Connecting Art and Mathematics Granada 2003 UNC, April 19, 2017 Connecting Art and Mathematics Torus-Knot_3,5 This talk is about my transition from a mathematician to a physicist to an engineer and to an artist. Carlo H. Séquin EECS Computer Sciences University of California, Berkeley

Basel, Switzerland M N G I grew up in Basel, CH, Granada 2003 Basel, Switzerland M N G I grew up in Basel, CH, >>> where I attended the M N G (a high school with emphasis in math and science). >>> During my university years I heard my math lectures in this 500 year old institute, where many famous mathematicians have lectured:

Jakob Bernoulli (1654‒1705) Logarithmic Spiral Granada 2003 Jakob Bernoulli (1654‒1705) such as Jakob Bernoulli, or . . . Logarithmic Spiral

Leonhard Euler (1707‒1783) Imaginary Numbers … or Leonhard Euler. Granada 2003 Leonhard Euler (1707‒1783) … or Leonhard Euler. From an early age on, I was fascinated with numbers … Imaginary Numbers

ISAMA 2004 Descriptive Geometry and during high-school I fell in love with geometry. In 11th grade we had a subject called Descriptive Geometry, where we constructed the intersection lines between two cylinders. -- I thought that this was very cool stuff! – Ever since, geometry has been in my blood stream! But I love to design and build things! So I became an experimental physicist: Optics and geometry are closely related.

Geometry in every assignment . . . Granada 2003 Geometry in every assignment . . . CCD TV Camera (1973) Soda Hall (1992) During the last 40 years, I have been fortunate to be associated with many challenging design projects that involved nifty geometry problems. My first job was at Bell Labs in Murray Hill. I was put into the group that had just invented CCDs, and over 3 years we developed the first all-solid-state TV camera. (The success of this brought me to Berkeley). There I teamed up with D. A. Patterson, and we built the first Reduced Instruction Set Computer on a chip. At that time, my students and I worked mostly on IC-layout programs which offer a lot of 2D geometry problems. Then we needed a new home for CS; and I transitioned from 2D CAD to 3D CAD. Later I teamed up with Paul Wright in ME on a project called “CyberCut – CyberBuild”. This started with optimized tool paths for 3-axis milling machines to make free-form surfaces such as the 3D Yin-Yang. Then came the 3D printers and additive layered manufacturing.. . RISC 1 MicroChip (1982) 3D-Yin-Yang (2000)

More Recent Designs and Models Granada 2003 More Recent Designs and Models During the last two decades, I have worked with artists and with mathematicians. The top row shows some mathematical visualization models fabricated on 3D printers. Top-left is a 3D Hilbert cube with 512 “elbows”. Top right is a special Klein bottle. The bottom row shows two larger sculptures that emerged from my collaboration with Brent Collins. Often I could integrate the necessary design work with classes on CAD or solid modeling that I taught. I consider this to be a very fortunate situation.

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

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; –often the surfaces in between look like soap film membranes. All photos by Phillip Geller

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 Collins’ sculptures and some well defined mathematical surfaces.

Scherk’s 2nd Minimal Surface (1834) ISAMA 2004 Scherk’s 2nd Minimal Surface (1834) In particular, with the classical minimal surface discovered by Scherk in 1834. This surface is formed by two planes intersecting in the z-axis, where 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.”

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”

“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 before we close the loop, 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”

Brent Collins: Hyperbolic Hexagon (1993) Granada 2003 Brent Collins: Hyperbolic Hexagon (1993) 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). Here is the specific picture that prompted me to pick up the phone and call Brent Collins out of the blue. -- 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 (i.e. when there are an odd number of saddles) 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 them?

Brent Collins’ Prototyping Process Granada 2003 Brent Collins’ Prototyping Process This is the kind of models that Brent normally builds before committing 2-3 months of carving such a shape from solid wood. He makes some kind of skeletal frame – here made from embroidery-hoops. Then he fleshes out the skeleton with wire meshing and beeswax, shown in black on the right. But this is time consuming and may take a couple of weeks for such a model. At that time, we had more phone conversations -- about once a week --, and every time we came up with more new and intriguing ideas. So there was no way that Brent could keep up with making models for all these ideas. Armature for the "Hyperbolic Heptagon" Mockup for the "Saddle Trefoil" Time-consuming ! (1-3 weeks)

Sculpture Generator 1, GUI Granada 2003 Sculpture Generator 1, GUI Computer graphics to the rescue!! == 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. This program was partly developed in 1996, when I spent the Fall semester here at UNC on a sabbatical.

