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Leonardo Meeting, San Francisco, May 12, 2008

Explore the fascinating world of knot sculptures designed by Leonardo. Discover the role of computers in aesthetic optimization and the creative process. From math-art connections to rapid prototyping, Leonardo's work showcases the beauty of knots in various forms. Experience the intricacy and complexity of his sculptures made from knots and learn about the different techniques used in their creation.

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Leonardo Meeting, San Francisco, May 12, 2008

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  1. Leonardo Meeting, San Francisco, May 12, 2008 Carlo H. Séquin EECS, Computer Science University of CA, Berkeley • NaughtyKnotty Sculptures

  2. Technical Designs … CCD Camera, Bell Labs, 1973 Soda Hall, Berkeley, 1994 RISC chip, Berkeley, 1981 “Octa-Gear”, Berkeley, 2000

  3. Since 1994: Aesthetic Designs … • What is the role of the computer in: • aesthetic optimization, • the creative process ?

  4. Collaboration with Brent Collins “Hyperbolic Hexagon II”

  5. “Sculpture Generator I ” GUI

  6. Math-Art Connection • When does a mathematical model become a piece of art ?

  7. Rapid Prototyping Model of the 24-Cell • Noticethe 3-foldpermutationof colorsMade on the Z-corp machine.

  8. Hamiltonian Cycles on 4D Cross Polytope

  9. Sculptures Made from Knots • 2004 - 2007:Knots as constructive building blocks.

  10. Tetrahedral Trefoil Tangle (FDM)

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

  12. Tetra Trefoil Tangle • Complex linking (two views)

  13. Platonic Trefoil Tangles • Take a Platonic polyhedron made from triangles, • Add a trefoil knot on every face, • Link with neighboring knots across shared edges.

  14. Icosahedral Trefoil Tangle • Simplest linking (type 1)

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

  16. Arabic Icosahedron

  17. Dodecahedral Pentafoil Cluster

  18. Realization: Extrude Hone - ProMetal • Metal sintering and infiltration process

  19. Sculptures Made from Knots More recently, I have been looking for sculptures where the whole piece is just a single knot. • Generate knots & increase their complexity in a structured, procedural way • Make aesthetically pleasing artifacts

  20. Many Different Ways . . . • I.Bottom-up knot construction • II. Fusing simple knots together • III.Top-down mesh infilling • IV. Longitudinal knot splitting

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

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

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

  24. Metal Sculpture at SIGGRAPH 2006

  25. A Knot Theorist’s View Thus our construction element should use a “more knotted thing”: e.g. an overhand knot: It is still just the un-knot !

  26. Recursion Step • Replace every 90° turn with a knotted elbow.

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

  28. Recursive Cubist Trefoil Knot

  29. A Knot Theorist’s View Thus our assembly step should cause a more serious entanglement: adjacent knots should entangle one another, or crossing strands should be knotted together . . . • This is just a compound-knot ! • It does not really lead to a complex knot !

  30. 2.5D Celtic Knots – Basic Step

  31. Celtic Knot – Denser Configuration

  32. Celtic Knot – Second Iteration

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

  34. Outline • I. Bottom-up knot construction • II.Fusing simple knots together • III.Top-down mesh infilling • IV. Longitudinal knot splitting

  35. Knot-Fusion • Combine 3 trefoils into a 9-crossing knot

  36. 3rd Generation of Trefoil-Sierpinsky

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

  38. INTERMEZZO:Homage toFrank Smullin (1943 – 1983)

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

  40. Frank Smullin (Nashville, 1981): Granny Knot Square Knot • “ The Granny-knot has more artistic merits than the square knot because it is more 3D;its ends stick out in tetrahedral fashion... ”

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

  42. Strands in the Granny-Knot-Lattice

  43. Granny-Knot-Lattice (Séquin, 1981)

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

  45. Next Recursion Step • Substitute the 8 crossings with 8 Granny-knots

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

  47. A Nice Symmetrical Starting Knot 4-fold symmetric Knot 819 (3,4) Torus Knot • Granny Knot with cross-connected ends

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