1. The Magic of 2-Manifold Sculptures Homage to Eva Hild
Florida 1999
LASER, Stanford, Feb.15, 2018
The Magic of 2-Manifold Sculptures Homage to Eva Hild
Eva Hild is one of a few artists whose work I not only admire, but who also inspires me to create my own intriguing, geometrical shapes.
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley

2. “Hollow” by Eva Hild, Varberg, 2006
Granada 2003
“Hollow” by Eva Hild, Varberg, 2006
Eva lives in Sweden and creates such beautiful, flowing surfaces with intricate connectivity.
I never met her personally; but became aware of her work on the Internet.

3. An large collection of ceramic creations & metal sculptures
Granada 2003
Eva Hild
An large collection of ceramic creations & metal sculptures
Her web-site shows a large and highly varied collection of 2-manifold sculptures – this means, a single, relatively thin surface with one or more rims or borders. Most of them are in ceramic, some of them in metal. Her sculptures are not only a pleasure to look at, but they invite mental exploration of questions such as: How many tunnels are there? How many separate rims are there? Is this a 2-sided surface, like a sheet of paper, or single-sided, like a Moebius band? These sculptures inspire me to try to create similar shapes. I do not possess the skills to create large ceramic pieces myself, so I create CAD models of such surfaces, and the more promising ones I then realize on a 3D printer.
Her creations come in a variety of styles:

4. Hild Sculptures: Different Styles
Granada 2003
Hild Sculptures: Different Styles
Some of her sculptures are collections of bulbous outgrowths, as shown on the left.
Others are characterized by intricately connected funnels of various sizes.
Both of these may have many rims or borders that are mostly circular in shape.
“Bulbs” & “Funnels” + many circular borders

5. Eva Hild Ceramic surfaces
Granada 2003
Eva Hild
Ceramic surfaces
Here is an example of the type of surface that interests and intrigues me the most:
It has some partial funnel shapes and many internal tunnels, but it also has this long, beautiful, undulating border curve.

6. More Ceramic Hild Surfaces
Granada 2003
More Ceramic Hild Surfaces
Here are a few more examples of this style of sculptures. I would like to make models of such shapes. But how do I do this? What CAD tool should I use? -- Most CAD tools that I am aware of do not give much support for such shapes. SolidWorks is great, if you want to design crankshafts or ventilation ducts. Maya is great, if you want to make shapes like a cuddly Teddy-bear.

7. Surface, connecting: Rims & Funnels & Tunnels
Granada 2003
Key Features
Rims Funnels Tunnels
First, I needed to identify the key features that define her sculptures:
The undulating border curves, which I call “rims”, marked in red,
If these curves are roughly circular or elliptical, I call them “funnels” and mark them in blue;
And then there are internal toroidal tunnels like you find in a donut; here are 3 of them, marked in green.
Surface, connecting: Rims & Funnels & Tunnels

8. Sculptures Defined by Key Features
Granada 2003
Sculptures Defined by Key Features
Here are two more-complex sculptures, on which I have marked many of these 3 types of key features.
The green “tunnel” construct can be used to define narrow constrictions, as well as the maximal cross-section of a bloated bulb.
Rims Funnels Tunnels
Marked features:

9. “NOME” Non-Orientable Manifold Editor
Granada 2003
“NOME” Non-Orientable Manifold Editor
Place key features: Rims, Funnels, Tunnels;
Connect their borders with surface patches;
Smooth the assembly with CC-subdivision;
Use offset-surfaces to thicken the 2-manifold;
Create finely tessellated B-rep (STL-file);
Send to 3D-Printer.
With the help of a couple of students, we are building a home-brewed CAD tool to make the design of such surfaces easier.
We call it NOME – standing for “Non-Orientable Manifold Editor” -- because single-sided surfaces like Moebius bands and Klein bottles are of particular interest to me. This tool allows us to place the key features mentioned into desired configurations and then connect the borders of the funnels and tunnels and rims with surface patches.
In the end, the crude, polyhedral assembly is smoothed with the Catmull-Clark subdivision process. It is then given some thickness; and a boundary representation of that object can be exported as an STL file, which can be sent to any 3D printer.

10. Computer-Aided Design Process
Granada 2003
Computer-Aided Design Process
Here is how this works for one of the simplest Eva Hild sculptures, called “Interruption.” On the left, I start with a picture of the Hild sculpture, and mark the 3 types of key features; then I use NOME to place simple ribbon elements that represent those features in 3D space.
In the middle column, you see the simple triangular and quadrilateral facets that connect the control polylines of the various features with one another to yield a polyhedral surface with the proper topology. After subdivision, the blue&red surface smoothly connects the various key features.
Now all we have to do is to form offset surfaces to obtain the desired material thickness and write out an STL file that we can send to a 3D-printer.
Modeling “Interruption”  D-print

11. Granada 2003
Room for Improvement
When I hold the first physical model in my hand, I typically see a few things that must be improved.
In my first stab at modeling “Interruption”, the tunnel at the lower left turned out to be pinched into a narrow oval.
I can fix this by inserting an additional, explicit tunnel at this location.
Here the tunnel is not as nicely rounded as in Hild’s original.

