1. Knotting Math and Art, USF, 2007
Granada 2003
Knotting Math and Art, USF, 2007
Analysis and Synthesis of Intuitively Conceived Geometrical Art
Collaboration between
Brent Collins, Gower, MO
Carlo H. Séquin, Berkeley, CA

2. Leonardo -- Special Issue
Granada 2003
On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art
With an Artist’s Statement by Brent Collins
George K. Francis with Brent Collins
Our collaboration was prompted by this article in Leonardo
Leonardo -- Special Issue

3. Brent Collins (SUNY Albany 1997)
Granada 2003
Brent Collins (SUNY Albany 1997)
Since 1995 I have had a close collaboration with Brent Collins whom you see here holding our first collaborative wood sculpture.
Today I want to describe to you a few of our collaborative pieces and the process by which they came into being.
“Hyperbolic Hexagon II”

4. Brent Collins: Stacked Saddles
Granada 2003
Brent Collins: Stacked Saddles
Long before we knew of each other Brent was carving (among other works) these kinds of sculptures from solid three trunks.
In the last segment of our talk, Brent will show you a collection of his works that gives you an idea of the breadth of his work.
Why are these sculptures particularly relevant to this talk ? Look at all the saddles in the interior of these sculptures ...

5. Scherk’s 2nd Minimal Surface
Granada 2003
Scherk’s 2nd Minimal Surface
Can be understood in relation to ...

6. Scherk’s 2nd Minimal Surface
Granada 2003
Scherk’s 2nd Minimal Surface
Normal
“biped”
saddles
Artistically only the core of this surface is interesting – consisting of a chain of holes and saddles.
Normally you just find ordinary bi-ped saddles as you would put on a horse,
but if you start with 3 planes intersecting in the z-axis then you get 3rd order saddles – so called monkey saddles.
Generalization to
higher-order saddles (monkey saddle)

7. “Scherk Tower” 5-story core Monkey saddles Thick surface “Flare” added
Granada 2003
“Scherk Tower”
5-story core
Monkey saddles
Thick surface
“Flare” added
Artistic interpretation of this basic topology. It uses 5 monkey-saddle stories from the core. Of course we have to thicken the surface.
And to make it more interesting some flare has been added to the overall shape.

8. “Hyperbolic Hexagon” by B. Collins
Granada 2003
“Hyperbolic Hexagon” by B. Collins
6 saddles in a ring
6 holes passing through symmetry plane at ±45º
= “wound up” 6-story Scherk tower
Discussion: What if …
we added more stories ?
or introduced a twist before closing the ring ?
The piece that prompted me to pick up the phone in late

9. Closing the Loop
straight or
twisted

10. “Solar Arch” (Collins and Séquin, 1997)
12 stories
4th-order saddles
270° twist

11. Brent Collins’ Prototyping Process
Armature for the "Hyperbolic Heptagon"
Mockup for the "Saddle Trefoil"
Time-consuming ! (1-3 weeks)

12. “Sculpture Generator I”, GUI

13. Design of “Solar Arch” branches = 4 storeys = 11 height = 1.55
flange = 1.00
thickness = 0.06
rim_bulge = 1.00
warp =
twist =
azimuth = 56.25
detail = 8

14. Breckenridge Competition 1997

15. “Solar Arch” is among 5 Finalists
2nd place

16. Collins’ Fabrication Process
Wood master pattern for sculpture
Layered laminated main shape
Example: “Vox Solis”

17. Slices through “Minimal Trefoil”
50%
30%
23%
10%
45%
27%
20% 5%
35%
25%
15% 2%

18. 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.

19. Emergence of the “Heptoroid” (1)
Assembly of the precut boards

20. Emergence of the “Heptoroid” (2)
Forming a continuous smooth edge

21. Emergence of the “Heptoroid” (3)
Smoothing the whole surface

22. The Finished “Heptoroid”
at Fermi Lab Art Gallery (1998).

23. “Scherk-Collins” Sculptures (FDM)

24. Fused Deposition Modeling (FDM)

25. Zooming into the FDM Machine

26. Hypersculpture: Family of 12 Trefoils
W=2
W=1
B= B= B= B=4

27. Extending the Notion of a “Saddle”
B= B= B=3
B = number of branches = the order of the saddles.
B = 1: A “one-leg saddle”?  just a simple band.

