1. University of California, Berkeley
Florida 1999
CS 284
Sculpture Generator I
Carlo H. Séquin
University of California, Berkeley
In Collaboration with
Brent Collins
Gower, Missouri
Roots of what Jimmy’s paper is all about

2. Brent Collins “Genesis” – Brent Collins at BRIDGES 2000 Florida 1999
Brent Collins is a professional artist living in Gower, MO, who has been carving abstract geometrical structures from solid wood blocks or from laminated assemblies.
“Genesis” – Brent Collins at BRIDGES 2000

3. Brent Collins: Early Sculptures
Granada 2003
Brent Collins: Early Sculptures
Brent Collins is an intuitive artist. He has no formal training in Math or geometry. Before we got in contact, these are the kinds of sculptures that he created. Some of these geometries are inspired by things you can find in nature. Many of his sculptures comprise minimal surfaces which form an intricate composition of tunnels and saddles. But still, this is pretty amazing – given that he had no math background… Because there are some neat math principles displayed in these shapes:
All photos by Phillip Geller

4. Collins’ Abstract Geometric Art
Beautiful symmetries
Graceful balance of the saddle surfaces
Superb craftsmanship
Intriguing run of the edges
What type of knot is formed ?
Mystery: one-sided or two-sided ?
What fascinates me about these creations…

5. Leonardo -- Special Issue
Granada 2003
Leonardo -- Special Issue
On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art
George K. Francis with Brent Collins
George Francis has analyzed Collins’ work from a mathematician’s point of view.
That is how I first became aware of Collins and his sculptures.

6. Brent Collins: Stacked Saddles
Granada 2003
Brent Collins: Stacked Saddles
Some simpler structures – mostly just towers of stacked saddles, possibly twisted.

7. Scherk’s 2nd Minimal Surface
Granada 2003
Scherk’s 2nd Minimal Surface
Normal
“biped”
saddles
A key element in all these sculptures is a sequence of holes and saddles. In the simplest case it is just a sequence of biped saddles that you would find on the back of a horse. If you take an infinite vertical stack of such saddles, they form the central part of what is known as Scherk’s second minimal surface. The artistically interesting central portion of this surface I am calling a Scherk tower.
This geometry 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”

8. 2nd and 3rd Order “Scherk Towers”
Granada 2003
2nd and 3rd Order “Scherk Towers”
Show these models!
Also: Scherk towers with 3rd and 4th order saddles.

9. “Hyperbolic Hexagon” by B. Collins
Granada 2003
“Hyperbolic Hexagon” by B. Collins
6 saddles in a ring
= “wound up” 6-story Scherk tower
6 holes passing through symmetry plane at ±45º
Discussion: What if …
we added more stories ?
or introduced a twist before closing the ring ?
A picture of this small wood sculpture prompted me to call Brent Collins on the telephone – and it resulted in a 45 intense discussion, and a 17-year long collaboration…. This piece has 6 of those biped saddles arranged in a toroidal ring. George Francis, a math professor at the University of Illinois, had already pointed out to Brent that this toroid could be a wound-up 6-story Scherk tower. So this made our phone conversation easier – since we had a common vocabulary. Our discussion started from the insight that to make a good 3D sculpture is not that easy, if you want to make it look good from all sides. The original Hyperbolic Hexagon has some weaknesses in that respect because from some angles it has some not so pleasing coincidences, because it is too symmetric: All 6 peripheral holes pass through the dominant plane at +/- 45 degrees. Perhaps this can be fixed by adding some twist into this toroid – as follows…

10. Closing the Loop straight or twisted “Scherk Tower”
Granada 2003
Closing the Loop
straight or
twisted
So we give the Scherk Tower some longitudinal twist before we close it into a loop. At right you can see the result without twist on top, and with 180 deg of twist at the bottom. It is less symmetric – but, I believe, more interesting.
In our discussion it also occurred to us that if you have an odd number of stories in the Scherk tower, you suddenly get Moebius surfaces! On the left, top and bottom saddles are both green; so when you join them: green meets green. But if you add one more story, the top saddle would be yellow; and if you join this with the green one at the bottom, you get a non-orientable, single-sided surface!
Also, as you add twist, edges no longer connect with themselves, but connect into other edges and form intertwined loops and knots. All this sounded rather intriguing, but was hard to visualize while talking on the phone.
“Scherk Tower”
“Scherk-Collins Toroids”

