1. 20-year Retrospective, May 20, 2017
Granada 2003
20-year Retrospective, May 20, 2017
Connecting Art and Mathematics
Torus-Knot_3,5
This talk is about my transition from a mathematician to a physicist to an engineer and to an artist.
Carlo H. Séquin
EECS Computer Sciences
University of California, Berkeley

2. Geometry in every assignment . . .
Granada 2003
Geometry in every assignment . . .
CCD TV Camera (1973) Soda Hall (1992)
During the last 40 years, I have been fortunate to be associated with many challenging design projects that involved nifty geometry problems.
My first job was at Bell Labs in Murray Hill. I was put into the group that had just invented CCDs, and over 3 years we developed the first all-solid-state TV camera. (The success of this brought me to Berkeley). There I teamed up with D. A. Patterson, and we built the first Reduced Instruction Set Computer on a chip. At that time, my students and I worked mostly on IC-layout programs which offer a lot of 2D geometry problems.
Then we needed a new home for CS; and I transitioned from 2D CAD to 3D CAD.
Later I teamed up with Paul Wright in ME on a project called “CyberCut – CyberBuild”. This started with optimized tool paths for 3-axis milling machines to make free-form surfaces such as the 3D Yin-Yang. Then came the 3D printers and additive layered manufacturing.. .
RISC 1 MicroChip (1982) D-Yin-Yang (2000)

3. More Recent Designs and Models
Granada 2003
More Recent Designs and Models
During the last two decades, I have worked with artists and with mathematicians.
The top row shows some mathematical visualization models fabricated on 3D printers. Top-left is a 3D Hilbert cube with 512 “elbows”. Top right is a special Klein bottle. The bottom row shows two larger sculptures that emerged from my collaboration with Brent Collins.
Often I could integrate the necessary design work with classes on CAD or solid modeling that I taught. I consider this to be a very fortunate situation.

4. Brent Collins (1997) “Hyperbolic Hexagon II”
Granada 2003
Brent Collins (1997)
Brent Collins is a wood sculptor, living on Gower MO, out in the nowhere, about half an hour north of Kansas City.
Here you can see him holding up “Hyperbolic Hexagon II” – our very first collaborative piece.
“Hyperbolic Hexagon II”

5. Sculpture Generator 1, GUI
Granada 2003
Sculpture Generator 1, GUI
Computer graphics to the rescue!! == To explore all these possibilities, I wrote a very special-purpose computer program. The only thing that it could model was such a chain of saddles and tunnels: I called it somewhat pompously: “Sculpture Generator 1”. Here you see its GUI. About 10 sliders define the geometry of this shape: the order and number of saddles, and their height; -- the width and thickness of the flanges, and the treatment of the edges: squarely cut off or rounded; -- and, most importantly, the amount of twisting and bending of the whole structure: For instance you can bend the Scherk tower into a full circle -- or just into an arch, as shown here.
This program was partly developed in 1996, when I spent the Fall semester here at UNC on a sabbatical.

6. Shapes from Sculpture Generator 1
Granada 2003
Shapes from Sculpture Generator 1
With this generator I could quickly create a whole lot of promising artistic geometries, by moving those sliders and picking some fancy colors and textures.
Some of these images, Brent liked enough, so that he was willing to spend 3 months of his life carving them at a 30 inch scale.

7. Profiled Slice through “Heptoroid”
Granada 2003
Profiled Slice through “Heptoroid”
One thick slice thru sculpture, from which Brent can cut boards and assemble a rough shape.
Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board.
From these Collins will precut boards
then assemble the complete shape
and fine tune and polish it.
He could not construct this! -- This is one slice of “Heptoroid”, which I designed for Brent using my sculpture generator. I then sent a dozen of such 3ft by 3ft blue prints to Brent, and he used a saber saw to cut these shapes out of 1-inch thick Mohagony boards.

8. Emergence of the Heptoroid (1)
Granada 2003
Emergence of the Heptoroid (1)
Here are the assembled cut-out pieces. In this way he obtains the proper rough shape that contains all the right symmetries. However, the surface exhibits strong stair-casing, and it takes him a few weeks to make the surface smooth.
Assembly of the precut boards

9. The Finished Heptoroid
Granada 2003
The Finished Heptoroid
This is what the result looked like: We called it “Heptoroid” because it has seven 4th-order saddles in a toroidal twisted loop.
It was exhibited at the Art Gallery at Fermi Lab near Chicago. The physicists there liked it a lot; but everybody saw something different, e.g.:
>>>
The geometry of a tokamak or stellarator; -- the head of the tunnel-boring machine used in digging the tunnel for their accelerator; -- or the inner shape for one of the elementary quark particles.
at Fermi Lab Art Gallery (1998).

