1. Artistic Geometry IAH Seminar April 20, 2017 Carlo H. Séquin
Granada 2003
IAH Seminar April 20, 2017
Artistic Geometry
Good art for me comes in 2 classes…
Carlo H. Séquin
EECS Computer Sciences
University of California, Berkeley

2. Great Art that I Simply Admire
Granada 2003
Great Art that I Simply Admire
First there is art that I simply can admire.
I would never attempt to improve on a cave painting like this…

3. Great Art that I Simply Admire
Granada 2003
Great Art that I Simply Admire
I would never attempt to improve on a classical painting like this either …
Michelangelo “Creation of Adam” Sistine Chapel
Michelangelo “Creation of Adam” Sistine Chapel

4. Great Art that I Simply Admire
Granada 2003
Great Art that I Simply Admire
These Dutch paintings also look perfect to me.
Van Gogh: “Starry Sky”
Vermeer: “Music Lesson”
Van Gogh: “Starry Sky”
Vermeer: “Music Lesson”

5. Great Art that I Simply Admire
Granada 2003
Great Art that I Simply Admire
And I would not know how to improve on Picasso’s “Guernica”
Picasso: “Guernica”

6. Great Art that I Simply Admire
Granada 2003
Great Art that I Simply Admire
The same is true for certain types of sculptures: I simply stand back and admire the artists who could do something like this.
Michelangelo: “David”
Auguste Rodin: “The Thinker”
Michelangelo: “David” Auguste Rodin: “The Thinker”

7. Art that Inspires Me to Do My Own
Granada 2003
Art that Inspires Me to Do My Own
On the other hand, Paul Klee is an artist who inspires me to try to do something similar on my own.
I would like to try to do something like that myself!

8. A Timid Attempt at Improvement
Granada 2003
A Timid Attempt at Improvement
But this is really cheating, since the left hand side of the new composite is also a Klee painting.
It is actually much harder than it looks to improve on a Klee painting!
– Let’s try something simpler first …
Paul Klee ???

9. Not My Plan to Improve on the Masters!
Florida 1999
Not My Plan to Improve on the Masters! ?
Or one could do some more sophisticated hacking with PhotoShop …
But this is NOT my plan to “improve” on the masters.
Picasso ? ? ?

10. Art that Inspires Experimentation
Granada 2003
Art that Inspires Experimentation
/artists/holland.shtml
Perhaps some of you have done some experiments where you tried to create some new art in the style of some famous artist.
There are even some web sites that allow you to do this -- based on a painting by Piet Mondrian…
On the web you can find several on-line painting programs that produce similar images:
EnchantedLearning.com Art and Artists
Abstract Geometric Canvas Art Print: Digitally created abstract canvas print.
Piet Mondrian
abstract-geometric-canvas-art-print/

11. Experiments in 3D Yin Yang Symbol MatriXArtbyDV Analogy in 3D
Granada 2003
Experiments in 3D
On the web you can also find many beautification attempts of the classical YinYang symbol.
But I wanted to take this into a new dimension – and literally go into the third dimension. So I gave this as a class exercise where I showed them the 2D symbol and asked them to “do this in 3D!” The student’s quickly understood that this meant to split a sphere into two parts along some S-0shaped surface that itself is composed of smaller spherical elements.
“Yin Yang - Original Modern Painting on Canvas, Acrylic, Abstract Yin Yang Decor, Yin Yang Art, Black and White Yin Yang, Yin Yang Silhouette” -- MatriXArtbyDV ($59)
Yin Yang Symbol MatriXArtbyDV Analogy in 3D

12. 3D Yin-Yang Stereolithography models by C. H. Séquin, 1999
Granada 2003
3D Yin-Yang
And this was my own solution at that time.
So, today I want to focus on my experiments with generated 3-dimensional geometry.
Stereolithography models by C. H. Séquin, 1999

13. “Re-creations” of Inspiring Art
Granada 2003
“Re-creations” of Inspiring Art
I have been doing some re-creations of inspiring art already when I was in high school.
Much inspiration came from Alexander Calder and his mobiles.
On the right is my own construction using ping-pong balls done almost 60 years ago.
Alexander Calder: Mobiles
C. H. Séquin, circa 1957