Some of the Parameters in “SC1” This slide shows more clearly some of the parameters used in this special-purpose program. With this program I could now generate very quickly a whole lot of intriguing shapes …

Generated Scherk-Collins Shapes On the left is an emulation of Brent’s “Hyperbolic Hexagon”; on the right is a variation that I call the “Minimal Trefoil”; and this was my own first optimized design. One key issue in programming this was: What should be the internal representation of these curvy shapes? Brent’s shapes were not truly minimal surfaces, and I wanted to keep good control over the resulting shapes.

Base Geometry: One “Scherk Story” ISAMA 2004 Base Geometry: One “Scherk Story” Taylored hyperbolas, hugging a circle The solution I chose was to model half a saddle as a stack of hyperbolas with varying curvature, whose heads always lie on the circular profile of a round tunnel shown in red. The hyperbolas are connected with triangle strips. This simple piece of geometry is defined just once, and is then replicated many times, stretched and twisted, and warped into a toroidal shape. Hyperbolic Slices  Triangle Strips Pre-computed -- then warped into toroid

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 a 30 inch scale.

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 then draws the next level of detail (as the protuberances mimicking solar flares, here) and then carves it out by hand. Wood master pattern for sculpture Layered laminated main shape Example: “Vox Solis”

Slices through “Minimal Trefoil” Granada 2003 Slices through “Minimal Trefoil” 50% 30% 23% 10% This is somewhat similar to what a 3D-printer does. The computer calculates the geometry of the individual slices – typically about 1/100” apart, And the machine then deposits the material in those places. But the complexity of a shape like this is about the limit of what Brent could design by hand. 45% 27% 20% 5% 35% 25% 15% 2%

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 construct this! -- This is one slice of “Heptoroid”, which I designed 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 Mohagony boards.

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

Emergence of the Heptoroid (2) Granada 2003 Emergence of the Heptoroid (2) As he does this, a continuous, broad rim emerges – 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 (which is 3/8 of a full turn) before closing the loop. Brent then continues to grind down this edge to form a smooth, thin surface. Forming a continuous smooth edge

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

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 their accelerator; -- or the inner shape for one of the elementary quark particles. at Fermi Lab Art Gallery (1998).

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

“Cohesion” (SIGGRAPH’2003 Art Gallery) Granada 2003 “Cohesion” (SIGGRAPH’2003 Art Gallery) Some of them were nice enough, so I sent them to Steve Reinmuth’s Bronze Studio in Eugene, OR. He converted them into bronze with a classical investment casting process, where the ABS plastic of the original was sublimated away in a hot kiln, and then replaced with bronze. This sculpture has only 2 monkey saddles. It is about a foot tall. Cast by Reinmuth Bronze Studio, Eugene, OR

Hyper-Sculpture: Family of 12 Trefoils Granada 2003 Hyper-Sculpture: 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 number of windings in the toroidal loop. – Let me explain this . . . W=1 B=1 B=2 B=3 B=4

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 with 7 stories 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. If I use an odd number of stories (7 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.

11 Stories, Monkey-Saddles, W=2: Granada 2003 11 Stories, Monkey-Saddles, W=2: Here is a more complex design. If you are good at looking at cross-eye stereo pictures (and you sit in the right place: central axis), you can see this in 3D. I did not quite believe myself that such geometries could be realized . . . cross–eye stereo picture

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, and then had Steve Reinmuth investment cast it in bronze and gold-plate it electrolythically.

