1. Leonardo Meeting, SETI Institute, Feb. 10, 2010
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Leonardo Meeting, SETI Institute, Feb. 10, 2010
Naughty Knotty Sculptures
I hope that when you heard about this talk – you did not come with the wrong expectations ...
Carlo H. Séquin
U.C. Berkeley

2. NOT This:
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So this talk is not about this: …

3. But This: Sculptures Made from Knots
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But This: Sculptures Made from Knots
But about this : Sculptures made from knots,
which in turn may raise some knotty problems in knot theory.
Knots as constructive sculptural building blocks.

4. Florida 1999
Technical Designs …
CCD Camera, Bell Labs, Soda Hall, Berkeley, 1994
My background: I have loved geometry since high school, and have been involved in CAGD for the last 30 years. At Bell Labs…
RISC chip, Berkeley, “Octa-Gear”, Berkeley, 2000

5. Since 1994: Aesthetic Designs …
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What is the role of the computer in:
aesthetic optimization,
the creative process ?
Since the mid 1990 also involved in Aesthetic design, and have interactive with several artists.
I tried to find some answers to questions like …
What role do computers have in the design of artistic objects? -- in conceptual, creative activities?

6. Collaboration with Brent Collins
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Collaboration with Brent Collins
In particular, I worked with Brent Collins, a sculptor living in Gower, MO -- For whom I designed geometrical shapes on the computer which he then built in wood. Here you see our first joint creation, which came out of a program that I created specifically for this purpose…
“Hyperbolic Hexagon II”

7. “Sculpture Generator I ” GUI
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It is Sculpture Generator In this project I extracted the mathematics hidden behind Brent’s sculptures.
– but this is the subject of another talk…

8. When does a mathematical model become a piece of art ?
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Math  Art Connection
When does a mathematical model become a piece of art ?
Today I want to explore the Math to Art connection and ask : …

9. Rapid Prototyping Model of the 24-Cell
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Rapid Prototyping Model of the 24-Cell
Notice the 3-fold permutation of colors Made on the Z-corp machine.
Here is a 3D model of a 4D regular polychoron, obtained by a perspective projection from 4D to 3D space. In 4 different colors are outlined 4 congruent Hamiltonian cycles that visit all 24 vertices of this 24-cell. -- It is kind of a pretty model, but not really an aesthetic sculpture!

10. 3 Hamiltonian Cycles on 4D Cross Polytope
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Here is a similar projection of a simpler regular 4D polychoron bounded by 16 tetrahedra. Again we can find 3 congruent Ham cycles on its edges.
Now, this bland computer diagram can be turned into something much more dramatic, if …

11. Hamiltonian Cycles on 4D Cross Polytope
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if we change the cross sections of the edges and turn them into a ribbon that actually “visits” the 6 vertices shown as small white balls.

12. PART A Knots as Constructive Building Blocks
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In this talk I want to focus in particular on knots and on sculptures made with knots. In part A …
Here are 4 interlocking trefoil knots; loosely interconnected -- more a plaything then an art object.

13. Tetrahedral Trefoil Tangle (FDM)
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Here is a tight tangle of knots that keeps its shape. From just one picture, it is hard to figure out what is going on…

14. Tetra Trefoil Tangles Simple linking (1) -- Complex linking (2)
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Tetra Trefoil Tangles
But if we color the 4 trefoils differently, you can see that they follow a simple linking scheme that results in a tight configuration.
On the right is a more complex linking, resulting in alternating over- and underpasses for each strand.
Simple linking (1) Complex linking (2)
{over-over-under-under} {over-under-over-under}

15. Complex linking (two views)
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Tetra Trefoil Tangle
Here is that complex linking in two views,
On left you can see one trefoil facing you;
On right: opposite view.
-- How do I conceive and construct such a thing?
Complex linking (two views)

16. Platonic Trefoil Tangles
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Take a Platonic polyhedron made from triangles,
Add a trefoil knot on every face,
Link with neighboring knots across shared edges.
Here is how …
Start with one of the Platonic solids, in this case, the Tetrahedron, add …