11. “Hyperbolic Heptagon” - Paper Skeleton
Granada 2003
“Hyperbolic Heptagon” - Paper Skeleton
So immediately after our phone call, we both started to build models of the “Hyperbolic Heptagon”… [[ Show yellow paper ring model ]].
This is the model that I constructed.

12. Brent Collins’ Prototyping Process
Granada 2003
Brent Collins’ Prototyping Process
And on the left you see what Brent built – from embroidery hoops!
Good idea: First build a model, before you invest much time and effort to start on a full-size sculpture -- which may take 2-3 month of carving and sanding and polishing.
However, even the beeswax model on the right took a couple of weeks to make. -- And we were generating 2-3 interesting new ideas every week in subsequent phone conversations.
There is simply not enough time to build models to check out all of them!
Armature for the Hyperbolic Heptagon
Mockup for the Saddle Trefoil
Time-consuming ! (1-3 weeks)

13. Sculpture Generator 1, GUI
Granada 2003
Sculpture Generator 1, GUI
Computer to the rescue – Let’s build virtual models!
Virtual prototyping provides a dramatic speed-up and lets you explore a much larger solution space!
This is the GUI of ScuGen. 1.
About 10 sliders define the geometry of this shape:
Order and number of sadles, their height, width, and thickness of the flanges, the amount of bending and twisting of the whole structure, and the treatment of the edges: square or rounded…

14. Base Geometry: One Scherk Story
Granada 2003
Base Geometry: One Scherk Story
Internal representation of geometry.
Hyperbolic Slices ==> Triangle Strips
Pre-computed -- then warped into toroid

15. Some of the Parameters in “SC1”

16. Generated Scherk-Collins Shapes
Typical generated forms.

17. The Basic Element Scherk’s 2nd minimal surface
3-story tower, trimmed, thickened
180 degrees of twist added

18. Toroidal Warp into Collins Ring
8-story tower
warped into a ring
360º twist added

19. A Plethora of Shapes

20. Edge Treatment
square, flat cut
semi-circular
bulging out

21. Embellishment of Basic Shape
color
texture
background

22. A Simple Scherk-Collins Toroid
Parameters: (genome)
branches = 2
stories = 1
height = 5.00
flange = 1.00
thickness = 0.10
rim_bulge = 1.00
warp =
twist = 90
azimuth = 90
textr_tiles = 3
detail = 8

23. Also a Scherk-Collins Toroid
branches = 1
stories = 5
height = 1.00
flange = 1.00
thickness = 0.04
rim_bulge = 1.01
warp = 360
twist = 900
azimuth = 90
textr_tiles = 1
detail = 20

24. A Scherk Tower (on its side)
branches = 7
stories = 3
height = 0.2
flange = 1.00
thickness = 0.04
rim_bulge = 0
warp = 0
twist = 0
azimuth = 0
textr_tiles = 2
detail = 6

25. V-art Virtual Glass Scherk Tower with
Granada 2003
V-art
Virtual Glass Scherk Tower with
Monkey Saddles (Radiance 40 hours)
Jane Yen
With this generator I can quickly create a whole lot of promising artistic geometries, -- and with some extra effort spent on rendering, we can now create virtual art and produce realistic looking images of these sculptural shapes.
But what does it take to make a real physical object?

26. How to Obtain a Real Sculpture ?
Prepare a set of cross-sectional blue prints at equally spaced height intervals, corresponding to the board thickness that Brent is using for the construction.

