Knotting Mathematics and Art University of Southern Florida

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) 2004 - 2007: Knots as constructive building blocks.

Tetrahedral Trefoil Tangle (FDM)

Tetra Trefoil Tangles Simple linking (1) {over-under-under} -- Complex linking (2) {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 ü u Take ü 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 - Pro. Metal sintering and infiltration process

Sculptures Made from Knots (2) 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. 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

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

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

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

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. 5 D Celtic Knots – Basic Step

Celtic Knot – Denser Configuration

Celtic Knot – Second Iteration

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

From Paintings to Sculptures Do something like this in 3 D ! 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 3 D; its ends stick out in tetrahedral fashion. . . ” Square Knot Granny Knot

Granny Knot as a Building Block Smullin: “Tetra. Granny” Four tetrahedral links, like a carbon atom. . . can be assembled into diamond-lattice. . . leads to the “Granny-Knot-Lattice”

Strands in the Granny-Knot-Lattice

Granny-Knot-Lattice (Sé (S quin, 1981)

A “Knotty” “ 3 D” 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. o o T h c u m ti y x e l p m o c !

A Nice Symmetrical Starting Knot Granny Knot with cross-connected ends 4 -fold symmetric Knot 819

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 Result after 2 more recursion steps Recursion step Mark crossings over/under to form alternating knot

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

2. 5 D Recursive (Fractal) Knot Trefoil Recursion Robert Fathauer: “Recursive Trefoil Knot”

Recursion on a 7 -crossing Knot . . . Map “the whole thing” into all meshes of similar shape Robert Fathauer, Bridges Conference, 2007

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

Recursive Figure-8 Knot 3 D 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 M. C. Escher FDM-model thin, colored FDM-model 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”