1. BID Celebration, December 9, 2015
Granada 2003
BID Celebration, December 9, 2015
Geometry & Design
a 60-year retrospective
Carlo H. Séquin
University of California, Berkeley
In the next 15 minutes I will try to give you a 60-year retrospective of my involvement with geometry and design.

2. Basel, Switzerland M. N. G. I grew up in Basel, CH,
Granada 2003
Basel, Switzerland
M. N. G.
I grew up in Basel, CH,
>>> where I attended the M. N. G. (a high school with emphasis in math and science).
>>> During my university years I heard my math lectures in this 500 year old institute, where many famous mathematicians have lectured:

3. Jakob Bernoulli (1654‒1705) Logarithmic Spiral
Granada 2003
Jakob Bernoulli (1654‒1705)
such as Jakob Bernoulli, or . . .
Logarithmic Spiral

4. Leonhard Euler (1707‒1783) Imaginary Numbers … or Leonhard Euler.
Granada 2003
Leonhard Euler (1707‒1783)
… or Leonhard Euler.
From an early age on, I was fascinated with numbers …
Imaginary Numbers

5. ISAMA 2004
Descriptive Geometry
and during high-school I fell in love with geometry. In 11th grade we had a subject called Descriptive Geometry, where we constructed the intersection-lines between two cylinders. -- I thought that this was very cool stuff!
[By the end of high-school my focus was on Geometry and Physics, in particular Optics, which has a lot to do with geometry! -- So I majored in Experimental Physics.]

6. Geometry in every assignment . . .
Granada 2003
Geometry in every assignment . . .
CCD TV Camera (1973) Soda Hall (1992)
In the 45 years since then, I have been fortunate to be associated with many challenging design projects that involved nifty geometry problems. At Bell Labs we built the first all-solid-state TV camera. --- When I came to Berkeley, together with Dave Patterson and our students we constructed the first R. I. S.- microcomputer chip Later, when we needed a new home for Computer Science, I stepped up from 2D geometry to 3D, mostly floor-plans and elevations. But I got involved with truly 3D, free-form geometry only after I teamed up with Paul Wright in ME, and as part of the CyberCut project helped develop optimal tool paths on milling machines for organically shaped surfaces.
RISC 1 MicroChip (1982) D-Yin-Yang (2000)

7. Recent Designs and Models
Granada 2003
Recent Designs and Models
During the last two decades, I have worked with artists and mathematicians and mostly focused on additive manufacturing such as 3D printing.
The results are mathematical visualization models, such as the 3D Hilbert cube, composed of 512 “elbows”, or the special double-Klein-bottle at top right.
I also designed a few large sculptures, as shown at the bottom; those resulted from a collaboration with Brent Collins.

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

9. Brent Collins: Hyperbolic Hexagon
Granada 2003
Brent Collins: Hyperbolic Hexagon
Six balanced saddles in a circular ring.
Inspired by the shape of a soap film suspended in a wire frame.
= Deformed “Scherk Tower”.
This collaboration was inspired by this piece that Brent had carved a couple of years earlier, called “Hyperbolic Hexagon”. Even though Brent is not a mathematician and conceived this shape almost intuitively, it is rather close to a minimal surface, which could be assumed by a soap film suspended in a suitable wireframe. The chain of six tunnels, surrounding the central opening can also be related to a classical minimal surface …

10. Scherk’s 2nd Minimal Surface (1834)
ISAMA 2004
Scherk’s 2nd Minimal Surface (1834)
… the second minimal surface discovered by Scherk. Its central part is a stack of alternating tunnels and saddles, beyond which it stretches of to infinity in all directions. This central part is the only artistically interesting portion of this surface, -- I call this a “Scherk Tower”.
The central part of this is a “Scherk Tower.”

11. Generalizing the “Scherk Tower”
Granada 2003
Generalizing the “Scherk Tower”
Normal
“biped”
saddles
In the simplest case, it is composed of simple biped saddles, as you would find on the back of a horse.
But the Scherk tower can be generalized to saddles of higher order, for instance 3rd-order “monkey saddles” -- with 3 valleys going down (one for the monkey’s tail) and 3 ridges going up between them.
Generalization to
higher-order saddles (“Monkey saddle”)
“Scherk Tower”