14. String Sculptures ?: Naum Gabo String Sculptures
Granada 2003
String Sculptures ?:
Naum Gabo also provided inspiration with his string sculptures made from plexi-glass and filament. On the right is a construction made from copper pipe and thin plastic hose.
Naum Gabo String Sculptures
C. H. Séquin, circa 1977

15. “Glowing TetraTangle”
Granada 2003
Orderly Tangles
This inspirational book by Alan Holden provided many recipes of how to make regular tangled objects out of wooden dowels.
It inspired me to build something out of the 4”-diameter cardboard tubes that carried the paper in the large Versatec plotters.
“Glowing TetraTangle”
C. H. Séquin (1983)
Alan Holden

16. Knots Chinese Button Knot C. H. Séquin (1994)
Granada 2003
Knots
Knots are a great inspiration to make artistic models – out of copper tubing or flexible drier ducts.
None of the experiments so far required any computers or CAD tools …
Chinese Button Knot
C. H. Séquin (1994)

17. Simple Geometry Max Bill FDM Models by C. H. Séquin (2000)
Granada 2003
Simple Geometry
This changed after 1995 when 3D printers became available to me.
Now I could make precise geometrical maquettes of my derivations.
One of the early “victims” whose work I could emulate was Max Bill – a Swiss sculptor.
Max Bill
FDM Models by C. H. Séquin (2000)

18. Ribbon Sculptures Altamont Collins: Pax Mundi (1994) Stelvio
Granada 2003
Ribbon Sculptures
Altamont
But my key inspiration and my key collaborator at that time was Brent Collins.
Here is a wood sculpture of his, and the kind of things it inspired me to do
-- and that is what I want to focus on today.
Collins: Pax Mundi (1994)
Stelvio

19. Toroids Twisted Hexagon Monkey Trefoil
Granada 2003
Toroids
Twisted Hexagon
And this is the sculpture that prompted me to start a collaboration with Brent Collins.
On the right you see some derived shapes; and how this was done is the main story of my lecture.
Monkey Trefoil
Brent Collins (1993) Hyperbolic Hexagon

20. Brent Collins (1997) “Hyperbolic Hexagon II”
Granada 2003
Brent Collins (1997)
Brent 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”

21. Brent Collins: Stacked Saddles
Granada 2003
Brent Collins: Stacked Saddles
Brent Collins has no mathematical training. He uses his intuition as well as rulers and compasses.
These are the kinds of sculptures that he created before we made contact.
One motif that stands out are the intricate compositions of tunnels and saddles – and often the surfaces in between look like soap film membranes.
All photos by Phillip Geller

22. The Math in Collins’ Sculptures
Granada 2003
The Math in Collins’ Sculptures
Collins works with rulers and compasses; any math in his early work is intuitive.
He is inspired by nature, e.g. soap films (= minimal area surfaces).
Prof. George Francis: “Connection to math. Minimal Surfaces!”
Brent is inspired by what he sees in nature, for instance soap films forming minimal surfaces.
George Francis is a mathematics professor at the University of Illinois. He was the first one to notice the connections between Collin’s sculptures and some well defined mathematical surfaces.

23. Scherk’s 2nd Minimal Surface (1834)
ISAMA 2004
Scherk’s 2nd Minimal Surface (1834)
In particular, the classical minimal surface discovered by Scherk in This surface is formed by two planes intersecting in the z-axis and the intersection line is then replaced by an infinite series of crisscrossing tunnels with smooth saddles between them.
Only the central portion of this surface is artistically interesting. The four flanges go off to infinity and become flat and rather boring. This central part I call a “Scherk Tower”.
The central part of this is a “Scherk Tower.”

24. 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”

25. “Scherk-Collins Toroids”
Granada 2003
Closing the Loop
straight or
twisted
Now we can take such a Scherk tower and bend it into a toroidal loop. If we close it into a straight loop without any twisting, we get the result at top right; – it is nice and symmetrical, but somewhat dull.
But if we give the tower a longitudinal twist of 180 degrees, as shown on the left, we obtain a less symmetric – but, I believe, more interesting, more dynamic result, shown at the bottom right.
“Scherk Tower”
“Scherk-Collins Toroids”

26. 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.
= Warped “Scherk Tower” (with 6 stories).
And this is the picture that prompted me to pick up the phone and call Brent Collins out of the blue. We had a very good 45-minute conversation, and this started a 20-year collaboration.
Brent called this wood sculpture “Hyperbolic Hexagon.” Because of the article by George Francis, we had a common language to discuss this shape over the phone, where we could not see each other or point to any jointly visible object. We both readily understood this as a 6-story Scherk tower closed into a toroidal loop.
In this first phone conversation, we discussed what might happen if we added a 7th story into this ring, or gave the Scherk tower an initial twist. Under some circumstances that surface then becomes single-sided, like a Moebius band, and the edges of the sculpture would form complicated knots. This was all intellectually challenging, -- but how do we know whether these geometries make beautiful sculptures that are worth 3 months of Collins’ time to carve it?