17. Icosahedral Trefoil Tangle
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Icosahedral Trefoil Tangle
Same principle applied to the icosahedron.
Simplest linking (type 1)

18. Icosahedral Trefoil Tangle (type 3)
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Icosahedral Trefoil Tangle (type 3)
And here is a more complex linking scheme:
Every trefoil links twice with ech one of its 3 neighbors.
Doubly linked with each neighbor

19. Arabic Icosahedron
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Final sculpture, named “Arab.Icosa” because it reminds me of some of the Moorish patterns seen in the windows of the Alhambra.

20. Dodecahedral Pentafoil Cluster
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We can also start with dodecahedron, and plaster a pentafoil on each face.

21. The result: Pentafoil cluster in metal. How do you fabricate this ? …
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The result: Pentafoil cluster in metal.
How do you fabricate this ? …

22. Realization by ProMetal (Ex One Co.)
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Realization by ProMetal (Ex One Co.)
Done by ProMetal, a subsidiary of ExOne Co.
with a 3D printing process using stainless steel powder.
The loose sintered steel matrix is then infiltrated with liquid bronze.
Metal sintering and infiltration process

23. “The Beauty of Knots” Make aesthetically pleasing artifacts!
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“The Beauty of Knots”
More recently, I have been looking for sculptures
where the whole piece is just a single knot.
Undergraduate research group in 2009
Make aesthetically pleasing artifacts!
What I have shown you so far have been ‘clusters’ of knots. Now let’s see what can be done with just a single knot.
In 2009: URAP -- Explore how simple knots can be turned into sculptures.

24. Flat (2.5D), uninspiring, lack of symmetry …
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Classical Knot Tables
We looked at some of the first simple knots in the classical knot table. When presented in this form, the knots are flat, they often don’t show their symmetries, and they are generally uninspiring…
Flat (2.5D), uninspiring, lack of symmetry …

25. Knot 5.2 But this is what you can do to them…
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But this is what you can do to them…
Here is the simple and unremarkable knot 5.2 turned into a rather nice, truly 3D sculpture.

26. Knot 6.1
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And here is Knot 6.1 after some tune-up

27. PART B Computer-Generated Knots
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PART B Computer-Generated Knots
Generate knots & increase their complexity in a structured, procedural way.
Explore several different methods…
I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
Over the last 4 years, I have also tried to generate aesthetically pleasing knots of high complexity with various computer programs. I will now briefly outline 4 methods that have yielded satisfactory results: --- The first one uses a bottom-up construction: I.e. -- I start with something simple, then add complexity…

28. A plane-filling Peano curve
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The 2D Hilbert Curve (1891)
A plane-filling Peano curve
The principle is illustrated here with the famous Hilbert Curve. -- It uses a recursive construction: each green elbow corner is replaced with the more complicated blue shape shown in middle … Pink corner on that … with purple shape in right image. Soon a dense array that fills the whole square. --> Do this in 3D!
Do This In 3 D !

29. Start with Hamiltonian path on cube edges and recurse ...
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“Hilbert” Curve in 3D (1999)
Start with Ham. … replace each of the 8 right-angle turns with the more complex structure shown on left. Then replace each right angle in it with a smaller copy of that same shape.
-- And perhaps recurse once more…
Replaces an “elbow”
Start with Hamiltonian path on cube edges and recurse ...

30. Jane Yen: “Hilbert Radiator Pipe” (2000)
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Flaws ( from a sculptor’s point of view ):
4 coplanar segments
Not a closed loop
Broken symmetry
Here is the third generation of this process done by one of my students in my modeling class.
Pretty, -- but not perfect form a sculptor’s view:
First it has sequences of 4 coplanar …
Second: not closed…

31. Metal Sculpture at SIGGRAPH 2006
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Here is a solution that eliminates all these flaws and has a maximal amount of symmetry.
Executed with the ProMetal sintering process described earlier.
… Knot Theorists …

32. It is still just the un-knot !
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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:
Knot theorists will not get too excited about this …
Just a simple loop!
They would prefer something that is truly knotted!

