly knotted and tangled together.**Two Linked Tori: Link 221**John Robinson, Bonds of Friendship (1979)**Borromean Rings: Link 632**John Robinson**Tetra Trefoil Tangles**• Simple linking (1) -- Complex linking (2) • {over-over-under-under} {over-under-over-under}**Realization: Extrude Hone - ProMetal**• Metal sintering and infiltration process**A Split Trefoil**• To open: Rotate around z-axis**Splitting Moebius Bands**• Litho by FDM-model FDM-modelM.C.Escher thin, colored thick**Knotty Problem**• How many crossings • does this “Not-Divided” Knot have ?**Recursive 9-Crossing Knot**9 crossings • Is this really a 81-crossing knot ?**Knot 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”**• Find maximal symmetry in 3D for simple knots. Knot 41 and Knot 61**Computer Representation of Knots**String of piecewise-linear line segments. • Spline representation via its control polygon. But . . .**Is the Control Polygon Representative?**You may construct a nice knotted control polygon,and then find that the spline curve it defines is not knotted at all ! • A Problem:**Unknot With Knotted Control-Polygon**• Composite of two cubic Bézier curves**Highly 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-Polygons**• The Result: • Control polygon has 12 crossings. • Compound Bézier curve is still the unknot!**An Intriguing Question:**First guess: Probably NOT Variation-diminishing property of Bézier curves implies that a spline cannot “wiggle” more than its control polygon. • Can an un-knotted control polygon • produce a knotted spline curve ?**Cubic Bézier and Its Control Polygon**Two “entangled” curves With “non-entangled” control polygons Convex hull of control polygon Region where curve is “outside” of control polygon Cubic Bézier curve**Two “Entangled” Bezier Segments “in 3D”**• NOTE: The 2 control polygons are NOT entangled!**The Building Block**Two “entangled” curves With “non-entangled” control polygons**Combining 4 such Entangled Units**• Use several units …**Control Polygons Are NOT Entangled …**• Use several units …**But This Is a Knot !**Knot 72