10. “Scherk-Collins” Sculptures (FDM)
Granada 2003
“Scherk-Collins” Sculptures (FDM)
Now that I had this wonderful playground of Sculpture Generator I, I could not wait for 2-3 months for Brent to carve another sculpture. Thanks to rapid prototyping via layered manufacturing, I could make many small sculptural maquettes in a matter of days.

11. “Cohesion” (SIGGRAPH’2003 Art Gallery)
Granada 2003
“Cohesion” (SIGGRAPH’2003 Art Gallery)
Some of them were nice enough, so I sent them to Steve Reinmuth’s Bronze Studio in Eugene, OR. He converted them into bronze with a classical investment casting process, where the ABS plastic of the original was sublimated away in a hot kiln, and then replaced with bronze.
This sculpture has only 2 monkey saddles. It is about a foot tall.
Cast by Reinmuth Bronze Studio, Eugene, OR

12. Hyper-Sculpture: Family of 12 Trefoils
Granada 2003
Hyper-Sculpture: Family of 12 Trefoils
W=2
Or I could make a “Hyper-Sculpture” consisting of a whole family of sculptures, which differed in only one parameter value between neighbors.
B is the number of branches in the saddles, and W is the number of windings in the toroidal loop. – Let me explain this . . .
W=1
B= B= B= B=4

13. 9-story Intertwined Double Toroid
Granada 2003
9-story Intertwined Double Toroid
Bronze
investment
casting from wax original made on 3D Systems’ “Thermojet”
… until I had a physical proof in hand. For this one I made a wax original on a 3D printer from 3D-Systems, and then had Steve Reinmuth investment cast it in bronze and gold-plate it electrolythically.

14. Snowsculpting Championships 2003
ISAMA 2004
Snowsculpting Championships 2003
Sculpture Generator I also created this shape, which was carved (with hand tools only) in a span of four days by this team headed by Stan Wagon, a professor of Mathematics and Computer Science at Macalester College in St. Paul, Minnesota, USA.
The team named their creation the “Whirled White Web” and they received the silver medal for it.
Unfortunately the sculpture collapsed only 45 minutes after judging had ended.
The Team: “Whirled White Web” (C. Séquin, S. Wagon, D. Schwalbe, B. Collins, S. Reinmuth)

15. “WWW” Wins Silver Medal

16. Inauguration Sutardja Dai Hall 2/27/09
Granada 2003
Inauguration Sutardja Dai Hall 2/27/09
And here is a bronze cast of that same piece -- displayed in the 3rd floor lobby of Sutardja Dai hall – the CITRIS building.
Now I want to tell you how we got from that first telephone conversation to this real physical object.

17. Evolving Trefoil
MWSU, Saint Joseph, 2013

18. Brent Collins’ Pax Mundi 1997: Wood, 30”diam.
2006: Commission from
H&R Block, Kansas City
to make a 70”diameter version in bronze.
My task: Define the master geometry.
CAD tools play important role!
This development started with this carved wood sculpture by Brent Collins, which he created in 1997.
In 2006 he received a commission from H&R Block in Kansas City … -- and I received a phone call: “Carlo, can you help?”
I was supposed to provide a detailed CAD model at the right size.
This raises the question: How do you model something like this ? …

19. SLIDE-GUI for “Pax Mundi” Shapes
Florida 1999
Good combination of interactive 3D graphics and parameterizable procedural constructs.
All this was then captured in a more modular program, constructed within the Berkeley SLIDE environment. This is a graphics program that my graduate students built in the 1990s. It has a good combination of interactive 3D graphics and parametrizable procedural constructs.
This new generator program has three columns of sliders controlling respectively: -- the sweep curve – the cross-section – and the application of it along the sweep. On display you see my final model of Pax Mundi.

20. Design of Smaller Two-Part Master
Granada 2003
Design of Smaller Two-Part Master
-- I had to break it apart into 2 smaller U-shaped pieces that were relatively flat.
Look at the blue part …
Alignment tabs for easy assembly

21. Machined Master Pattern #2
Granada 2003
Machined Master Pattern #2
This is what it looks like at full scale as it came off the NC machine.
However, Steve then had to cut this part in half again – so it would fit into his kiln; and the second bigger U-shape, he even cut into 3 parts!