27. 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 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.

28. 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 the 30 inch scale.

29. Collins’ Fabrication Process
Granada 2003
Collins’ Fabrication Process
How does Brent create his wood sculptures? Often he uses some “manual layered manufacturing.” He constructs cross sections at regular intervals and cuts them out of 1” thick wood boards, and those pieces are then put together with industrial strength glue. Then the rough shape is smoothed out and on that surface he would then draw the next level of detail and carve it out by hand.
Wood master pattern for sculpture
Layered laminated main shape
Example: “Vox Solis”

30. 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 do this! -- This is one slice I designe 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 Mahagony boards.

31. 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

32. Emergence of the Heptoroid (2)
Granada 2003
Emergence of the Heptoroid (2)
As he does this, one can see a continuous broad rim emerge – about an inch wide. . It travels around the loop 8 times before it gets back to the starting point, because for this sculpture, the Scherk-tower has been given a twist of 135 degrees (3/8 of a full turn) before closing the loop.
Brent then continues to grind down this edge to a smooth, narrow surface.
Forming a continuous smooth edge

33. Emergence of the “Heptoroid” (3)
Granada 2003
Emergence of the “Heptoroid” (3)
In this phase the sense of touch becomes very important. Also, the glue lines between the different layers of wood are a good indication whether the surface is nicely curved.
Smoothing the whole surface

34. 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 the accelerator; -- or the inner shape for one of the elementary quark particles.
== This is also the process by which “Hyperbolic Hexagon II” had been constructed.
at Fermi Lab Art Gallery (1998).

35. “Scherk-Collins” Sculptures (FDM)
Granada 2003
“Scherk-Collins” Sculptures (FDM)
Now that I had this wonderful playground of Sculpture Generator, 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.

36. Cohesion SIGGRAPH’2003 Art Gallery
Granada 2003
Cohesion
Some of them I then sent to Steve Reinmuth’s Bronze Studio in Eugene, OR, where they got converted into bronze by at classical investment casting process.
This sculpture has only 2 monkey saddles. It is about a foot tall.
SIGGRAPH’2003 Art Gallery

37. Hypersculpture: Family of 12 Trefoils
Granada 2003
Hypersculpture: 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 # of windings in the toroidal loop. – Let me explain that . . .
W=1
B= B= B= B=4

38. Going more than once around the loop
Granada 2003
Going more than once around the loop
W = 380°
W = 560°
W = 720°
On the left is a toroidal loop that sweeps through slightly more than 360 degrees. In the middle it goes around the loop 1.5 times. And at right it makes two full turns. Now If I use an odd number of stories (5 in this case), and just the right amount of twisting, I can create a structure that has no self-intersections.
… results in an interwoven structure.

39. 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.

40. Stepwise Expansion of Horizon
Granada 2003
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.
So this exploration with the sculpture generator is like a hike in the Swiss mountains: As you climb the first set of hills you then see the next ridge; on that ridge you can then see some high mountains; from the top of which you can finally see the Matterhorn. Each experiment stimulated ideas for some sort of extension of the generator program – like going twice around the loop. And a few minutes of programming allowed me to extend the range of possible shapes.
Swiss Mountains

41. Sculpture Generator 1 as a Playground
Granada 2003
Sculpture Generator 1 as a Playground
The computer becomes an amplifier / accelerator for the creative process.
I think this is the bottom line why we should embrace the use of the computer even in the art world.
Often in the art world: Bigger is Better!. So most artists dream to see some of their creations at a much larger scale.
– Well, in 2003 we got an opportunity to build such a sculpture 12 feet tall . . .

42. V-art Virtual Glass Scherk Tower with
Granada 2003
V-art
Virtual Glass Scherk Tower with
Monkey Saddles (Radiance 40 hours)
Jane Yen
Here is another design coming out of the sculpture generator – but this is only a virtual design.
But I was also fortunate enough to get a larger sculpture out of this – in a very durable material.