12. An Improved Model Two more tunnels added.
Granada 2003
An Improved Model
Two more tunnels added.
Here is my second try: I added two more tunnels to yield more circular openings as in Hild’s original.
In the top-center image, I have added the first few red quadrilaterals of the desired connecting surface, and at top right, the yellow facets complete the connecting surface.
In the bottom row, you see this surface after smoothing, then thickened with some offset, and finally the resulting 3D print.
Now the shape is more gracefully rounded.

13. “Hollow” by Eva Hild, Varberg, (2006)
Granada 2003
“Hollow” by Eva Hild, Varberg, (2006)
Double-sided (orientable)
Number of borders b = 1
Genus g = 2 (after closure)
2-hole torus with 1 puncture
With the emerging NOME tool in my hand, I was ready to tackle a more complex Hild surface, like this one, called “Hollow.”
For the mathematically inclined, here is the topological analysis of this sculpture: This is a double-sided surface with a single border. When you glue a suitably warped disk to this border, you obtain a simple two-hole torus. So this sculpture is a 2-hole torus of genus 2 with a single puncture.

14. Understanding “Hollow”
Granada 2003
Understanding “Hollow”
But this sculpture actually gave me a very hard time to gain a full understanding. For a long time, I could only find images from the front-side, as shown in the top row. == Eventually I managed to get an image of the backside by using Google Street-View and maneuvering into a position, from where I could see the greenish reflection in a large window behind the sculpture. And with some enhancements through Photoshop, I could obtain the image at bottom, left.
This was good enough to make a crude clay model to help me capture the connectivity of this surface.
Only more recently, did I find a picture of a ceramic model of this sculpture (bottom center) that gave me a clear view of the backside of this sculpture. Fortunately, it confirmed my own, earlier interpretation.
Front and back views & clay model

15. Modeling “Hollow” with NOME
Granada 2003
Modeling “Hollow” with NOME
Now I could model it in NOME. Top left, you see the key features that I placed to define this sculpture, and below it, the surface patches that I stitched between them.
In the middle is the merged surface, which was subdivided 3 times, and then thickened.
On the right are photos of the resulting 3D print.
Features – surface – D-print

16. Modeling Hild’s “Whole”
Granada 2003
Modeling Hild’s “Whole”
Here is the same process applied to another Hild sculpture of moderate complexity.
Again I made a clay model to gain a clear understanding of the connectivity of this surface.
With this, I was then able to identify the key features defining this sculpture: They are the 2 yellow “cross tunnels” and a 3-period “Gabo curve“ forming the undulating ribbon surrounding the core.
With this sculpture, I would like to point out an important difference between Hild’s way of creating a sculpture and my own, computer-assisted approach: …
Original – clay model – key features – 3D-print

17. Two Different Approaches
Granada 2003
Two Different Approaches
Eva Hild: -- incremental (organic) growth -- avoid strict symmetry -- no clear plan  surprising results My CAD approach: -- clear plan of overall final structure -- find maximal symmetry to reduce design work: -- green: D3-symm; overall: D1d=C2h
Hild grows her ceramic surfaces incrementally, a few inches at a time, -- in a process that may take weeks, or even months. She does not start out with a clear overall plan, but develops each sculpture as she gradually builds them. She does not aim for symmetry; she prefers more organic, less rigid shapes.
In contrast, I first look for overall symmetry; it reduces the overall amount of definition-work that I have to do. I only need to design part of a sculpture, and the rest then follows by repeating, or mirroring, the initial geometry. The green 3D-print model has 3-fold symmetry in its outer rim, and overall bilateral mirror-symmetry.

18. Extending the Paradigm
Granada 2003
Extending the Paradigm
Now, there is an advantage to a computer-based, procedural approach: With the key features defined for this sculpture, I can then readily make new sculptures by placing a different number of these features, and by choosing different symmetries. Here I show 3 models that started out, respectively, with 1, 2, and 3 cross tunnels, and are surrounded by Gabo curves that have 2, 3, and 4 periods, respectively. The bottom row shows the results.
=== In the model on the right, the 3 cross-tunnels are standing in a straight row…
1, , cross-tunnels
2, , Gabo undulations

19. 3 Cross-Tunnels In a 3-fold symmetrical configuration
Granada 2003
3 Cross-Tunnels
In a 3-fold symmetrical configuration
But these cross-tunnels can also be placed in a circular, 3-fold symmetrical configuration, and then be surrounded with a 3-period Gabo curve, to obtain a higher degree of symmetry.
To turn this into an attractive, and more dramatic, stand-alone sculpture, I added a conical stem at the bottom.