22. Spruing the Wax Parts for Casting
Granada 2003
Spruing the Wax Parts for Casting
These wax parts are then enhanced with the red sprues and runners and the funnels into which the molten bronze will be poured.
The smaller ones also serve as air-venting holes.

23. Ceramic Slurry Shell Around Wax Part
Granada 2003
Ceramic Slurry Shell Around Wax Part
This whole thing then gets dipped repeatedly into plaster slurry to make a ceramic shell.

24. Casting with Liquid Bronze
Granada 2003
Casting with Liquid Bronze
A close-up view of the casting process.

25. Freeing the Bronze Cast
Granada 2003
Freeing the Bronze Cast
When the parts have cooled down, the ceramic shell is smashed away.
The sprues and runners need to be cut away, and some first surface cleaning and smoothing is done.

26. Granada 2003
The “Growing” Ribbon
Three pieces are put back together into the larger horse-shoe part made on the NC machine This would not have fit into Steve’s kiln!

27. Assembly Completed Here the whole ribbon has been assembled.
Granada 2003
Assembly Completed
Here the whole ribbon has been assembled.
Then it needs to get smoothed and polished. – and provided with a patina.

28. Applying Patina to a Bronze Sculpture
Granada 2003
Applying Patina to a Bronze Sculpture
And finally it is provided with some patina. This is created by a combination of heat and chemistry: flame torch in one hand, spray flask in the other,
Steve Reinmuth, Bronze Studio, Eugene OR

29. Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin
Granada 2003
Finished “Pax Mundi” in the courtyard of the headquarters of H&R Block.
This was my first experience with building a large, permanent sculpture.
It would not have been possible without a well-orchestrated collaboration between these 3 individuals.
Team effort: Brent Collins, Steve Reinmuth, Carlo Séquin

30. Many Different Viae Globi Models
Roads on a sphere
I could now make more “curvy” Viae Globi models (i.e “roads on a sphere”).
In the middle is Maloja, inspired by a Swiss mountain road.
On the left is Stelvio, a pass route in Italy with 40+ hair-pin curves.
At right is Altamont, a high-way pass into the Central Valley in CA with multiple parallel lanes.
Altamont
Stelvio
Maloja

31. Florida 1999
Chinese Button Knot (Knot 940) Bronze, Dec Carlo Séquin cast & patina by Steve Reinmuth
Here is the same 9-crossing knot – known as the Chinese button knot – after having been cast in bonze.

32. Assembly of Music of the Spheres
Granada 2003
Assembly of Music of the Spheres
The fabrication process was pretty much the same as for “Pax Mundi”.
Close to final assembly of “Music of the Spheres”. Photo conveys a sense of the size and of the physical labor involved in building such a sculpture.

33. Illuminated Music of the Spheres
Granada 2003
Illuminated Music of the Spheres
And it looks even more spectacular at night!
Photo by Phillip Geller

34. 6-inch mode on FDM machine
ISAMA 2004
6-inch mode on FDM machine

35. Yet Another Medium: Stone
Granada 2003
Yet Another Medium: Stone
And I am also getting experience with outputting a virtual design into yet another medium: Stone !
That is what the sculpture looked a few days ago before it was packaged up and put on a ship from China to Oakland.
“The Three Pillars of Engineering” Math – Materials – Physics(Science) Sponsored by Paul Suciu (EECS alum)

36. Spring, 2012

37. ISAMA 2004
Contents of the crate

39. Installation, May 31, 2012

42. A plane-filling Peano curve
Granada 2003
The 2D Hilbert Curve (1891)
A plane-filling Peano curve
Another example to exercise this analogy from 2D to 3D
Do This In 3 D !

43. Construction of 3D Hilbert Curve

44. Granada 2003
Hilbert-Cube-512 (2006)
Can I do the next generation 4096 “ells” on the new FDM machine we have in the basement of SD hall ?

45. A Gridded Model of Trefoil Knottle
ISAMA 2004
A Gridded Model of Trefoil Knottle
And here is a physical model of the central picture; -- made on a FDM machine.

46. Klein Bottle with S6 Symmetry
ISAMA 2004
Klein Bottle with S6 Symmetry
This is still just a Klein bottle. We can make higher-genus non-orientable surfaces!