Extension of Concept Allow different kinds of “stretching” … Granada 2003 Extension of Concept Going more then once around the loop, was an extension of the original concept, which allowed me to make new interesting shapes. Here is another simple extension of the basic concept. After a toroid has been defined, we can apply some non-uniform scaling, stretching the sculpture preferentially in the vertical dimension and somewhat less in either of the two horizontal dimensions. Allow different kinds of “stretching” …

Extending the Paradigm: Totem 3 Granada 2003 Extending the Paradigm: Totem 3 This then results in some sculpture that are reminiscent of “totem poles”. On the left is the original ABS plastic maquette; on the right is the bronze investment cast made by Steve Reinmuth. Bronze Investment Cast

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 exploratory extension of 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 further extension of the generator program – like going twice around the loop. And often just a few minutes of programming were required to allowed me to extend the range of possible shapes. Swiss Mountains

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. Now, I want to give you a second example where a special piece of art led to a computer program …

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 this carved wood sculpture by Brent Collins, which he created 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?” I was supposed to provide a detailed CAD model at the right size. This raises the question: How do you model something like this ? …

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 this issue in 1998. It was clear that “Sculpture Generator 1” could not produce such a shape! I needed a new program. But first I had to figure out what is the conceptual model behind Pax Mundi.. .

Sculptures by Naum Gabo Brent told me that this ribbon was embedded in the surface of a sphere. The curve also reminded me of sculptures by Naum Gabo, which I had seen as a student in a visit to Paris. In these sculptures, 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 the seam of tennis- or base-ball;  “2-period Gabo curve.”

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; the North pole is on top, the 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 because of the inherent symmetry, 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!).

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

Pax Mundi Revisited Can be seen as: “Amplitude modulated, 4-period Gabo curve” Now, with this new understanding of things, I could characterize Pax Mundi as an “Amplitude-modulated 4-period Gabo curve.” Two pairs of lobes get closer to the poles than the other two. Here the globe lies on its side, the poles are at the left and right extrema. (Brent was not too excited about my description of his master piece!).

Progressive Sweeps Sculpture is not just a mathematical curve. Granada 2003 Progressive Sweeps Sculpture is not just a mathematical curve. There is some substance; it has volume. Define shape by sweeping a cross section along a given 3D space curve. But this sculpture is not just a line … There is some substance to it; it has some volume. A convenient way to describe this shape is by sweeping a crescent-like profile along the given Gabo curve.

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 parametrizable 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. On display you see my final model of Pax Mundi.

Modularity of Gabo Sweep Generator Sweep Curve Generator: Gabo Curves as B-splines: Cross Section Fine Tuner: Paramererized shapes: Sweep / Twist Controller: How is cross section applied? Modularity is a key ingredient of a good modeling program. Here is what the 3 sets of controls do: -- the first one defines the sweep curve; -- the 2nd one fine tunes a parametrized cross section. The middle one was used for Pax Mundi. -- The 3rd bank of sliders controls how this cross section is applied to the sweep curve. This last one is particularly important, and in many commercial CAD tools, it is implemented in an unsatisfactory way.

Azimuth / Twist Control Granada 2003 Azimuth / Twist Control Controls applied to the 2-period Gabo curve: If implemented properly, it gives you a lot of control! In these examples, we always have the same sweep curve, but the cross section is applied differently as it sweeps along that curve. On the left the orientation of the cross section is determined by the local curvature of the sweep curve. In the other 3 images I use a minimum-torsion approach that minimizes local twisting; but I can still control the azimuth of the cross section and have it lie mostly tangential to the underlying sphere, or perpendicular to it. In addition, I can at will add additional twisting as shown on the right. These were the programming capabilities that I had in my hands when Brent asked for my help. Natural orientation with Frenet frame Torsion Minimization: Azimuth: tangential / normal 900° of twist added.

Target Geometry (2007) Constraints: Bronze, 70” diameter Granada 2003 Target Geometry (2007) Now, for the construction of the final CAD model, these were all the images that I had available, when I was given the task of scaling up this fine wood sculpture to a 6-foot bronze sculpture. >>> In addition, there were some hard constraints: … { LIST }. -- The middle 3 constraints did not allow me to simply scale up the geometry of the wood sculpture; I had to make the ribbon slimmer! == Now let me give you a glimpse of what it takes to realize this as bronze sculpture: Constraints: Bronze, 70” diameter Less than 1500 pounds Less than $50’000 Maintain beauty, strength Minimize master geometry

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.

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.