28. Another Extension …
Allow different kinds of “stretching” …

29. Extending the Paradigm: “Totem 3”
Bronze Investment Cast

30. “Cohesion”
SIGGRAPH’2003 Art Gallery

31. Going more then once around the loop ...
w = 380°
w = 560°
w = 720°
… results in an interwoven structure.

32. 11 Stories, Monkey-Saddles, W=2:
cross – eye stereo picture

33. 9-story Intertwined Double Toroid
Bronze
investment
casting from wax original made on 3D Systems’ “Thermojet”

34. Stepwise Expansion of Horizon
Playing with many different shapes and
experimenting at the limit of the domain of the sculpture generator,
stimulates new ideas for alternative shapes and generating paradigms.
Swiss Mountains

35. Sculpture Generator 1 as a Playground
The computer becomes an amplifier / accelerator for the creative process.
Another occasion where Sculpture Generator 1 became invaluable ...

36. Snowsculpting Championships 2003
The Team: “Whirled White Web” (C. Séquin, S. Wagon, D. Schwalbe, B. Collins, S. Reinmuth)

37. Day 1
Removing lots of snow …

39. Day 2: Making a Torus

40. End of Day 2
The Torus

41. Day 3, am: Drawing Flanges

42. Day 3, pm: Carving Flanges, Holes

43. Day 4: Geometry Refinement

45. End of Day 4: Desired Geometry

46. Day 5, am: Surface Refinement

47. Official Team Photo

48. Judgement Time: “Whirled White Web”

49. 12:40 pm ° F

50. 12:41 pm ° F

51. “WWW” Wins Silver Medal

52. Large and Durable Sculpture !
Need a material more permanent than snow ...

53. Another Inspiration: Brent Collins’ “Pax Mundi” (1997)

54. Keeping up with Brent ...
Pax Mundi cannot be done with Sculpture Generator I
Needs a more general program !
First: Need to understand what is going on  

55. Sculptures by Naum Gabo
Pathway on a sphere:
Edge of surface is like seam of tennis- or base-ball;
==> 2-period Gabo curve.

56. 2-period “Gabo Curve”
Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used (symmetry!).

57. 4-period “Gabo Curve”
Same construction as for as for 2-period curve

58. “Pax Mundi” Revisited
Can be seen as: Amplitude modulated, 4-period Gabo curve

59. SLIDE-GUI for “Pax Mundi” Shapes
Granada 2003
Good combination of interactive 3D graphics and parameterizable procedural constructs.
Good combination of interactive 3D graphics and parameterizable procedural constructs.

60. 2-period Gabo sculpture
Tennis ball – or baseball – seam used as
sweep curve.

61. “Viae Globi” Family (Roads on a Sphere)
periods

62. “Via Globi 5” (Virtual Wood)
Wilmin Martono

63. Extension: Free-form Curve on a Sphere
Interpolating Circle Splines
on the plane and on the sphere
Carlo Séquin
Jane Yen
(2001)

64. Many Different “Viae Globi” Models

65. Paradigm Extension: Sweep Path is no longer confined to a sphere!
“Music of the Spheres” (Brent Collins)

66. 2006: Commission for a Big Sculpture!
Scale up original “Pax Mundi” (to 6ft diam.)
Less than 1500 pounds
Budget 50’000 $
Due in 4 months ( Nov. 2006)
Collaboration: Collins, Reinmuth, Séquin
My task: Create the digital file for a mold master

67. Target Geometry

68. Emulation; Define Master Pattern
Master to make a mold from.
Use 4 copies.

69. Subdivide into Two Master Segments

70. Joe Valasek’s CNC Milling Machine
Styrofoam milling machine

71. Machined Master Pattern #2

72. (Cut) Master  Silicone Rubber Mold

73. Mold  Several (4) Wax Copies

74. Spruing the Wax Parts for Casting

75. Ceramic Slurry Shell Around Wax Part

76. Shell Ready for Casting

77. Casting with Liquid Bronze

78. Bronze Cast Slowly Cooling Off

79. Cracking the Ceramic Shell

80. The Freed Bronze Cast

81. Assembly of Pieces

82. The “Growing” Ribbon

83. The Completed Assembly

84. Grinding the Welded Seams, Polishing the Surface

85. Applying Patina

86. Front Door
H&R Block Building

87. The Final Destination

88. Installation Process

89. Steve Tightening the Bolts

90. Brent Polishing Our Baby

93. “Pax Mundi II” at H&R Block Headquarters

95. QUESTIONS ? Our Next Commission 10 ft diameter
amber-tinted polyester resin
to be hung in an atrium below skylight