47. 125 Tetras in 25 Projected 5-Cells

49. Inspiration: Henk van Putten
ISAMA 2004
Inspiration: Henk van Putten
In 2013 at the Bridges art exhibit, I was intrigued by these simple geometrical sculptures by Henk van Putten.
They comprise a very modular, intriguing geometry.
I wondered what it would be like if you could experiment with such modules in real time in a tangible, physical manner.
-- like a special kind of LEGO blocks that you can snap together to form these and many other but similar sculptures.
In the right-hand image, the modularity is rather obvious, but where is it in the “Borsalino” on the left?
“Borsalino” “Interaction”
Sculptural forms put together from a few modular shapes

50. The Wonders of Rapid-Prototyping
ISAMA 2004
The Wonders of Rapid-Prototyping
E R=1.0
Here are the solid models of these two parts. They were built on the old FDM machine in Etcheverry. We also need some kind of snap-together joint. We experimented with two different geometries,
A beveled one, and a square one. On the right you see the assembled Borsalino.
Of course we “cooked” more parts than just the minimum number needed for this sculpture…
C R=2.4142
Two modular components can form the Borsalino

51. Parts Catalog So Far 2 types of end-caps; 3 curved connectors
ISAMA 2004
Parts Catalog So Far
Here you see the family of parts we had up to this point. We have end-caps of two different radii.
On the left you see the older type of flanges, tapered at 45º; on the right the preferred square, rectilinear flanges, with a separate connector ring.
The one on the right also has the internal cathedral ceiling, so that it could be built entirely without support.
-- We also have the two curved connectors to go with the two different sizes of end-caps to make complete Borsalinos, as shown before.
In addition we made a very small connector part (right-most) which is simply one quarter of the larger end-cap, so we can make turns through other angles than180* at this tighter radius.
2 types of end-caps; 3 curved connectors

52. Triply Twisted Rhombic Borsalino
ISAMA 2004
Triply Twisted Rhombic Borsalino
The last example as a sculpture. I think this would look good at a large scale, carved from stone, or made from gleaming sheet metal!
Sculpture!

53. Inspiration: Paul Bloch
ISAMA 2004
Inspiration: Paul Bloch
But I managed to make smooth closure for 2.5 turns in our helical spiral.
-- as an emulation of Paul Bloch’s “After Wright”.
This employed two straight pieces, two 45-degree curved connectors of the mid-sized radius, and 2 rhombic curved connectors!
“After Wright” (Guggenheim, NYC)

54. Inspiration: Jon Krawczyk
ISAMA 2004
Inspiration: Jon Krawczyk
Seen at nd street (near Harrison) in San Francisco you can see this sculpture by Jon Krawczyk.
This led to a few two-legged sculptures made form our LEGO-Knot blocks.
303 2nd Street, San Francisco

55. “Pas de Deux”
ISAMA 2004
And also to this more involved free-form sculpture, called “Pas de Deux”, shown from two different angles.

56. Real Knots: Figure-8 Knot (4_1)
ISAMA 2004
Real Knots: Figure-8 Knot (4_1)
The Figure-8 knot can be made with S4 symmetry with a vertical C2 axis, and with up-down-up-down glide symmetry around the equator. Because this knot is its own mirror image, I could NOT use the helical pieces we had. So I used the planar connector pieces shown in blue and cyan. Again, I needed an special connector piece to interconnect the four lobes in a graceful manner; but now I needed two mirror pairs, shown in magenta and red.
Two new pieces (magenta, red) for smooth closure

57. Tetra-Tangle of Four Bow-Tie Links

58. Making Sculptures Glow …
ISAMA 2004
Making Sculptures Glow …
And this is the desired effect.

59. KB-modules of Many Shapes
Granada 2003
KB-modules of Many Shapes
Here are the actual realizations of these four designs.
The were all fabricated on a FDM machine from Stratasys. (It takes about 6-8 hours per pair, and costs about 20$ a piece).
4 different version of the basic cube-corner module

60. Two Different Cube Frames
ISAMA 2004
Two Different Cube Frames
Here are two different genus 10 cube frames formed by placing different components at the 8 corners and with different orientations.
I also show examples of two different mounts – on the corner of the cube, -- the other supported by one of the cube edges.
---- But just building cube frames gets boring after a while …
Many possible placements of the KB modules at cube corners