Design of Smaller Two-Part Master Granada 2003 Design of Smaller Two-Part Master -- I had to break it apart into 2 smaller U-shaped pieces that were relatively flat. Look at the blue part … Alignment tabs for easy assembly

Machined Master Pattern #2 Granada 2003 Machined Master Pattern #2 This is what it looks like 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!

(Cut) Master  Silicone Rubber Mold Granada 2003 (Cut) Master  Silicone Rubber Mold The raw styrofoam parts then get coated with this blue plastic paint to yield a smooth surface. Next, these master-geometry modules are used to make a silicon rubber mold (half of it is visible in white).

Mold  Several (4) Wax Copies Granada 2003 Mold  Several (4) Wax Copies Then this (white) rubber mold is used to cast FOUR identical positive copies in (brown) wax.

Spruing the Wax Parts for Casting Granada 2003 Spruing the Wax Parts for Casting These wax parts are then enhanced with the red sprues and runners and the funnels into which the molten bronze will be poured. The smaller ones also serve as air-venting holes.

Ceramic Slurry Shell Around Wax Part Granada 2003 Ceramic Slurry Shell Around Wax Part This whole thing then gets dipped repeatedly into plaster slurry to make a ceramic shell.

Taking the Shell out of the Kiln Granada 2003 Taking the Shell out of the Kiln These shells get fired in a hot kiln. In that process, the wax runs out, leaving a cavity of the desired shape for the bronze.

Shell Ready for Casting Granada 2003 Shell Ready for Casting Here is an empty shell positioned for the casting.

Granada 2003 The Pour Liquid bronze is poured into the shell.

Casting with Liquid Bronze Granada 2003 Casting with Liquid Bronze A close-up view of the casting process.

Freeing the Bronze Cast Granada 2003 Freeing the Bronze Cast When the parts have cooled down, the ceramic shell is smashed away. The sprues and runners need to be cut away, and some first surface cleaning and smoothing is done.

Assembling the Segments Granada 2003 Assembling the Segments Now the assembly can begin. Here is a first weld.

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!

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.

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

Front Door of the ... H&R Block Building Granada 2003 Front Door of the ... January 2007: Installation in Kansas City:…, lifting it off the truck… Just barely fit through the door…1.5” clearance -- another thing that one should remember to check!! H&R Block Building

Steve Reinmuth, Bronze Studio, Eugene OR Granada 2003 Steve Reinmuth, Bronze Studio, Eugene OR Steve Reinmuth, the key man in the realization of Pax Mundi Here he is tightening the bolts during the installation. Reinmuth Bronze Studio, Inc. 3295 Meadow Lane Eugene, OR, 97402 http://www.reinmuth.com/

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

Extension: Free-form Curve on a Sphere Spherical Spline Path Editor (Jane Yen) And again, there are paradigm extensions. We don’t have to limit ourselves to up/down undulations around the equator, as in the Gabo curve. We can allow free-form curves on a sphere. Jane Yen wrote a nice program based on “Circle Splines” that can make smooth undulating curves (shown in yellow) through just a sparse set of interpolated points (shown in blue). With this … Smooth interpolating curve through sparse data points

Many Different Viae Globi Models Roads on a sphere I could now make more “curvy” Viae Globi models (i.e “roads on a sphere”). In the middle is Maloja, inspired by a Swiss mountain road. On the left is Stelvio, a pass route in Italy with 40+ hair-pin curves. At right is Altamont, a high-way pass into the Central Valley in CA with multiple parallel lanes. Altamont Stelvio Maloja

Paradigm Extension: Sweep Path is no longer confined to a sphere! Granada 2003 Paradigm Extension: Sweep Path is no longer confined to a sphere! The next extension was to allow the path to deviate from the sphere surface. Now we can make paths that correspond to mathematical knots because we can do over- and under-crossings. Chinese Button Knot

Florida 1999 Chinese Button Knot (Knot 940) Bronze, Dec. 2007 Carlo Séquin cast & patina by Steve Reinmuth Here is the same 9-crossing knot – known as the Chinese button knot – after having been cast in bonze.