27. Collins’ Fabrication Process
Granada 2003
Collins’ Fabrication Process
Here is one way ny which Brent builds his wood sculptures…
With ruler and compasses he constructs slices through the desired geometry at different levels, and cuts out the various layers from wood boards, which he then glues together.
In this way he made himself this outer part of a torus surface on which he then could sketch the undulating pathway on the right, creating these undulations reminiscent of the protuberances on the solar surface.
Hence the name: “Vox Solis” -- Voice of the Sun.
Wood master pattern for sculpture
Layered laminated main shape
Example: Vox Solis

28. “Vox Solis” by Brent Collins
Granada 2003
“Vox Solis” by Brent Collins
This eventually leads to this final bronze sculpture.

29. Our First “Joint” Sculpture
Granada 2003
Our First “Joint” Sculpture
Six monkey saddles in a ring with no twist
(like Hyperbolic Hexagon)
azimuth = –30°, flange 1.5
(aesthetics)
size, thickness
(fabrication consideration)
The proposed geometry coming out of the generator.

30. Brent Collins “Hyperbolic Hexagon II”
Granada 2003
Brent Collins
Brent Collins, seen here holding our first collaborative piece.
“Hyperbolic Hexagon II”

31. Inauguration Sutardja Dai Hall 2/27/09
Granada 2003
Inauguration Sutardja Dai Hall 2/27/09
Master was used to make a mold.
Once a mold for this sculpture was in existence, it was not too expensive to get a copy cast for myself.
You can see that copy in the 3rd floor lobby of Sutardja Dai hall.

32. Slices through “Minimal Trefoil”
Granada 2003
Slices through “Minimal Trefoil”
50%
30%
23%
10%
Brent builds some of his structures in layers. Hed has very good 3D visualization skills, and he can sketch out the necessary layers not just for completely trivial shapes.
Still, something like this “Trefoil” is probably the limit what he can do.
These are the layers that you would get if you sliced this Minimal Trefoil shape – which came out of my sculpture generator.
Also, this is a conceptual view of a layered manufacturing process to be discussed later!
45%
27%
20% 5%
35%
25%
15% 2%

33. 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.
But Brent could never hope to manually generate this kind of slice geometry!
This is one of a dozen actual drawings that I sent to Brent. It was produced in my sculpture generator. It represents the geometry on one of the wooden boards from which Brent then built our second collaborative piece: the Heptoroid. The 5 superposed traces show the geometry of this sculpture at top & bottom of the board, as well as at levels ¼ ½ and ¾ in the board.
Brent uses a saber saw to cut these shapes out of inch thick boards. He chose to cut perpendicular to the board; so he cut around the outermost contours.

34. Emergence of the Heptoroid (1)
Granada 2003
Emergence of the Heptoroid (1)
Brent cuts out the 12 profiles from wood boards, and laminates them together with industrial-strength wood glue. In this way he obtained this rough shape which contains all the right symmetries, but show strong stair-casing on its surface.
Assembly of the precut boards

35. Emergence of the Heptoroid (2)
Granada 2003
Emergence of the Heptoroid (2)
Then he grinds down the stair-casing to create a smooth surface.
Here you see a continuous rim emerge – about an inch wide.
It travels around the loop 8 times before it gets back to the starting point.
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

36. Emergence of the Heptoroid (3)
Granada 2003
Emergence of the Heptoroid (3)
In another month of work, Brent continues to reduce the thickness of the surface and to make it as smooth as possible.
The glue lines are additional nice cues – in addition to the haptic feedback from running the hand over the surface
Smoothing the whole surface

37. The Finished Heptoroid
Granada 2003
The Finished Heptoroid
The finished sculpture. We called it “Heptoroid” because it has seven 4th-order saddles in a toroidal twisted loop.
at Fermi Lab Art Gallery (1998).