20. Free-form surfaces offer a bigger modeling challenge!
Granada 2003
“Wolly” by Eva Hild
Now, let’s look at one of Hild’s more complex and more convoluted, free-form sculptures. This one has no symmetry whatsoever! ==
How can I possibly capture a thing like this ??
Free-form surfaces offer a bigger modeling challenge!

21. 2-sided, single border, genus 4
Granada 2003
Topology of “Wholly”
The first step is always to figure out the topology and connectivity of the given surface.
Here it is captured in the relatively simple model at the bottom. This is an orientable surface of genus 4 with a single border.
Geometrically, it can be seen as a chain of 8 side-by-side tunnels.
2-sided, single border, genus 4

22. A Flexible, Parameterized Model
Granada 2003
A Flexible, Parameterized Model
The top row shows a basic polyhedral CAD-model that yields the proper topology for “Wholly.”
But it would be overwhelming to ask the user to move all individual vertices of this model into “appropriate” locations to recreate “Wholly.” -- We need a higher level of control. Here I defined the 9 cross sectional planes, 7 of which go through the walls between pairs of adjacent tunnels, and 2 more at the ends.
Each of these sections can now be non-uniformly stretched, rotated, and shifted. A result of such coordinated edits is shown in the bottom row. Each of the 9 cross sections remains planar and symmetrical, but their sizes and positions define the shapes and orientations of the tunnels between them. The resulting shape in the lower right starts to show the flavor of Hild’s “Wholly” sculpture.
Polyhedral model, with high-level edit-controls

23. First, Not Very Successful Attempt
Granada 2003
First, Not Very Successful Attempt
But, with the modeling approach described above, it was actually quite difficult and tedious to obtain good results.
Often the tunnels were squashed into narrow slits, and the undulations of the rim often ended up in sharp cusps.
This was the first model of “Wholly” that I sent to a 3D printer; and clearly, it was not very successful …
An FDM model of “Wholly” ?

24. Second Attempt with Same Set-Up
Granada 2003
Second Attempt with Same Set-Up
But such a model helps to find out where I needed to focus my attention.
By more carefully fine-tuning the polyhedral model, I was able to obtain a nicer looking model.
Wholly_A2: a more carefully tuned model

25. Comparison Wholly_A1 Wholly_A2 Tunnels still not very round
Granada 2003
Comparison
Wholly_A1
Wholly_A2
Tunnels still not very round
Here is a direct comparison between the two models: Clearly there is an improvement.
But it is still difficult to make the tunnels really round!
== So I tried a different approach…

26. Granada 2003
Trying a New Approach
The new modeling approach directly manipulates 8 parameterized, circular tunnels, which are the inner parts of tori, and thus remain nicely rounded.
I can vary their sizes and the angles between them, and the program keeps them in appropriate contact.
Combining 8 partial toroids into a flexible chain

27. Parameterization (2D concept proof)
Granada 2003
Parameterization (2D concept proof)
This simpler, 2D demonstration shows this concept in action:
== I can adjust the azimuth angle of the green circle;
== I can also vary its radius;
== And I can then alter any other radius or azimuth – for instance the purple one.
Radii, heights, azimuth angles, tilt are adjustable.
Automatic adjustment of tunnel/tunnel separation.

28. Resulting 3D CAD Model Wholly_B1
Granada 2003
Resulting 3D CAD Model
This is how I see the resulting 3D CAD model, as I fine tune the parameters that define the radii and heights of the 8 toroids, and the angles between adjacent ones.
Wholly_B1

29. Model after Clean-up Wholly_B1
Granada 2003
Model after Clean-up
And this is the model that then comes off an inexpensive 3D printer, -- propped-up in the proper position;. it demonstrates some further improvement.
However: For such truly free-form surfaces, it is still surprisingly difficult to make a good CAD model. For such shapes, the incremental, physical approach that Eva Hild uses is clearly superior. -- So, Eva does not have to fear competition form me anytime soon!
=== But her sculptures also inspire a different kind of quest: -- This sculpture is a 2-sided, orientable surface; -- and so are all the other ones that I have analyzed! -- I find it intriguing that Hild’s incremental creation process never seems to have resulted in a single-sided surface like a Moebius band.
Wholly_B1

