1. EECS Computer Science Division University of California, Berkeley
ISAMA 2004
G4G9
The Beauty of Knots
Carlo H. Séquin
To a mathematician, knots are closed, non-self-intersecting curves, embedded in 3D space. Their topological properties are locked-in by this embedding and cannot be changed by geometrical deformation.
EECS Computer Science Division
University of California, Berkeley

2. Classical Knot Tables Flat (2.5D), uninspiring, lack of symmetry …
ISAMA 2004
Classical Knot Tables
Simple knots are typically depicted in this classical knot table, which tabulates them by the minimal number of crossings with which each knot can be projected onto a plane. When presented in this form, the knots look flat, at best 2.5-dim, and these tables often don’t show their full symmetry. In short, this depiction is rather uninspiring…
Flat (2.5D), uninspiring, lack of symmetry …

3. ISAMA 2004
Trefoil Knot
But knots _do_ lend themselves to make beautiful artwork. Here are two realizations of the simplest possible sculpture - the trefoil. -- On the left there is a small soapstone sculpture from Kenya;
On the right there is: “Venus” by Hugh Cunningham, located on Toronto's Yorkville Avenue.
[See also: ]

4. Figure-8 Knot Bronze, Dec. 2007 Carlo Séquin
ISAMA 2004
Figure-8 Knot Bronze, Dec Carlo Séquin
Here is an artistic realization of the next more complicated knot, the 4-crossing Figure-8 knot.
It was cast in bronze by Steve Reinmuth in Eugene OR, and it won 2nd prize at the AMS Exhibit of Mathematical Art, in 2009 in Washington DC.
2nd Prize, AMS Exhibit 2009

5. “The Beauty of Knots” Undergraduate research group in 2009
ISAMA 2004
“The Beauty of Knots”
Undergraduate research group in 2009
What is the most symmetrical configuration?
What is the most 3-dimensional configuration?
Make aesthetically pleasing artifacts!
Based on these encouraging beginnings, I started a small undergraduate research project In 2009, called “The Beauty of Knots”. In this group we explored how simple knots could be turned into sculptures. Specifically, for many knots at the beginning of the knot tables, we asked: What is …
But the main goal was to make beautiful artifacts!

6. Some Results Emphasizing Symmetry
ISAMA 2004
Some Results Emphasizing Symmetry
Here are some early results where the goal was to maximaize the symmetry of a knot.
Knot 7.7 has 2-fold rotational symmetry.
Knot 7.4 has D2 symmetry, three mutually perpendicular C2 rotation axes.
Knot Knot 74

7. Knot Transformations Bring out special, desirable qualities:
ISAMA 2004
Knot Transformations
Bring out special, desirable qualities:
A graceful “evenly-spaced” curve: Minimize electrostatic repulsion potential on a flexible wire.
Tightest configuration: Pull tight a rope of fixed diameter without self-intersections.
The least “wiggly” curve: Minimize the arc-length integral of curvature squared.
The most 3D-filling configuration: Wrap knot around a sphere or a cylinder; Turn configuration inside out (point inversion); Play with wires, alu-foil, pipe cleaners!
Since this was a CS group, we discussed may procedural approaches of making “better” knots…
--- To make a graceful, evenly-spaced curve, one might…
But this is not a good topic for a 15 minute presentation. I will write about it in my paper for the gift book.
Bottom line: -- Non-computer methods were most effective on small knots.

8. ISAMA 2004
Knot 52
But if one steps away from simply using a cylindrical tube and is willing to be creative in deforming the knot, one can get rather pretty results.
Here is the simple and unremarkable knot 5.2 turned into a truly 3D sculpture. -- Perhapsit is not easy to see the connection to the knot-diagram at right. But turn the diagram upside-down in your mind, and then drop the two shoulder lobes down over its sides -- and you get the basic structure of the sculpture.

9. Knot 61 Here is a result for Knot 6.1.
ISAMA 2004
Knot 61
Here is a result for Knot 6.1.
This knot has 2-fold rotational symmetry. But the knot curve does not have any point lying on that symmetry axis; so a new way of suspending it had to be found.
Here it is shown hugging a vertical pole like a koala bear.

10. “Signature Knot” for G4G9
ISAMA 2004
“Signature Knot” for G4G9
I felt, we needed a “signature knot” for the this Gathering for Gardner #9.
Obviously it had to be NINE-crossing knot !
– but which one ?
Has to be a 9-crossing knot … -- but which one ?