Definition of Sweep Path (hugging 4 different spheres) Granada 2003 Definition of Sweep Path (hugging 4 different spheres) A more recent design by Brent Collins that required this particular paradigm extension. It is called “Music of the Spheres”, because the sweep path deviates 3 times from the outer sphere and then loops around 3 smaller inner spheres indicated by black lines. In this process, it forms a topological trefoil knot. Brent again got a commission to do this in bronze at a large scale! Music of the Spheres (Brent Collins)

Partitioning; Joint Design Granada 2003 Partitioning; Joint Design Again I needed to minimize the number and sizes of the master molds. Here, the minimal geometry that needs to be defined comprises 1/3 of the whole sculpture, because of its 3-fold symmetry. Clearly 1/3 part on the left is too large a piece to be handled by Steve. Even 1/9th of the sculpture was still too large to fit in Steve Reinmuth’s kiln. So I divided the sculpture into 18 pieces. And again I provided some alignment stubs – one of which will also serve for mounting. Alignment stubs 1/3 = unique geometry 18 pieces: fit in kiln!

Assembly of Music of the Spheres Granada 2003 Assembly of Music of the Spheres The fabrication process was pretty much the same as for “Pax Mundi”. Close to final assembly of “Music of the Spheres”. Photo conveys a sense of the size and of the physical labor involved in building such a sculpture.

Installation at MWSU, Feb. 2013 Granada 2003 Installation at MWSU, Feb. 2013 Just finishing the installation at MWSU Steve Reinmuth Brent Collins

Illuminated Music of the Spheres Granada 2003 Illuminated Music of the Spheres And it looks even more spectacular at night! Photo by Phillip Geller

Granada 2003 More Recent Endeavors Modeling two different classes of sculptures By Charles Perry and by Eva Hild. More recently I have been focusing on sculptures by Charles Perry, quite well know on the East coast of the USA, and by Eva Hild, a Swedish artist. I am now working on some CAD tools that allow me to generate sculptures in the style of these two artists. Since Perry’s ribbon sculptures are not all that different from Collins’ ribbon sculptures in the challenges that they offer, Let me now focus on Eva Hild’s sculptures.

A large collection of ceramic creations & metal sculptures Granada 2003 2-Manifolds by Eva Hild A large collection of ceramic creations & metal sculptures Eva Hild is a Sweedish artist who creates large ceramic and metal sculptures in the form of organically undulating surfaces. They present interesting mathematical puzzles. What are these sculptures from a mathematician’s point of view? Most of them are smooth, thin sheets of material. Mathematicians call this a 2-manifold. Some questions: Are these surfaces 2-sided like a sheet of paper -- or perhaps single-sided like a Moebius band or a Klein bottle. How many borders they have, and what is their genus?

Topological Analysis Surface Classification Theorem: Granada 2003 Topological Analysis Surface Classification Theorem: All 2-manifolds embedded in Euclidean 3-space can be characterized by 3 parameters: Number of borders, b: # of 1D rim-lines; Orientability, σ: single- / double- sided; Genus, g: specifying “connectivity” . . . The “Surface Classification Theorem” states: All 2-manifolds embedded in Euclidean 3-space can be characterized by 3 parameters: The number of its borders, b: i.e., # of 1D rim-lines; its orientability, that is: whether the surface is single- or double-sided; and its connectivity, specified by its genus, g, -- the latter essentially says how many closed-loop curves you can draw on this surface without splitting it into different countries. A sphere is of genus 0; a simple doughnut has genus 1; and a blob with n tunnels or with n handles attached has genus n. Determining the first 2 parameters is normally not too difficult. But determining the genus can be tricky! genus 0 genus 2 genus 4

Determining the Euler Characteristic Granada 2003 Determining the Euler Characteristic Another approach to find the genus: χ = V – E + F = Euler Characteristic How many cuts to obtain a single connected disk? Disk: χ = 1; every ribbon-join lowers χ by 1;  thus “Tetra” ribbon frame: χ = –2 From this: Genus = 2 – χ – b for non-orientable surfaces; Genus = (2 – χ – b)/2 for double-sided surfaces. Sometimes it is simpler to determine first the Euler Characteristic, and then calculate the genus from that. One way to find the E.C. is to draw a mesh onto the surface and then apply the formula that E.C. is equal to # of V-E+F. But often there is a short-cut; particularly for sculptures that are composed from many interconnected ribbon elements: We simply count the number of ribbons that we have to cut, until we are left with a jagged piece of surface with a single rim, that is topologically equivalent to a disk. The Euler characteristic of our original surface is then 1 minus the # of cuts made. When we do this to the “Tetra” frame we have to make 3 cuts, so its E.C is -2. From the E.C. we can then calculate the genus of the surface according to these two formulas.

