Knotting Mathematics and Art University of Southern Florida, Nov.3, 2007 Naughty Knotty Sculptures Carlo H. Séquin U.C. Berkeley  Knotty problems in knot theory

Sculptures Made from Knots (1) : Knots as constructive building blocks.

Tetrahedral Trefoil Tangle (FDM)

Tetra Trefoil Tangles Simple linking (1) -- Complex linking (2) {over-over-under-under} {over-under-over-under}

Tetra Trefoil Tangle (2) Complex linking -- two different views

Tetra Trefoil Tangle Complex linking (two views)

Octahedral Trefoil Tangle

Octahedral Trefoil Tangle (1) Simplest linking

Platonic Trefoil Tangles u Take a Platonic polyhedron made from triangles, u Add a trefoil knot on every face, u Link with neighboring knots across shared edges. u Tetrahedron, Octahedron,... done !

Icosahedral Trefoil Tangle Simplest linking (type 1)

Icosahedral Trefoil Tangle (type 3) Doubly linked with each neighbor

Arabic Icosahedron

Dodecahedral Pentafoil Cluster

Realization: Extrude Hone - ProMetal Metal sintering and infiltration process

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 sculptures where the whole piece is just a single knot and which also involve some “interesting” knots.

Outline I. Bottom-up assembly of knots II. Top-down mesh infilling III. Longitudinal knot splitting

The 2D Hilbert Curve (1891) A plane-filling Peano curve Do This In 3 D !

“Hilbert” Curve in 3D Start with Hamiltonian path on cube edges and recurse... Replaces an “elbow”

Jane Yen: “Hilbert Radiator Pipe” (2000) Flaws ( from a sculptor’s. point of view ): 4 coplanar segments Not a closed loop Broken symmetry

Metal Sculpture at SIGGRAPH 2006

A 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 Step Replace every 90° turn with a knotted elbow.

Also: Start from a True Knot e.g., a “cubist” trefoil knot.

Recursive Cubist Trefoil Knot

A 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 Step

Celtic Knot – Denser Configuration

Celtic Knot – Second Iteration

Recursive 9-Crossing Knot Is this really a 81-crossing knot ? 9 crossings

From Paintings to Sculptures Do something like this in 3D ! Perhaps using two knotted strands (like your shoe laces).

INTERMEZZO: Homage to Frank Smullin (1943 – 1983)

Frank Smullin (1943 – 1983) Tubular sculptures; Apple II program for calculating intersections.

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 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-Lattice

Granny-Knot-Lattice (Squin, 1981) Granny-Knot-Lattice (Séquin, 1981)

A “Knotty” “3D” Recursion Step Use the Granny knot as a replacement element where two strands cross...

Next Recursion Step Substitute the 8 crossings with 8 Granny-knots

One More Recursion Step Now use eight of these composite elements; connect; beautify. Too much complexity !

A Nice Symmetrical Starting Knot Granny Knot with cross-connected ends 4-fold symmetric Knot 8 19

Recursion Step Placement of the 8 substitution knots

Establishing Connectivity Grow knots until they almost touch

Work in Progress... Connectors added to close the knot

Outline I. Bottom-up assembly of knots II. Top-down mesh infilling III. Longitudinal knot splitting

Recursive Figure-8 Knot Recursion step Mark crossings over/under to form alternating knot Result after 2 more recursion steps

Recursive Figure-8 Knot Scale stroke-width proportional to recursive reduction

2.5D Recursive (Fractal) Knot Robert Fathauer: “Recursive Trefoil Knot” Trefoil Recursion

Recursion on a 7-crossing Knot Robert Fathauer, Bridges Conference, Map “the whole thing” into all meshes of similar shape

From 2D Drawings to 3D Sculpture Too flat ! Switch plane orientations

Recursive Figure-8 Knot 3D Maquette emerging from FDM machine

Recursive Figure-8 Knot 9 loop iterations

Outline I. Bottom-up assembly of knots II. Top-down mesh infilling III. Longitudinal knot splitting

A Split Trefoil To open: Rotate around z-axis

Split Trefoil (side view, closed)

Split Trefoil (side view, open)

Another Split Trefoil How much “wiggle room” is there ?

Trefoil “Harmonica”

An Iterated Trefoil-Path of Trefoils

Splitting Moebius Bands Litho by FDM-model FDM-model M.C.Escher thin, colored thick

Split Moebius Trefoil (Séquin, 2003)

“Knot Divided” by Team Minnesota

Knotty Problem How many crossings does this Not-Divided Knot have ?

A More General Question u Take any knot made from an n-sided prismatic cord. u Split that cord lengthwise into n strands. u Cut the bundle of strands at one point and reconnect, after giving the bundle of n strands a twist equivalent of t strand-spacings (where n, t are mutually prime). u How complex is the resulting knot ?

Conclusions u Knots are mathematically intriguing and they are inspiring artistic elements. u They can be used as building blocks for sophisticated constellations. u They can be extended recursively to form much more complicated knots. u They can be split lengthwise to make interesting knots and tangles.

Is It Math ? Is It Art ? it is: “KNOT-ART”