Art Math X, Boulder, CO, June 2005

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Art Math X, Boulder, CO, June 2005

• Tangled Knots

Carlo H. Séquin U.C. Berkeley
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The idea for this paper ...

• Breckenridge, CO, January 2005,
• International Snowsculpting Championships
• Team Minnesota constructs Knot Divided.
• (S. Wagon, D. Schwalbe, R. Seeley, C. Séquin, J.
  Sullivan)

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Formal Submission, July 2004

• KNOT DIVIDED

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KNOT DIVIDED

• Can a  DIVIDED KNOT  be  NOT DIVIDED ?We start
  with the simplest knot the overhand knot, also
  known as the trefoil or pretzel knot,  -- which
  we then split lengthwise along the whole strand
  that forms the three loops. But there is a
  twist that may lead to surprises the knotted
  strand is a triply twisted Moebius band!Thus
  the question Does our cut separate the
  structure into two pieces, or does it form a
  single, highly knotted strand?

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Final Model

• Thicker beams
• Wider gaps
• Less slope

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Intermediate Construction Model

• The first milestone a Moebius band with 3/2
  flips

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TUESDAY, Jan. 25, 2005
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WEDNESDAY, Jan. 26, 2005
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THURSDAY, Jan. 27, 2005
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FRIDAY, Jan. 28, 2005
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SATURDAY, Jan. 29, 2005
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Knot Divided by Team Minnesota
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What Is It Mathematically ?

• A split, triply-twisted Moebius band
• in the form of a trefoil knot.

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Splitting Moebius Bands

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

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Split Moebius Trefoil (Séquin, 2003)
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What would happen if the original band were
double-sided?

• gt True split into two knots !
• Probably tangled result
• How tangled is it ?
• How much can the 2 parts move ?
• Explore these issues, and others ...

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How to Use Knots in Constructive Ways

• as building blocks ...
• for symmetrical assemblies.
• Explore artistic possibilities !

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Outline of Presentation

• Splitting Orientable Knotted Bands
• Symmetrical Assemblies of Knots
• Infinite Knot Lattices
• Recursive Knots
• Space-Filling Knots

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Is It Math ?Is It Art ?

• it isKNOT-ART

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PART I Splitting Trefoils

• Take a simple trefoil knot
• Split it lengthwise
• See what happens ...

How much wiggle room ?
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Trefoil Stack
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An Iterated Trefoil-Path of Trefoils
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Splitting the Knot into 3 Strands

• 3-deep stack

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Another 3-Way Split

• Parts are different, but maintain 3-fold symmetry

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Split into 3 Congruent Parts

• Change the twist of the configuration!
• Parts no longer have C3 symmetry

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PART II Linking Knots

• Use knots as constructive building blocks

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Tetrahedral Trefoil Tangle (FDM)
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Tetra Trefoil Tangles

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

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Tetra Trefoil Tangle (2)

• Complex linking -- two different views

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Tetra Trefoil Tangle

• Complex linking (two views)

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Octahedral Trefoil Tangle
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Octahedral Trefoil Tangle (1)

• Simplest linking

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Platonic 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 !

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Arabic Icosahedron
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Icosahedral Trefoil Tangle

• Simplest linking (type 1)

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Icosahedral Trefoil Tangle(Type 3)

• Doubly linked with each neighbor

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Arabic Icosahedron, UniGrafix, 1983
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Arabic Icosahedron
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Dodecahedral Pentafoil Cluster
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Realization Extrude Hone - ProMetal

• Metal sintering and infiltration process

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PART III Knot Lattices

• Can we build knot latticesthat are infinitely
  extensible in 3D ?
• This is what got me into 3D Graphics !A talk by
  Sculptor Frank Smullin ...

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Frank Smullin (1943 1983)

• Tubular sculptures
• Apple II program for
• calculating intersections.

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Frank Smullin

• The Granny knot has more artistic merits than
  the square knot because it is more 3Dits ends
  stick out in tetrahedral fashion...

Square Knot
Granny Knot
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Granny Knot as a Building Block

• 4 tetrahedral links ...
• like a carbon atom ...
• can be assembled intoa diamond-lattice ...

... leads to the Granny-Knot-Lattice ?
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Granny Knot Lattice (1981)
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The Strands in the G.K.L.
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The Use of Figure-8 Knots

• Figure-8 knot can also have 4 lobes
• sticking out in tetrahedral directions.

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Figure-8 Knots in Diamond Lattice Cell
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A Denser Lattice of Figure-8 Knots
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Dense Figure-8 Knot Lattice

• Model made with selective laser sintering.

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My Conceptual 3D-CAD Tools
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PART IV Recursive Knots

• Another way of generating aesthetic complexity

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2.5D Celtic Knots Basic Step
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Celtic Knot Denser Configuration
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Celtic Knot Second Iteration
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True 3D Knot Configuration

• Start from cubist trefoil knot . . .

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Recursion Step

• Replace every 90 turn with a knotted elbow.

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Recursive Cubist Trefoil Knot
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PART V Space-filling Knots

• Can we pack knots so tightly
• that they fill all of 3D space ?
• Ian Stewart, Mathematical Recreations,
• Scientific American, Nov. 1995

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4 (convoluted) Trefoils Make a Cube

• Cubes stack up to fill space

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A Simpler, More Elegant Solution

• Three congruent
• interlocking trefoils
• make a hexagonal
• prismatic block.

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A Simpler, More Elegant Solution

• Outer and middle floor plans ?

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Planning the Space Partition of the Tiling
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A Simpler, More Elegant Solution

• Three congruent
• interlocking trefoils
• make a hexagonal
• prismatic block.
• Styrofoam model.

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Minimal Solution

• Squash a trefoil into a 3-sided prism
• or make cuts in a foam-rubber block.

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Minimal Solution

• Make cuts in a foam-rubber block.
• NOT what we think of when we want to fill space
  with knots!
• ? They all should mutually link.

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What We Would Really Like ...

• Stacking cubes or prisms to fill space is a
  cheap way to fill space with knots!
• Neighboring knots should mutually link,so that
  the fabric of space holds together.

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Toroidal Tile (Linking Unknots)

• The Basic Tile

Assembly of 5 Tiles
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Extensible Linkage of Toroidal Tiles
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Conclusions

• Knots are interesting artistic elements in
  themselves.
• They can be split lengthwise to make intriguing
  configurations.
• They can be used as building blocks for
  sophisticated constellations.

Art Math Knot-Art