12. “Scherk-Collins Toroids”
Granada 2003
Closing the Loop
straight or
twisted
We can now take such a Scherk tower and bend it into a toroidal loop. If you close a 4-story Scherk tower into a loop without any twisting, you get the result at top right; – and if you do this with a six story tower, you get Brent’s original “Hyperbolic Hexagon”.
But, we can also give the tower a longitudinal twist of 180 degrees, as shown on the left, before closing it into a loop. Then we obtain a less symmetric – but, perhaps a more dynamic result, as shown at the bottom right. You probably can see that there are endless possibilities to choose the number of stories, the branching of the saddles, and the amount of twist. -- But which combinations will make good sculptures?
“Scherk Tower”
“Scherk-Collins Toroids”

13. Sculpture Generator 1, GUI
Granada 2003
Sculpture Generator 1, GUI
To explore all these possibilities, I wrote a very special-purpose computer program. The only thing that it can model is such a chain of saddles and tunnels: Here you see its user interface. 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.

14. Shapes from Sculpture Generator 1
Granada 2003
Shapes from Sculpture Generator 1
Here is a bunch of shapes popping out of “Sculpture Generator I” as one moves those sliders and adds some fancy colors and textures.
With this generator I could quickly create a whole lot of promising artistic geometries. --- Some of which Brent liked enough, so that he was willing to spend 2 months of his life carving them at the 30 inch scale.

15. Profiled Slice through “Heptoroid”
Granada 2003
Profiled Slice through “Heptoroid”
One thick slice thru the 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.
To make this possible, I let the computer calculate slices through the intended sculpture -- one inch apart, and I then sent such 3ft by 3ft blue prints to Brent.
Brent used a saber saw to cut these shapes out of 1-inch thick Mohagony boards.

16. Emergence of the Heptoroid (1)
Granada 2003
Emergence of the Heptoroid (1)
Brent then laminates the dozen or so wood boards with industrial-strength glue. In this way he obtaines the proper overall shape that contains all the right symmetries. But the surface shows strong stair-casing.
Assembly of the precut boards

17. Emergence of the Heptoroid (2)
Granada 2003
Emergence of the Heptoroid (2)
After a few weeks of chiseling and grinding, a smooth surface starts to appear. Here you see a continuous broad rim emerging – about an inch wide. It travels around the loop 8 times before it gets back to the starting point. Brent continues to grind down the wood volume to a smooth, thin surface with a narrow, well-defined edge.
[For this sculpture, the Scherk-tower has been given a twist of 135 degrees (3/8 of a full turn) before closing the loop.]
Forming a continuous smooth edge

18. The Finished Heptoroid
Granada 2003
The Finished Heptoroid
And this is the result: We called it the “Heptoroid” because it has seven 4th-order saddles in a twisted toroidal loop.
It was exhibited at the Art Gallery at Fermi Lab near Chicago; and the physicists there liked it a lot; (everybody saw something different . . .)
--- But artists dream of seeing their sculptures at a much larger scale. – Well, in 2003 we got an opportunity to build such a sculpture 12 feet tall . . .
at Fermi Lab Art Gallery (1998).

19. Snowsculpting Championships 2003
ISAMA 2004
Snowsculpting Championships 2003
However, it was carved out of snow! This was done at the annual snow-sculpting competition in Breckenridge, Colorado.
It took 5 people 5 solid days to create this sculpture … But it held up less than 1 hour beyond jurying time …
“Whirled White Web” (C. Séquin, S. Wagon, D. Schwalbe, B. Collins, S. Reinmuth)

20. “WWW” Wins Silver Medal
Granada 2003
Still, the jury stuck by their decision reached 45 minutes earlier – and awarded the silver medal to the “Whiled White Web”!
Nevertheless, you may understand why I prefer to work with more durable materials now!

21. ISAMA 2004
Here is another shape that also came out of my sculpture generator. In this case I made a small 6-inch model on an FDM machine, and then sent it to Dingli Stone Carving Art Co., in SE China. I told them to scale it up by a factor of 8 and carve it from solid granite.

22. Yet Another Medium: Stone
Granada 2003
Yet Another Medium: Stone
After a few weeks I got a picture showing the work in progress.
(BTW: This is the same place that also carved the MLK monument in Washington, DC. – also starting with a small model)
Progress picture from Dingli Stone Carving Art Co., SE China

23. A few months later, this is what they shipped to us a in crate.
ISAMA 2004
A few months later, this is what they shipped to us a in crate.

24. Granada 2003
Here we just finished installation. Paul Suciu, standing in the middle, sponsored this project. He is a former EECS student of Prof. David Hodges.
Everybody is very happy that we could assemble the sculpture without scratching the beautiful, polished surface.