30. Orientability 2-sided, 1-sided, orientable non-orientable
Granada 2003
Orientability
Cooling tower
Moebius band
So, I wanted to see, how the typical elements in Hild’s sculptures can be combined into non-orientable surfaces.
As a quick refresher: Here are some basic geometrical shapes and their topological orientabilities.
Cylinders and “cooling towers” are orientable and have distinct insides and outsides; but the Moebius band on the right is single-sided.
Hild’s “Interruption” is clearly 2-sided as shown by the red and green paint.
We can make a single-sided surface by starting with a “Dyck disk.” This is a double funnel with two stubs going off in opposite directions. When we connect these stubs into a closed loop, we obtain a non-orientable surface. This is a Klein bottle with a puncture.
Hild: “Interruption”
Dyck loop
2-sided, sided, orientable non-orientable

31. ISAMA 2004
Klein Bottles
A quick reminder: This is the classical Klein bottle, where inside and outside seamlessly merge into one another.
On the right is a picture of Cliff Stoll in Berkeley, who sells Klein bottles in a wide range of sizes.
-- But the toroidal tunnels that you find in these shapes are not part of Eva Hild’s repertoire of 2-manifold sculptures.
Klein bottle (1882) Professor Felix Klein Cliff Stoll, Berkeley

32. Dyck Loops 3, 5, 7 Dyck disks in a cycle
Granada 2003
Dyck Loops
However, we can obtain the same topological result, if we replace that toroidal handle with two or more Dyck disks.
Whenever we form ring with an odd number of Dyck disks, we will obtain a single-sided surface, equivalent to a Klein bottle with multiple punctures.
The bottom row shows loops with 3, 5, and 7 such disks.
== However, these surfaces would not be taken for a true Hild surface, -- they are too regular!
3, , Dyck disks in a cycle

33. A Less Regular Dyck Ring
Granada 2003
A Less Regular Dyck Ring
5 Dyck funnels + a “bulb”
Eva Hild typically avoids rigid symmetry and makes the tunnels and lobes in a sculpture of somewhat different sizes. So, here, I scaled subsequent instances of Dyck disks by 10% and let this logarithmic spiral sweep through only 300 degrees. The remaining 60 degrees are then filled-in with a bulbous element, as found in Hild’s sculpture “Hollow.”
This provides a convenient way to connect two Dyck disks with rather different diameters.
CAD model D-print

34. Final Take: “Pentagonal Dyck Cycle”
Granada 2003
Final Take: “Pentagonal Dyck Cycle”
After a little bit of fine-tuning in our NOME editor, this lead to this “Final Take” of the “Pentagonal Dyck Cycle”.
This sculpture made it into the Nominees’ Gallery of the Bridges 2017 Art Exhibit.
Made it into the Nominees’ Gallery of the Bridges Art Exhibition.

35. Tetrahedral configuration of six “4-stub Dyck funnels”
Granada 2003
Not a “Hild Sculpture”
I know, that my computer-assisted process most directly leads to regular, symmetrical surfaces that are not easily confused with Eva’s creations. So, let me push symmetry and topology to new extremes! Here, I start with “4-stub Dyck funnels.” These are disks with two stubs going off on either side. I place six of these across the edges of a tetrahedron, and I connect all 24 stubs with 12 curved tunnels. The result is is a single-sided surface of genus 14. (This is equivalent to the connected sum of 7 Klein bottles).
It may not be beautiful in the traditional sense, but it is definitely intriguing, and I was very happy when I found this highly symmetrical configuration. == In fact, I was so happy, that I wanted to find out what might happen, if I start with 24 such Dyck funnels and put them on the edges of a rhombic dodecahedron …
Tetrahedral configuration of six “4-stub Dyck funnels”

36. Dodecahedral-Cluster of Genus 50
Granada 2003
Dodecahedral-Cluster of Genus 50
24 4-stub funnels,
48 tunnels,
24-fold symmetry,
single-sided,
genus 50.
And this is what happens ! This thing has 48 tunnels and 24-fold symmetry. It is a single-sided surface of genus 50, and is thus equivalent to the connected sum of 25 Klein bottles with 24 punctures.
It took 132 hours (5.5 days) to build this model on a LULZBOT 3D-printer. Internal support removal took several more hours.

37. Conclusions Thank you, Eva Hild ! You are a wonderful inspiration.
Granada 2003
Conclusions
Thank you, Eva Hild !
You are a wonderful inspiration.
And while my models do not come close to the natural beauty of your ceramic surfaces, playing with the key elements of your sculptures has lead to some different, and intriguing surfaces.
In conclusion, I would thus like to say: Thank you Eva Hild!

38. Eva Hild: Snow Sculpture (2011)
Granada 2003
Eva Hild: Snow Sculpture (2011)
Eva also did a marvelous snow sculpture! -- Questions ?
Eva Hild