43. ISAMA 2004
It started with this small 6-inch model made on an FDM machine. This was send it to Dingli Stone Carving Art Co., in SE China, and we told them to scale it up by a factor of 8 and carve it from granite.

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

45. Granada 2003
Spring, 2012
And a few months later a couple of boxes arrived on a container ship.

46. And this is what we found in one of the crates,
ISAMA 2004
And this is what we found in one of the crates,
And the other one contained a cylindrical base …

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

48. Because this elegant form has a very finely crafted surface.
Granada 2003
Because this elegant form has a very finely crafted surface.

49. Granada 2003
The craftsmanship in this work is absolutely wonderful. It looks like a metal cast!

50. 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.
Check it out on the 6th-floor terrace of Sutardja Dai Hall at UC Berkeley.

51. The Viae Globi Series (Roads on a Sphere)
Granada 2003
The Viae Globi Series
(Roads on a Sphere)
Another example how one special piece of art led to a computer program, which then allowed me to make a whole series of sculpture designs that all seem to belong to the same family.
I want to give you one more example where a special piece of art . . .

52. 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 a carved wood sculpture by Brent Collins, which he did 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?”

53. How to Model Pax Mundi ... Already addressed that issue in 1998:
Pax Mundi could not be done with Sculpture Generator I
Needed a more general program !
Used the Berkeley SLIDE environment.
First: Needed to find the basic paradigm   
Fortunately I had already addressed that issue in It was clear that “SG1” could not produce such shapes! . . .
First I had to figure out what is the conceptual model behind Pax Mundi.. .

54. Sculptures by Naum Gabo
It reminded me of sculptures by Naum Gabo, which I had seen as a student in a visit to Paris.
The edge forms an undulating pathway on a sphere.
I call these types of curves “Gabo curves”.
Pathway on a sphere:
Edge of surface is like seam of tennis- or base-ball;
 “2-period Gabo curve.”

55. 2-period “Gabo Curve”
Here is how I define them in my computer program: The bluish rectangle on the right is a Mercator projection of the surface of the Earth. The equator is the horizontal line in the middle; North pole is on top, South pole at the bottom.
In this domain I defined an undulating curve that crosses the equator a specified number of times. Here we have two full waves.
The curve is a B-spline, and I only needed 3 parameters (shown by the small blue arrows) to control the shape of the curve, its amplitude, and the width of the lobe.
Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used (symmetry!).

56. 4-period “Gabo Curve” Same construction as for as for 2-period curve
And if we cram 4 complete periods around the equator, the result would look like this.
Same construction as for as for 2-period curve

57. Pax Mundi Revisited
Can be seen as: “Amplitude modulated, 4-period Gabo curve”
Now, with this new view of things, I could characterize Pax Mundi as an “amplitude-modulated 4-period Gabo curve.”
Brent was not too excited!. Also: Advice from some of his friends: Don’t get involved with this computer nerd! You will loose your creativity and no longer be able to do your own art. Fortunately, he was not too concerned about this.

58. 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 parameterizable 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.

59. Modularity of Gabo Sweep Generator
Sweep Curve Generator:
Gabo Curves as B-splines
Cross Section Fine Tuner:
Paramererized shapes
Sweep / Twist Controller
Modularity: A key ingredient of a good modeling program.

60. Emulation; Define Master Pattern
Granada 2003
Emulation; Define Master Pattern
Master to make a mold from.
Alignment tab
First, I made an effort to extract the minimal amount of “master geometry” that had to be defined in detail, so that the whole sculpture could be assembled from several copies from it. This turned out to be 1/4 of the whole sculpture. I defined this geometry as a curvy sweep through space (shown on the right), and also provided this segment with extra alignment tabs, so that the 4 pieces could be joined with perfect alignment.
Use 4 copies.

61. Joe Valasek’s NC Milling Machine
Granada 2003
Styrofoam milling machine
Steve Reinmuth found a person with a NC milling machine… This is used to mill the master geometry from high-density styro-foam.
Unfortunately, the gantry of this machine had a clearance of only 14 inches, and it could not handle my complicated, bulky 3D part.

62. Machined Master Pattern #2
Granada 2003
Machined Master Pattern #2
This is one such part 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!

63. 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!

64. 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.