Eva Hild’s 2-Manifold Sculptures Granada 2003 Eva Hild’s 2-Manifold Sculptures But let’s look at some examples. This sculpture called “Hollow” is in Vaarberg, Sweden. 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. (One can cut the 2 horizontal tunnels in the front and in the back to reduce this to a disk with a bunch of holes or “punctures.”) “Hollow” (Eva Hild) FDM Maquette 1 border; single-sided; genus 2

Eva Hild Sculptures “Whole” (Eva Hild) FDM Maquette Granada 2003 Eva Hild Sculptures Here is another sculpture called “Whole” and my recreation. 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 1 border; single-sided; genus 2

Complex 2-Manifold Sculpture Granada 2003 Complex 2-Manifold Sculpture But it is not so easy to make a parameterized procedural model of her more complex and more irregular sculptures! Here is something that I am currently struggling with: How can I capture this very free-form, asymmetrical shapes? “Wholly” by Eva Hild (Sweden)

Parameterized Control Mesh Granada 2003 Parameterized Control Mesh Coarse control mesh attached to 9 panels that can be moved and scaled individually Surface after 3 levels of CC - subdivision Here is one way by which I try to tackle this: First I need to understand the underlying topology – the basic genus 4 topology is shown on top. Then I form a parameterized control mesh that captures this topology– introducing higher-level controls to reduce the number of DOFs to adjust the geometry. Here I tied the many control points to just nine panels that can be individually moved and scaled. Now I am working with a couple of students to create the proper data structures and graphics utilities. There are plenty of intriguing challenges ahead!

Developers of SLIDE Jordan Smith (2001) Jane Yen (2007) Granada 2003 Developers of SLIDE At this point, I would like to note, that I could not do all this by myself! Here are two of my star students Jordan Smith and Jane Yen who were instrumental in the development of SLIDE. Jordan Smith (2001) Jane Yen (2007)

URAP Program at UCB Sean Liu (graduated May 2016) Granada 2003 URAP Program at UCB But now I am retired and have no more PhD students. However, I still work with several undergraduate students through a program called: Undergraduate Research Apprenticeships. With me, they get to learn basic solid modeling, computer-aided design, and may write some program extensions to one of our home-brewed modeling systems (SLIDE, NOME), and then create some sculpture models of their own. >>{Sean Liu create a program to pack as many equilateral triangles as possible into a hyperbolic surface – always placing 7 around each vertex.} Sean Liu (graduated May 2016)

Bridges Baltimore, July 2015 Florida 1999 Bridges Baltimore, July 2015 Large, “7-Around” Hyperbolic Disks Sean Liu create a program to pack as many equilateral triangles as possible into a hyperbolic surface – always placing 7 around each vertex. We presented this paper at the annual Bridges conference devoted to Math and the Arts, where we were able to pack more than 2000 triangles in this particular fashion. Sean Jeng Liu, Young Kim, Raymond Shiau, Carlo H. Séquin University of California, Berkeley

Having Fun in Your Work Fun with geometry ! 1998 2002 2009 Granada 2003 Having Fun in Your Work Fun with geometry ! I hope, that from my presentation you could see that I am having fun with what I am doing. Let me then conclude my talk with a little life-philosophy: 1998 2002 2009

The Secret of a Happy Life: Granada 2003 The Secret of a Happy Life: Do something that you really like to do . . . . . . and get paid for doing it! What is the secret of a happy life: . . . >>> … >>> And it _IS_ possible to get into this state. I have been in it for about 30 years … But you have to find out what it is that you are really passionate about! – And that is one important function (!) of your years in college! Look around, sample many different things, see what feels right for you. Then keep this in mind whenever you look for your next job. ==> I wish you all a successful career – and more importantly: a happy and fulfilling life!

ISAMA 2004 QUESTIONS ? ? My grand children are already picking up that same spirit!