25. Granada 2003
You should go and check it out on the 6th floor terrace of Sutardja Dai Hall!
-- The craftsmanship in this work is absolutely wonderful. It looks like a metal cast!

26. I am sure it took many, many hours of sanding and polishing.
Granada 2003
I am sure it took many, many hours of sanding and polishing.

27. The Design Process Designing is the main activity of engineers:
Granada 2003
The Design Process
Designing is the main activity of engineers:
We are designing geometric shapes, mechanisms, motors, algorithms, data-bases, wearable computers, user interfaces …
I have now designed and built things for about 60 years:
Mostly geometrical stuff: IC-chips, buildings, sculptures, toys …
Occasionally, I pause and look over my own shoulder and ask myself: How does a design come to be?
Designing is a main activity of engineers. -- I have now designed and built things for about 60 years: mostly geometrical stuff: IC-chips, buildings, sculptures, or toys…
Occasionally, I pause and look over my own shoulder and ask myself: How does a design come to be ??

28. The Design Process EECS: Architecture:
Granada 2003
The Design Process
In 1989: E 298B-4: “Interdisciplinary Seminar on Design Theory and Methodology” A. Agogino (ME), W. Ibbs (CE), C. Séquin (CS).
Translate the languages used in different domains, e.g.:
EECS:
Architecture:
Top-down
Form
follows
Function
In 1989 Alice Agogino organized an inspiring and influential “Interdisciplinary Seminar on Design Theory and Methodology” where we tried to distill out how different disciplines understand and teach the design process. We found, for example, that in EECS we distinguish top-down and bottom-up design phases. On any real design task, one will typically alternate between the two modes and end up with a “meet-in-the-middle” approach, also called Yoyo Design. --- In architecture, however, they turn this diagram through 90 degrees and distinguish between “form-follows-function” and “function-follows-form;” and these 2 purist approaches are mitigated by the constraints imposed by the surrounding environment, by the budget, and by code requirements.
Yoyo Design
Bottom-up
Environment, Constraints

29. Granada 2003
The Design Cycle
Overall, Design is an iterative process, which has been depicted in a hundred different ways – many of which you can see on the Web, – and several of which you have heard described in some BID seminars. But, in the end, the basic concept is pretty much the same for all of them: -- You take your best guess – you build it or simulate it – you evaluate it – you make improvements – and you go around the cycle again.
So the structure of the Design Process is not much of a mystery; it is mostly constrained optimization.
A plethora of variations can be found on the web . . .

30. The Real Miracle: Creativity !
Granada 2003
The Real Miracle: Creativity !
Where do novel ideas come from?
How do we introduce a paradigm shift that cannot be found just through small optimization steps?
How do we learn to think “outside the box”?
The real miracle is “Creativity”: Where do novel ideas come from? How can we introduce a paradigm shift that cannot be found by just small optimization steps. How do you learn to think outside the box?
--- In this domain there is much less of a consensus!

31. Creativity according to W. Shockley
Granada 2003
Creativity according to W. Shockley
Here is one wild model, proposed by Nobel-prize winner Bill Shockley: You find this diagram on the inside cover of his text book on mechanics.
We do not have time to study this now. I just want to make the point that ideas about where creativity comes from are much more divergent, and recommendations as to what is needed to enhance it differ widely.
From inside cover of “mechanics”

32. Granada 2003
CREATIVITY
My own strong belief is that creativity is strongly linked to a playful state of mind. (You don’t invent new combat moves in a crisis situation).
But when you are relaxed, with extra time at your disposal, then your brain can make novel connections.
PLAY

33. CHS at Play ? Art-Math Conference “Bridges” Winfield, Kansas, 1999
Granada 2003
CHS at Play ?
Art-Math Conference “Bridges” Winfield, Kansas, 1999
Therefore, I like to play with things and to build “toys” and fun stuff. At the end of the graduate course in Creative Geometric Modeling in 1985, we put on an “art exhibit”.
>>> And in 1998 I helped to launch the annual Art-Math conference called “Bridges.”
1985: Exhibit at Conclusion of “Creative Geometric Modeling”

34. The Bridges Conference
ISAMA 2004
The Bridges Conference
Mathematical Connections
in Art, Music, and Science the largest, best-established, annual Math / Art conference in the world
Bridges, now in its 18th year, has evolved into the largest and best-established, Math-Art conference.
It’s main mission is to make Mathematics more attractive and less “scary” to the general public.