38. Extending the Paradigm: “Totem 3”
Granada 2003
Extending the Paradigm: “Totem 3”
So let’s look how the basic paradigm of the Scherk Collins Toroids can be extended:  Simple stretching of the original round shape, can lead to something that looks more like totem pole.
This slide also shows one other important step: Around 2000 Steve Reinmuth figured out how to turn the plastic maquettes coming off the FDM machine into bronze sculptures. Lost-ABS process … ABS sublimates at a few hundred centigrades…
Bronze Investment Cast

39. “Cohesion” SIGGRAPH’2003 Art Gallery
Granada 2003
“Cohesion”
Another example of a dirfect conversion from ABS plastic into bronze… (design consists of just two monkey saddles)
SIGGRAPH’2003 Art Gallery

40. Sculpture Design branches = 4 storeys = 11 height = 1.55 flange = 1.00
Granada 2003
Sculpture Design
branches = 4
storeys = 11
height = 1.55
flange = 1.00
thickness = 0.06
rim_bulge = 1.00
warp =
twist =
azimuth = 56.25
mesh_tiles = 0
textr_tiles = 1
detail = 8
bounding box:
xmax= 6.01,
ymax= 1.14,
zmax= 5.55,
xmin= -7.93,
ymin= -1.14,
zmin= -8.41
Another design that lead to a large sculpture (10 feet diameter). Here you can see the relevant parameters

41. Granada 2003
Two Modules Assembled
The modules and how they go together to form a ring.

42. Master Module for “Millennium Arch”
Granada 2003
Master Module for “Millennium Arch”
Here is the master module – full size! From an NC-milling machine, coated with wax to make a smooth surface for a mold in silicone rubber.

43. Fabrication of “Millennium Arch”
Granada 2003
Fabrication of “Millennium Arch”
From this master module, a negative mold is made.
In this mold 6 copies are then cast in this yellow polyester.
The mold for the key module
A polyester segment cast

44. Two Times Three Modules
Granada 2003
Two Times Three Modules
2x3 modules assembled for easier shipping – and so that it would fit through the door!
Laid out on the floor of the atrium where it was supposed to go.

45. Merging the Two Half-Circles
Granada 2003
Merging the Two Half-Circles
And there the two halves were epoxied and fused together.

46. and refining the surface.
Granada 2003
Curing the bond,
and refining the surface.

47. Brent Collins with “Millennium Arch”
Granada 2003
Brent Collins with “Millennium Arch”
The whole 10-foot diameter circle assembled.

48. Granada 2003
Installation

49. “Millennium” Arch by Night
Granada 2003
“Millennium” Arch by Night
And there it hangs under the skylight.
This was my first experience with building large permanent sculptures. Through these two pieces, Brent got some more good exposure ==> A new commission for 2 sculptures for a science building..

50. Granada 2003
Sculpture Generator #2
Kyu Simm, and undergraduate researcher with me in Fall 2010 helped me to program this Sculpture Generator #2 that can fit a whole saddle chain along an arbitrary B-spline. He took my Sculpture Generator I and ported it to a more modern environment. (C++ prog lang. OpenGL-graphics system, AntTweakBar –GUI, )
It can handle arbitrary spline curves; but has a special high-level module to optimize a symmetrical trefoil knot. Here is a first design …

51. First Maquette some bad spots! from FDM machine
Granada 2003
First Maquette
some bad spots!
When I thought we were close to a final design, I made a first maquette made on the FDM machine.
This showed that there were some problems that the computer rendering did not reveal:
Thin spots at the saddle centers. Needed some program fixes.
from FDM machine

52. Granada 2003
Refined Maquette
Flaws corrected. Final design except for flange thickness.

53. Final Design for “Evolving Trefoil”
Granada 2003
Final Design for “Evolving Trefoil”
Final design,

54. After Cleaning away Support
Granada 2003
After Cleaning away Support
Three identical modules merge from the cleaning bath.
[[ show models ]]

55. Three Modules Assembled
Granada 2003
Three Modules Assembled
That is how they go together to form half the final sculpture.

56. Simulation of “Evolving Trefoil”
Granada 2003
Simulation of “Evolving Trefoil”
No fabrication pictures yet. Completion also planned for late december 2012.
So for now, the closest we can come to experiencing “Evolving Trefoil” is this artistic computer rendering done by Charles de Rousiers, a coworker with Prof. Ramamoorthi in the CS Division. This was done more than a year ago in the early phase of my design process of this sculpture.
Stay tuned – more pictures will come to my web site.
I hope this gave you a glimpse how an inspirational model can lead to a procedural description -- which then can lead to a whole lot of related artifacts.