33. Replace every 90° turn with a knotted elbow.
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Recursion Step
OK - so we replace each ellbow with a simple overhand knot…
Replace every 90° turn with a knotted elbow.

34. Also: Start from a True Knot
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Also: Start from a True Knot
And to get a true knot, we start from a knotted loop in the first place.
e.g., a “cubist” trefoil knot.

35. Recursive Cubist Trefoil Knot
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Result : “Recursive cubist trefoil knot.”

36. A Knot Theorist’s View This is just a compound-knot !
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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:
adjacent knots should entangle one another, or
crossing strands should be knotted together . . .
Knot theorists may still not be overly excited:
Just a ‘compound knot’ - a series of small knots on a rope. To fix this … our assembly step should …

37. 2.5D Celtic Knots – Basic Step
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Let’s look at such a procedure in 2.5 Dim, by making a recursive Celtic knot:
On the left: basic recursion step: a simple crossing is replaced by a tangle of 9 crossings.
Then we repeat that step on each of those 9 crossings.

38. Celtic Knot – Denser Configuration
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Here I pushed all the subunits together to obtain a tighter configuration.

39. Celtic Knot – Second Iteration
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And then I use this more complex element at each of the 9 crossings. -- And we could recurse further!

40. Outline I. Bottom-up knot construction
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I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
But let’s look at some different methods!
A second approach fuses simple knots together

41. Combine 3 trefoils into a 9-crossing knot
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Knot-Fusion
Arrange 3 trefoils to touch as shown;
Then perform a cross-over linking where they touch. Make sure that the result is unicursal (a single loop)! --- It is !!
Then recurse!
Combine 3 trefoils into a 9-crossing knot

42. Sierpinski Trefoil Knot
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Here is the next generation, and it is still unicursal.

43. Close-up of Sierpinski Trefoil Knot
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Here is an oblique close-up of one of the corners.

44. 3rd Generation of Sierpinski Knot
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Once I was convinced that everything was OK, I actually built one on our fused deposition modeling machine. This is how it comes out of the machine,…

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And this is what it looks like once the support scaffolding has been removed.

46. From Paintings to Sculptures
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From Paintings to Sculptures
Do something like this in 3D !
Perhaps using two knotted strands (like your shoe laces).
But now, I would like to do something like this to obtain a truly 3-D result structure.
Perhaps based on the knot with which you lace your shoes …

47. A “Knotty” “3D” Recursion Step
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A “Knotty” “3D” Recursion Step
Now back to our main theme, how to make a truly 3D complex recursive knot. The Granny knot is nicely 3-dimensional, so let’s start from that, and replace every crossing with yet another Granny knot.
Use the Granny knot as a replacement element where two strands cross ...

48. Substitute the 8 crossings with 8 Granny-knots
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Next Recursion Step
Now we apply that same substitution again at all 8 crossings in the left figure. The result still has 4 open ends in the same places as the original knot on the left; so we can place this more complex configuration at all 8 crossings…
Substitute the 8 crossings with 8 Granny-knots

49. One More Recursion Step
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One More Recursion Step
Too much complexity !
Now use eight of these composite elements;
connect;
beautify.
On right: 8 complexes in the proper locations, but not yet wired together…
Because it became clear to me, that the result would not be aesthetically pleasing -- far too complex!
There is much room here for more work!

50. Outline I. Bottom-up knot construction
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I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
Now I will describe another recursive approach that focuses on the ‘meshes” rather than the crossings.