11. “Signature Knot” for G4G9
ISAMA 2004
“Signature Knot” for G4G9
… a 9-crossing knot:
Knot 940
the same Knot !
It has 3-fold symmetry!
The one I like the most is Knot Its diagram from the knot tables is show at left. What this diagram does not tell you is …
-- that this knot has 3-fold symmetry, you realize this only
-- when you build this knot from pipe-cleaners and then play with it.
Then you also discover other remarkable symmetries…

12. Knot 940: “Chinese Button Knot”
ISAMA 2004
Knot 940: “Chinese Button Knot”
This knot also has interesting 3-D properties. In its tightest configuration it forms a rather nice spherical cluster, and because of this property, it can serve as a button on clothing. Hence it is also known as the “Chinese Button Knot.”
It has interesting 3D properties !

13. Knot 940: Chinese Button Knot
ISAMA 2004
Knot 940: Chinese Button Knot
Here is an instanciation of this knot: as a cubistic solid model -- which I later fabricated on an FDM machine.
This is a rapid-prototyping machine that can build almost any shape one thin layer at a time piled on top of one another.

14. Knot 940: Chinese Button Knot
ISAMA 2004
Knot 940: Chinese Button Knot
Here it is shown as a loose sculpture made from tubing copper, and from copper wire - for the smaller one.
It shows some remarkable “sperical” symmetry.

15. ISAMA 2004
Chinese Button Knot (Knot 940) Bronze, Dec Carlo Séquin cast & patina by Steve Reinmuth
This symmetry is also exploited in this small bronze sculpture.
I want to again acknowledge Steve Reinmuth, who did the casting and created the exquisite patina.

16. ISAMA 2004
Knot 940 in Ribbon Form
Here is our sugnature knot again, this time constructed from translucent plastic bands.
This configuration will be the subject of some hands-on constructivist activities on Saturday afternoon at Tom Rodgers’ house.
Will be the subject of some hands-on “constructivist activities” on Fri./Sat. pm.

17. From Simple Knots to Complicated Knots
ISAMA 2004
From Simple Knots to Complicated Knots
But onwards to much more c omplicated structures…
Here is a sculpture for which I used a computer to help me design it. It is the 3D version of the famous Hilbert curve. -- It looks like a complicated knot - but it is not!
- It is the “unknot” -- just a simple unknotted loop.
Saturday afternoon Chaim Goodman Strauss will lead an effort to build one of those from steel plumbing parts -- at a much bigger scale!
“Hilbert Cube 512” – looks complicated … but it is not; -- just a simple, unknotted loop!

18. Generating Complicated Knots
ISAMA 2004
Generating Complicated Knots
Is there a procedure to make knots of arbitrary complexity…?
Perhaps by fusing simple knots together…
Perhaps by applying recursive techniques…
Start with: 2.5D - Celtic Knots
But, if we want to use the computer to make some truly complicated knots. What kind of procedure should we use?
Perhaps …
Yes -- to both of the above. Let’s start with 2.5D Celtic Knots…

19. 2.5D Celtic Knots – Basic Step
ISAMA 2004
2.5D Celtic Knots – Basic Step
Here is a recursive procedure to make a Celtic knot:
On the left you see the basic recursion step: A simple crossing of a red and a green strand is replaced by a tangle of 9 crossings.
-- Here is that famous again !
Then we repeat that step on each of those 9 crossings. => and we obtain a pattern with NINE times NINE crossings

20. Celtic Knot – Denser Configuration
ISAMA 2004
Celtic Knot – Denser Configuration
Here I pushed all the subunits together to obtain a tighter configuration. But overall it is still a complicated crossing of a red and a green strand, so I can use it again on the original pattern…

21. Celtic Knot – Second Iteration
ISAMA 2004
Celtic Knot – Second Iteration
Here I use this more complex element at each of the 9 original crossings.
-- And we could recurse even further!
-- but the projector here probably would not handle it …

22. Another Approach: Knot-Fusion
ISAMA 2004
Another Approach: Knot-Fusion
Here is another approach at making complicated knots:
Arrange 3 trefoils to touch with the tips of their lobes as shown;
Then perform a cross-over linking where they touch. Make sure that the result is unicursal, i.e., that it is a single loop! -- You can trace your eyes along it and convince yourself that it is !
Then we recurse!
Combine 3 trefoils into a 12-crossing knot

23. Sierpinski Trefoil Knot
ISAMA 2004
Sierpinski Trefoil Knot
Here is the next generation, and it is still unicursal.
Trust me!

