ZURICH, Aug. 2002

1
ZURICH, Aug. 2002

• ART -- MATH
• Carlo H. Séquin
• University of California, Berkeley

2
Focus of Talk

• How can we use the visualization power offered
  by computer graphics and by computer-controlled
  rapid prototyping for the design of geometrical
  sculptures?

3
Outline

• Background (Why?)
• Collaboration with Brent Collins
• Parameterized Sculpture Families
• Sculpture Optimization

4
I am a Designer
5
Roots of My Passion for Sculpture

• My love for geometry and abstract
  sculptureemerged long long before I learned to
  play with computers.
• Thanks to Alexander Calder, Naum Gabo,Max
  Bill, M.C. Escher, Frank Smullin, ...

6
Leonardo -- Special Issue
On Knot-Spanning Surfaces An Illustrated Essay
on Topological Art With an Artists Statement by
Brent Collins
George K. Francis with Brent Collins
7
Brent Collins Early Sculptures
All photos by Phillip Geller
8
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 ?

9
Hyperbolic Hexagon II (wood)
Brent Collins
10
Brent Collins Stacked Saddles
11
Scherks 2nd Minimal Surface
Normal biped saddles
Generalization to higher-order saddles(monkey
saddle)
12
Hyperbolic Hexagon by B. Collins

• 6 saddles in a ring
• 6 holes passing through symmetry plane at 45º
• wound up 6-story
  Scherk tower
• What would happen,
• if we added more stories ?
• or introduced a twist before closing the ring ?

13
Closing the Loop
straight or twisted
14
Collins - Séquin Collaboration

• Discuss ideas on the phone
• Exchange sketches
• Vary the topological parameters
• But how do you know whether it is beautiful ?
  Need visual feedback.
• Making models from paper strips is not good
  enough.
• A key problem is making the sculpture look good
  from all sides !

15
Brent Collins Prototyping Process
Mockup for the "Saddle Trefoil"
Armature for the "Hyperbolic Heptagon"
Time-consuming ! (1-3 weeks)
16
Collins Fabrication Process

• Building the final sculpture (2-3 months)
• Take measurements from mock-up model,transfer
  parallel contours to 1 boards.
• Roughly precut boards, leaving registration
  marksand contiguous pillars for gluing boards
  together.
• Stack and glue together precut boards,remove
  auxiliary struts.
• Fine-tune overall shape,sand and polish the
  surface.
• A big investment of effort !

17
Collins Fabrication Process
Wood master patternfor sculpture
Layered laminated main shape
Example Vox Solis
18
Sculpture Generator, GUI
19
Sculpture Generator I

• Prototyping Visualization tool
  forScherk-Collins Saddle-Chains.
• Slider control for this one shape-family,
• Control of about 12 parameters.
• Main goal Speed for interactive editing.
• Geometry part is about 5,000 lines of C
• 10,000 lines for display user interface.

20
Scherk-Collins Sculptures
21
The Basic Element
Scherks 2ndminimal surface
3-story tower,trimmed, thickened
180 degreesof twist added
22
Toroidal Warp into Collins Ring
8-story tower
warped into a ring
360º twist added
23
A Plethora of Shapes
24
Edge Treatment
square, flat cut
semi-circular
bulging out
25
Embellishment of Basic Shape
color
background
texture
26
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 360.00
• twist 90
• azimuth 90
• textr_tiles 3
• detail 8

27
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

28
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

29
1-story Scherk Tower

• branches 5
• stories 1
• height 1.35
• flange 1.00
• thickness 0.04
• rim_bulge 0
• warp 58.0
• twist 37.5
• azimuth 0
• textr_tiles 8
• detail 6

30
180º Arch Half a Scherk Toroid

• branches 8
• stories 1
• height 5
• flange 1.00
• thickness 0.06
• rim_bulge 1.25
• warp 180
• twist 0
• azimuth 0
• textr_tiles e
• detail 12

31
Main Goal in Sculpture Generator I

• Real-time Interactive Speed !
• Cant afford surface optimizationto obtain true
  minimal surfaces
• also, this would be aesthetically too limited.
• ? Use closed-form hyperbolic approximation.

32
V-art
VirtualGlassScherkTowerwith MonkeySaddles(R
adiance 40 hours) Jane Yen
33
How to Obtain a Real Sculpture ?

• Prepare a set of cross-sectional blue printsat
  equally spaced height intervals,corresponding
  to the board thickness that Brent is using for
  the construction.

34
Slices through Minimal Trefoil
50
10
23
30
45
5
20
27
35
2
15
25
35
Profiled Slice through the Sculpture

• One thick slicethru Heptoroidfrom which Brent
  can cut boards and assemble a rough
  shape.Traces represent top and bottom,as
  well as cuts at 1/4, 1/2, 3/4of one board.