65. 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

66. 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

67. An even larger collection of ceramic creations & metal sculptures
Granada 2003
Eva Hild
An even larger collection of ceramic creations & metal sculptures
Here is a glimpse of my most recent endeavor, started last summer. I became enamored with the work of Eva Hild. She is a Swedish artist who creates large ceramic and metal sculptures in the form of organically undulating surfaces.
I started out by trying to understand and classify such sculptures from a mathematician’s point of view: whether they are 2-sided like a sheet of paper or single-sided like a Moebius band or a Klein bottle. How many borders they have, and what their genus is (this is a measure of their connectivity. But then I wanted to create my own models of some of her simpler surfaces. Here are some examples…

68. Eva Hild’s 2-Manifold Sculptures
Granada 2003
Eva Hild’s 2-Manifold Sculptures
This sculpture called “Hollow” is in Vaarberg, Sweden. It is large enough for children to climb around on it. 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. But what is the best way of modeling this? >> Funnels, tunnels, rims …
“Hollow” (Eva Hild) FDM Maquette
Topology: 1 border; single-sided; genus 2

69. Modeling Hild Sculptures
Granada 2003
Modeling Hild Sculptures
Find salient features: Rims (red), Funnels (blue), Tunnels (green)
Left: Hild: “Interruption” overlaid with markings of special features that seem to come up in most of her sculptures.
These elements are: red rims, blue funnels, green tunnels. I started to build a generator that provides these elements in an easy to use, parameterized form.
Middle: The first step then is to place some such elements in an appropriate way to match the given sculpture.
Right: This is the surface that results when these elements are smoothly connected.

70. Eva Hild Sculptures “Whole” (Eva Hild) FDM Maquette
Granada 2003
Eva Hild Sculptures
Here is another sculpture called “Whole” and a maquette fro an inexpensive printer.
>>> When trying to re-create a not-completely trivial sculpture, I first try to make a rough clay model that captures the basic topology.
>>> Based on that I then find the best way to place some strategic tunnels and funnels and surround them with an appropriate rim to obtain a good model.
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

71. “Hild-like” Sculptures?
Granada 2003
“Hild-like” Sculptures?
Extending the paradigm . . .
I mentioned earlier possible paradigm extensions. Here I took the Hild sculpture that you just saw, which 3 ups and down in the surrounding Gabo rim, and logically extended it to more tunnels and a rim that has 6 ups and downs.

72. Single-sided “Hild” Sculptures
Granada 2003
Single-sided “Hild” Sculptures
All(?) Hild sculptures seem to be 2-sided.
Creating single-sided sculptures (Moebius bnds) by connecting an odd number of “Dyck disks.”
But then I was looking for a more systematic way of deviating from Hild’s designs. It turns out that all her sculptures that I analyzed in this respect turn out to be 2-sided surfaces. So I asked myself, can a make a shape that at first glance looks like one of her sculpture, but which is a Moebius-like single-sided surface?
The left image show a structure called a “Dyck disk”.
If the two funnels are connected with a tube, as shown in the middle, then we obtain a single-sided surface. -- But this does not look very Hild-like; she constructs no such toroidal loops.
On the right, I have connected THREE Dyck disks into a cycle; this also creates a single-sided surface; and this one looks much more like a possible Hild sculpture.

73. Granada 2003
Rings of 5 and 7 Dyck Disks
Actually, any odd number of Dyck disk closed into circular loop will create a single-sided surface. Here you see maquettes with 5 and with 7 Dycke disks. (The one on the left is the way it comes off a 3D printer with the support scaffolding still in place.
The question arises, at what point does it stop to be possibly seen as a Hild sculpture.

74. Complex 2-Manifold Sculpture
Granada 2003
Complex 2-Manifold Sculpture
Here is a much more typical Hild sculpture. For something like this, it is much more difficult to figure out how to make a parameterized, procedural model. This is something that I am currently struggling with. I envision plenty of fun challenges ahead!!
“Wholly” by Eva Hild (Sweden)

75. Granada 2003
Conclusions
I am grateful to artists who through their work inspire me to discover my own creations in a somewhat similar style.
In a few instances, a collaboration evolved that allowed us to create things that neither of us could have done alone.
Bridging the gap between the arts and the math/engineering world can be very rewarding.
So, in conclusion I like to say that …

76. ISAMA 2004
QUESTIONS ? ?