24. Close-up of Sierpinski Trefoil Knot
ISAMA 2004
Close-up of Sierpinski Trefoil Knot
Here is an oblique close-up of one of the corners of a somewhat smoother, geometrically optimized version.

25. 3rd Generation of Sierpinski Knot
ISAMA 2004
3rd Generation of Sierpinski Knot
Once I was convinced that everything was OK, I actually built one of these 120-crossing knots on our fused deposition modeling machine. This is how it comes out of the machine,…

26. ISAMA 2004
And this is what it looks like once the support scaffolding has been removed.

27. Another Approach: Mesh-Infilling
ISAMA 2004
Another Approach: Mesh-Infilling
...
Map “the whole thing” into all meshes of similar shape
Now I show you a 3rd recursive approach to making ever more complicated knots. This approach was pioneered by Bob Fathauer. I call it mesh-infilling. You take a knot such as the 7-crossing knot shown in the 2. field from the left and see whether any portion of it ressembles its convex hull around the overall shape. In this case it does: the red hexagon on the left shows up four times inside the 7-crossing knot. So now you map the whole knot into each of these four regions, and reconnect the wiring at the peripheries of these regions. Then you recurse.
Robert Fathauer, Bridges Conference, 2007

28. 2.5D Recursive (Fractal) Knot
ISAMA 2004
2.5D Recursive (Fractal) Knot
Here is another beautiful example of this approach. This design is based on the trefoil knot. The trefoil knot can be diagrammed in many different ways as shown on the left. Robert picked the middle representation and rounded the loops to perfect circles; then the recursion step becomes easy. In this case we only place two copies of the knot into the parent knot. 5 recursive steps leads to the beautiful display shown on the right. -- It is beautiful, but kind of flat . To get back to the theme at the beginning of my talk: --- can we do something more 3D ?
Trefoil Recursion
3 views step
Robert Fathauer: “Recursive Trefoil Knot”

29. Recursive Figure-8 Knot (4 crossings)
ISAMA 2004
Recursive Figure-8 Knot (4 crossings)
Mark crossings over/under, form alternating knot
Result after 2 more recursion steps
With this goal in mind, let’s start with the Figure-8 knot. 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 overall “eye” shape. Now, as our general recursion step, we map the whole knot just once into this inner portion.
-- At right: the result of two more recursion steps.
Now all we have to do, is define alternating over- and under passes to make this a true alternating knot.
Recursion step

30. Recursive Figure-8 Knot
ISAMA 2004
Recursive Figure-8 Knot
In addition, I also scale down the stroke width proportional to the recursive reduction,
so that this process could continue ad infinitum!
But the result is still rather flat … as in all previous recursive designs…
Scale the stroke-width proportional to recursive reduction

31. From 2D Drawings to 3D Sculpture
ISAMA 2004
From 2D Drawings to 3D Sculpture
To make it more 3D, I turn the loop plane through 90* after every recursion step. The plane for the new inner loop is always placed at right angles to the previous two planes. Now we get a truly 3-dimensional structure, with none of the coordinate planes unjustly favored.
Too flat ! Switch plane orientations

32. Recursive Figure-8 Knot 3D
ISAMA 2004
Recursive Figure-8 Knot 3D
Result as it comes out of the FDM machine. The gray material is the scaffolding that allows the yellow curve to be built in a truly 3D configuration.
Maquette emerging from FDM machine

33. Recursive Figure-8 Knot
ISAMA 2004
Recursive Figure-8 Knot
And here it is, -- freed from scaffolding, and mounted and photographed as if it were a truly monumental sculpture.
You may wonder is this Art or is this Mathematics ?…
9 loop iterations

34. Is It Math ? Is It Art ? it is: “KNOT-ART”
ISAMA 2004
Is It Math ? Is It Art ?
it is: “KNOT-ART”
This brings to mind a joke by Mike Twohy.
The modest artist says …
“Joe Six-Pack” replies…
-- And my answer is: “ It is Knot-Art! “