36
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)

37
Another Joint Sculpture

• Heptoroid

38
Heptoroid ( from Sculpture Generator I )
Cross-eye stereo pair
39
Emergence of the Heptoroid (1)
Assembly of the precut boards
40
Emergence of the Heptoroid (2)
Forming a continuous smooth edge
41
Emergence of the Heptoroid (3)
Smoothing the whole surface
42
Advantages of CAD of Sculptures

• Exploration of a larger domain
• Instant visualization of results
• Eliminate need for prototyping
• Create virtual reality pictures
• Making more complex structures
• Better optimization of chosen form
• More precise implementation
• Rapid prototyping of maquettes

43
SFF (Solid Free-form Fabrication)
Monkey- Saddle Cinquefoil
44
Various Scherk-Collins Sculptures
45
Fused Deposition Modeling (FDM)
46
Looking into the FDM Machine
47
Zooming into the FDM Machine
48
Séquins Minimal Saddle Trefoil

• Stereo-lithography master

49
Séquins Minimal Saddle Trefoil

• bronze cast, gold plated

50
Minimal Trefoils -- cast and finished by Steve
Reinmuth
51
Brent Collins Trefoil
52
Part III
Developing Parameterized Sculpture Families
53
Family of Symmetrical Trefoils
W2
W1
B1 B2 B3
B4
54
Close-up of Some Trefoils
B1 B2
B3
Varying the number of branches, the order of the
saddles.
55
Higher-order Trefoils (4th order saddles)
W1 (Warp)
W2 ?
56
Exploring New Ideas W2

• Going around the loop twice ...

resulting in an interwoven structure.
57
9-story Intertwined Double Toroid
Bronze investment casting from wax original
made on 3D SystemsThermojet
58
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
59
Note
The computer becomesan amplifier /
acceleratorfor the creative process.
60
Inspiration Brent Collins Pax Mundi
61
Keeping up with Brent ...

• Sculpture Generator I can only do warped Scherk
  towers,not able to describe a shape like Pax
  Mundi.
• Need a more general approach !
• Use the SLIDE modeling environment(developed at
  U.C. Berkeley by J. Smith)to capture the
  paradigm of such a sculpturein a procedural
  form.
• Express it as a computer program
• Insert parameters to change salient aspects /
  features of the sculpture
• First Need to understand what is going on ?

62
Sculptures by Naum Gabo

• Pathway on a sphere
• Edge of surface is like seam of tennis ball
• ? 2-period Gabo curve.

63
2-period Gabo curve

• Approximation with quartic B-splinewith 8
  control points per period,but only 3 DOF are
  used.

64
4-period Gabo curve

• Same construction as for as for 2-period curve

65
Pax Mundi Revisited

• Can be seen as Amplitude modulated, 4-period
  Gabo curve

66
SLIDE-UI for Pax Mundi Shapes
67
Viae Globi Family (Roads on a Sphere)
L2 L3 L4
L5
68
Via Globi 3 (Stone)
Wilmin Martono
69
Via Globi 5 (Wood)
Wilmin Martono
70
Via Globi 5 (Gold)
Wilmin Martono
71
Extending the Paradigm

• Try to Expand the Sculpture Family
• Aim for more highly convoluted paths,
• maintain high degree of symmetry.
• Need a better tool to draw on sphere

72
Circle Splines on the Sphere

• Examples from Jane Yens Editor Program

73
Via Globi -- Virtual Design
Wilmin Martono
74
Maloja -- FDM part

• A rather winding Swiss mountain pass road in the
  upper Engadin.

75
Stelvio

• An even more convoluted alpine pass in Italy.

76
Altamont

• Celebrating American multi-lane highways.

77
Lombard

• A very famous crooked street in San Francisco
• Note that I switched to a flat ribbon.

78
Part IV
Using Virtual Shapes and Physical 3D Models for
Sculpture Optimization
79
Another Inspiration by B. Collins
80
Collins Conceptual Design
SWEEP CURVE (FOR DOUBLE CYLINDER) IS COMPOSED OF
4 IDENTICAL SEGMENTS, FOLLOWS THE SURFACE OF A
SPHERE.
81
Reconstruction / Analysis (v1)
FROM THE FDM MACHINE
AWKWARD ALIGNMENT
82
Further Explorations (v2 add twist)
83
A More Complex Design (v3)
84
Verification with 3D Model (v4)
GALAPAGOS-4 (SIDE
VIEW)
85
Fine-tuned Final(?) Version (v5)
86
Galapagos-6 in the Making
87
Galapagos-6 (v6)
88
Conclusions (1)

• Virtual Design / Prototyping is a novel medium
  (to artists).
• It can play an important role -- even for
  traditional sculptors
• it can save time and labor, and
• allows to tackle sculptures of a complexitythat
  manual techniques could not conquer.

89
Conclusions (2)

• The computer is not only a great visualization
  and prototyping tool,
• it also is a generator for new ideas and
• an amplifier for an artists inspiration.

90
Conclusions (3)

• Rapid prototyping (layered fabrication)must now
  be considered a new facetin the spectrum of MM
  technologies.
• It provides tangible (high-quality haptic)output
  for objects with which usersmay want to
  interact.
• Even for sculptures(intended primarily for
  visual enjoyment)the physical maquette
  disclosessubtle geometrical features that
  arenot visible in the virtual rendering.

91
Acknowledgements

• Brent Collinsfor his inspiring artworkand many
  stimulating discussions.
• Jordan Smith,Jane Yen,Human Meshkin,for
  developing some of the software modules that I am
  using in my work.

92
Questions ?
THE END
93
SPARE

94
Conclusions (3)

• What makes a CAD tool productive for this kind
  of work ?
• Not just virtual clay,
• partly procedural
• fewer parameters that need to be set.
• Keep things aligned, joined
• guarantee symmetry, regularity,
• watertight surfaces.
• Interactivity is crucial !

95
Some of the Parameters in SC1