61. A Selection of Super-Bottles built
Granada 2003
Q U E S T I O N S ?
A Selection of Super-Bottles built

62. Rhombic Dodecahedron σ=1, genus 22
ISAMA 2004
Rhombic Dodecahedron
Here is the largest “Super-Duper Bottle” that I have built so far. It combines all the KBM modules that I have: the six 4-way parts and 8 cube corners.
It is based on the edge-graph of the rhombic dodecahedron, and results in a single-sided surface of genus 22.
σ=1, genus 22

63. Continuum, Space Museum, Washington D.C. Charles Perry, 1976, bronze
Granada 2003
Continuum, Space Museum, Washington D.C. Charles Perry, 1976, bronze
On the other hand, understanding and modeling Perry’s “Continuum”, which can be seen in front of the Space Museum in Washington, was much more difficult.
Continuum II Singapore, 1986

64. Granada 2003
Charles O. Perry
One of my heros! A rich collection of topological sculptures!
Here is a large collection of 2-manifold sculptures by Charles Perry, all of them are in metal. He calls them “topological sculptures”. He had an active interest in this branch of mathematics.

65. Modeling “Tetra” by Charles Perry (3)
Granada 2003
Modeling “Tetra” by Charles Perry (3)
Maquette of Perry’s “Tetra”
In this case, based on the paper model and an annotated sculpture image, I made a better geometrical model from 4 metal hoops connected with scotch tape. And from this model, I later developed a parameterized computer model.
>>> This then allowed me to make a model on a 3D-printer.
Overall, modeling “Tetra” was not too hard. It is relatively simple, and I had many good images.
Annotated sculpture image
Metal-rings plus scotch-tape model
CAD model of Perry’s “Tetra”

66. Modifications of Perry’s Tetra Sculpture
Granada 2003
Modifications of Perry’s Tetra Sculpture
4 half-twists, two-sided half-twists, single-sided

67. Complex 2-Manifold Sculpture
Granada 2003
Complex 2-Manifold Sculpture
But it is not so easy to make a parameterized procedural model of her more complex and more irregular sculptures!
Here is something that I am currently struggling with: How can I capture this very free-form, asymmetrical shapes?
“Wholly” by Eva Hild (Sweden)

68. A large collection of ceramic creations & metal sculptures
Granada 2003
2-Manifolds by Eva Hild
A large collection of ceramic creations & metal sculptures
Eva Hild is a Sweedish artist who creates large ceramic and metal sculptures in the form of organically undulating surfaces. They present interesting mathematical puzzles. What are these sculptures from a mathematician’s point of view? Most of them are smooth, thin sheets of material. Mathematicians call this a 2-manifold. Some questions: Are these surfaces 2-sided like a sheet of paper -- or perhaps single-sided like a Moebius band or a Klein bottle. How many borders they have, and what is their genus?

69. Eva Hild’s 2-Manifold Sculptures
Granada 2003
Eva Hild’s 2-Manifold Sculptures
But let’s look at some examples. This sculpture called “Hollow” is in Vaarberg, Sweden. On the right is a 10 inch maquette made on a low-end 3D printer. This sculpture is single-sided, has 1 border, and a genus of two. (One can cut the 2 horizontal tunnels in the front and in the back to reduce this to a disk with a bunch of holes or “punctures.”)
“Hollow” (Eva Hild) FDM Maquette
1 border; single-sided; genus 2

70. Eva Hild Sculptures “Whole” (Eva Hild) FDM Maquette
Granada 2003
Eva Hild Sculptures
Here is another sculpture called “Whole” and my recreation.
Hild’s sculptures are made by hand in an incremental, organically growing way, and typically all the lobes and tunnels are slightly different in size. My computer generated models on the other hand impart as much symmetry as possible. In this way I have to design only a smaller fraction in detail, and the rest of the shape is then obtained from mirroring and rotation operations.
“Whole” (Eva Hild) FDM Maquette
1 border; single-sided; genus 2

71. 3 double-tunnels, wrapped with a Gabo-3 curve
My Own Designs
3 double-tunnels, wrapped with a Gabo-3 curve

72. Single-Sided Surfaces
Non-orientable surfaces that look like Hild sculptures

73. Single-sided surface formed by 5 Dyck disks in a ring
Pentagonal Dyck Cycle
Single-sided surface formed by 5 Dyck disks in a ring