35. My Favorite Annual Conference: 2014
ISAMA 2004
My Favorite Annual Conference: 2014
BRIDGES Art …
Last year the conference was held in Seoul, Korea, in the newly opened, Gwacheon National Science Museum. This I a fabulous place!

36. ISAMA 2004
“LEGO®” Knots (2014)
The main talk I gave there was on “LEGO-Knots”. This was a project where I built a set of modular, tubular parts that matched the LEGO Duplo building blocks in cross section.
These curved parts allow the construction of various models of mathematical knots and links, as well as the creation of free-form constructivist sculpture models.
A modular system of building blocks matched to the LEGO Duplo parts.

37. “LEGO-Knots” My initial building blocks Henk van Putten (2013)
ISAMA 2004
“LEGO-Knots”
Here is a close-up on the first few parts that I built on an FDM machine. They were just enough to emulate two sculptures by Hank van Putten, which were the inspiration for this project…
My initial building blocks Henk van Putten (2013)

38. Making Sculptures Glow …
ISAMA 2004
Making Sculptures Glow …
And since the parts are hollow, I can also thread Xmas lights into them and make them glow.
This is particularly relevant with the Holiday Season approaching!

39. ? Modualarity ! Can it also be applied to more “free-form” shapes?
Granada 2003
Modualarity !
Can it also be applied to more “free-form” shapes?
Can we start with a Klein bottle module and form connected sums of Klein bottles resulting in single-sided surfaces of higher genus ?? ?
Since then I have been hooked on modularity! And I wondered: Can modularity also be applied to more free-form shapes? Can we start with a Klein bottle module and form connected sums of Klein bottles resulting in single-sided surfaces of higher genus ?
On the right is a glass structure created by Cliff Stoll, who lives here in Berkeley and sells Klein bottles of all kinds. -- However, this “Double Klein Bottle” is not actually a single-sided surface, but a double-sided torus of genus 1 with self-intersections.
Kiva Ford: “Worlds smallest Klein bottle”
Cliff Stoll

40. Connected Sum of Two Klein Bottles
Granada 2003
Connected Sum of Two Klein Bottles
Here are two ways of making a true single-sided surface of genus-4 that corresponds to the connected sum of 2 Klein bottles. On the left is a rather integral solution that would be difficult to assemble from snap-together parts.
On the right in the bottom row you see two identical tubular parts that incorporate the characteristic mouth of a classical Klein bottle. If you join them together as shown on top, you get the desired non-orientable surface of genus 4. If, instead, you connect them as shown on the bottom right, you obtain an orientable 2-hole torus of genus 2.
An “integral” solution A “modular” solution

41. Modular “Super-Bottles”
Granada 2003
Modular “Super-Bottles”
Some key components … A C
To push the modularity concept even further, I designed a few different tubular junction parts with three arms that emerge with right angles between them. In all four cases shown, two of the arms expose one side of the surface, while the third arm exposes the opposite side. These components can readily be placed at the corners of a cube. D S

42. Granada 2003
Genus 10 Super-Bottle
Eight of these corner components form a complete cube frame. The 8 corner pieces can be arranged in a few hundred different ways, and in most cases the surface will be single-sided and non-orientable and have a genus of In a few special cases, the parts can be arranged so that the 2-sidedness of the surface is maintained, and we then get a torus of genus 5.
Based on a cube frame; 4 different corner components.

43. Genus 8 Super-Bottle Based on the frame of a 3sided prism.
Granada 2003
Genus 8 Super-Bottle
Here I used the same corner components; but with the addition of 6 curved connector pieces bending through 30 degrees, I can now form the edge-frame of a 3-sided prism.
This will form a single-sided surface of genus 8 in most cases; but it can also be arranged into a 4-hole torus of genus 4.
=== At this point, I would like to thank Chris Myers in the Invention Lab for his help in building all the necessary parts on the Stratasys U-Print machine.
Based on the frame of a 3sided prism.

44. Granada 2003
BID Lab Opening, May 14, 2004
As a conclusion, here is a slide from a talk that I gave at the BID Lab Opening a decade ago.
My talk had the pompous title: “Design, Technology, and the Human Experience”. I looked at the role of the computer in the creative process and in aesthetic optimization, which then led into a discussion how BID might help bridge the gap between Art and Engineering. I think that all of you, who have participated in BID over the last few years, can attest to the fact that there is no longer a gap between these two disciplines; they have been fused together quite intimately.