51. Recursive Figure-8 Knot (4 crossings)
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Recursive Figure-8 Knot (4 crossings)
Mark crossings over/under to form alternating knot
Result after 2 more recursion steps
A stylized view of the Fig.8 knot is drawn so that the inner portion of it is a scaled-down copy of the whole shape. Now as our general recursion step we map the whole knot into this inner portion.
At right the result of 2 more recursion steps.
Now we define alternating over- and under passes…
Recursion step

52. Recursive Figure-8 Knot
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Recursive Figure-8 Knot
And scale the stroke width …
So the process could continue ad infinitum!
Scale the stroke-width proportional to recursive reduction

53. 2.5D Recursive (Fractal) Knot
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2.5D Recursive (Fractal) Knot
Robert Fathauer has used this approach and has generated some beautiful displays…
This design is based on the trefoil knot…
He picked the middle representation and rounded the loops to perfect circles; then the recursion step becomes easy.
Trefoil Recursion
3 views step
Robert Fathauer: “Recursive Trefoil Knot”

54. From 2D Drawings to 3D Sculpture
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From 2D Drawings to 3D Sculpture
But the result is too flat …just as my first Fig.8 knot design.
To make it more 3D, I turn the loop plane after every recursion step…always placing it at right angle to the previous two planes.
Too flat ! Switch plane orientations

55. Recursive Figure-8 Knot 3D
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Recursive Figure-8 Knot 3D
Result as it comes out of the FDM machine.
Maquette emerging from FDM machine

56. Recursive Figure-8 Knot
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Recursive Figure-8 Knot
And here itr is, -- freed from scaffolding and mounted and photographed as if it were a real monumantal sculpture.
9 loop iterations

57. Outline I. Bottom-up knot construction
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I. Bottom-up knot construction
II. Fusing simple knots together
III. Top-down mesh infilling
IV. Longitudinal knot splitting
The last approach is to take a knot and split it longitudinally and see what happens.

58. Splitting Moebius Bands
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Splitting Moebius Bands
You probably know that when you split a triply-twisted Moebius band, you obtain a trefoil knot.
Escher demonstrated that very nicely!
But what happens when you similarly split a trefoil knot ?
Litho by FDM-model FDM-model M.C.Escher thin, colored thick

59. To open: Rotate one half around z-axis
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A Split Trefoil
Well it depends again on the amount of twist in the trefoil; if it is a multiple of 360*, then it will divide into two separate trefoils.
And if we employ 3 full turns, and design the goemetry very carefully, we can actually move the two resulting trefoil strands appart to some limited amount.
To open: Rotate one half around z-axis

60. Split Trefoil (side view, closed)
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Here is a side-view of this sculpture in closed configuration

61. Split Trefoil (side view, open)
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Here it is opened by about 30 degrees.

62. Split Moebius Trefoil (Séquin, 2003)
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But if we split a trefoil with a twist equal to an odd multiple of 180*, then we obtain a single knot with double the length of the strand.

63. “Knot Divided” by Team Minnesota
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This topology was used as a design motif in the international snow-sculpting championships in Breckenridge, CO, January 2005.
It was 12 feet tall and its title was …
-- with the intended double meaning.

64. does this Not-Divided Knot have ?
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Knotty Problem
Like a giant rollercoaster … need to go twice through the original knot…
Here is some homework for the Knot theorists: after the split, what kind of a knot is this ?
How many crossings does it have ? …
How many crossings
does this Not-Divided Knot have ?

65. Is It Math ? Is It Art ? it is: “KNOT-ART”
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Is It Math ? Is It Art ?
it is: “KNOT-ART”
So you may wonder: Is this math or is this art ? Reminds me of this joke by Mike Twohy:
… Joe six-pack answers … “NOT”

66. Figure-8 Knot Bronze, Dec. 2007 Carlo Séquin
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Figure-8 Knot Bronze, Dec Carlo Séquin
To finish, I would like to leave you with two images that I definitely consider to be art:
Sometimes it is enough to just take a very simple knot and to work with the geometrical form of the generating curve, and the cross section of the sweep, and the finish of the material surface, to turn it into a piece of art.

67. Florida 1999
Chinese Button Knot (Knot 940) Bronze, Dec Carlo Séquin cast & patina by Steve Reinmuth
Here is an artistic treatment of a more complex knot, the knot 9_40, known as…
Thanks to Steve Reinmuth who does the casting and creates these exquisite patinas.