Schwarz P surface – the math

Minimal surfaces have been studied for over 200 years.  The research began when Joseph-Louis Lagrange asked a very simple question around 1760: “What does a surface bounded by a given curve look like, when it has smallest surface area?” This was a hard problem to study — roughly speaking, the mathematics of minimizing surface area leads to a partial differential equation of  the surface. The tools to study such equations had not yet been developed. In fact, the first mathematical conjectures about minimal surfaces were made from the careful observations of soap film by the physicist Joseph Plateau (published in 1873). Over the years many aspects of the problem have been solved, and most recently progress has been made using the tools of geometric measure theory.

When looking at at surface with minimal area, it turns out that smaller pieces of it must also have minimal area with respect to their boundaries. This means they must locally look like a saddle, not a hill nor a bowl (since we could reduce the area by chopping off the hill or filling in the bowl). Just like a saddle surface, minimal surfaces look the same from both of their sides. This makes sense when we consider soap films: the surface tension of a soap film is in equilibrium at every point, the forces pulling to one side must balance the forces which pull to the other side.

In differential geometry, the term mean curvature measures the bending of a surface at a point. For minimal surfaces, it must be zero. Mathematically speaking, minimal surfaces are defined to have locally minimal surface area — small pieces of them can always be realized as a soap film. This local definition means minimal surfaces are independent of the boundary problem, and so mathematicians are also interested in infinitely large minimal surfaces without boundary.

The Rueben H. Fleet Science Museum asked me to model a Schwarz P surface. This is a triply periodic minimal surface, meaning it has translational symmetries in three independent directions. The Schwarz P surface was originally described by Hermann Schwarz (1890) and his student Edvard Neovius (1883). More examples of triply periodic (and other) minimal surfaces were found by Alan Schoen in 1970. The Schwarz P surface is a genus 3 surface that fills space. It is part of a huge family of minimal surfaces, for example

So what does the Schwarz P surface look like? Imagine two interconnected thickened cubic lattices. The Schwarz P surface lies on the intersection of these thickened lattices. In the picture to the left, one lattice lies inside the yellow surface, the other in the spaces between it. (Thanks to the minimal surface archive at Indiana University for these pictures.)

It’s a tricky surface to visualize. Another way to see it is to first find the minimal surface for a 4-gon with corners at the vertices of a regular octahedron. Then extend this surface, by rotating copies of it by 180 degrees about the boundary lines. In the figure on the left you can see one such 4-gon in the center top. Imagine rotating it about one of the edges. Keep going. You can see parts of 6 copies of the 4-gon about the vertex in the figure. Now keep repeating this process. Eventually you get to the triply periodic surface shown above.

Joining models with magnets

Part of the design challenge for the models for the Taping Shape* exhibit, was to find a way to join them together. I was inspired by Jason Cantarella’s Decomposition of a Cube Manipulative which uses small magnets to join the pieces together. These magnets are 3mm (diameter) x 3mm (height) cylindrical rare earth magnets.

To make holes for the magnets I made a cylinder of height 6.4mm and radius 1.6mm. I knew I needed to use the Boole tool to create the holes. Most importantly, I had to make sure to that the holes perfectly aligned on different pieces. Cinema 4D has a wonderful Array tool, which I used to create an array of four cylinders centered at the origin. I adjusted the radius of the array until the cylinders were perfectly placed on the pair-of-pants model. The 6.4mm height of the cylinders allowed me to position the models above or below the array, so the height of each hole was precisely 3.2mm.

I then moved the different models (or the array) around the origin to get four holes perfectly placed in each rim of the pair-of-pants, ring and caps models.  The photo below shows the ring system for the pair-of-pants with the cylinder array ready for the Boole tool on the left. The ring on the right is ready to go.After printing, I found that the magnets fit snugly and would not come out. If you are worried about this, Jason used a little JB Weld epoxy. (He suspects that you could also use superglue.)

Putting the magnets in was nearly impossible. However Jason’s magnet insertion tools were just awesome. They allowed me to seat the magnets into the little holes, and helped me keep track of which end of the magnet went where. I strongly recommend printing the $$+$$ and $$–$$ magnet insertion tools in different colors to help with this. I put down one tool to check a print, then picked the other one up instead, messing up the placement of the magnets. (I discovered the hard way that the magnets really don’t come out…)

For the pair-of-pants and caps models, I alternated the $$+$$ and $$–$$ ends of magnets around the rims of the pair-of-pants. I did this in a consistent way, for example the $$+$$ was always at the front and back of the pants.  For the rings, the plain ones should be aligned the same alternating way on the top and bottom rims. The ones with plus/minus signs or 90 degrees should have the arrangement rotated by 90 degrees.

The end result? Models which snap together in a satisfying way. The reason for the rings should now be clear. Without them, the models connect in only two possible orientations. With them, the models can be snapped together in four different ways.

*The Taping shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587).  (The InforMath Project is a partnership between San Diego State University and several museums at the Balboa Park, including the Rueben H. Fleet Science Center .)

Constructing a pair-of-pants

Constructing a pair-of-pants surface was easy and difficult all at once. I used Cinema 4D to create the surface. I did this by using the Subdivision Surface tool on a cube which I had extensively edited. The photos below show the image before and after I applied the tool.

It took a long time to get the cube just right. I took a rectangular prism, then used the Knife tool to slice the top and bottom faces of the cubes. From there, I extruded both the top and the legs. To get the right shape around the middle, I used the Knife tool and the Close Polygon tool extensively. It was quite tricky to find the right shape for the legs, hip and waist of the pants. Roughly speaking, the Subdivision Tool takes midpoints of edges and faces, then moves these to a carefully defined weighted average. There is a nice Numberphile movie where the folks at Pixar explain this here.

The next step was to use the Boole Tool with cubes (in a number of different ways) to cut out the pair-of-pants in the middle, and the rounded caps at the ends. I then selected the entire pair-of-pants surface, and used the Extrude Tool with caps to thicken it by 5mm on the inside. I finished the pants by Optimizing (to make sure all the overlapping vertices were taken care of), and by making sure all the normal vectors were pointing outwards (so the surface would print). I repeated these steps for the rounded caps as well. The final objects looked great and printed easily on the MakerBot 2X printer with supports but no raft. You can see small holes for magnets in the rims of the pants. I’ll explain how (and why) I added these in the next post.

Once I had the regular pair-of-pants figured out, I made a “bent” pair-of-pants as well. I simply took the edited cube used to make it, then edited it some more.  I shortened the “waist” area of the pants, and lengthened the “torso” area, before extruding outwards. The dimensions of the “bent torso” square matched those of the squares for the legs. This ensured that the “bent waist” circle would match those of the legs. I also used the Knife and Close Polygon tools to make the bend at the waist less extreme. I then extruded, optimized and checked the normals of the surfaces as before.

Finally, I made a ring system for the models. This was easy to do — I simply took the regular pair-of-pants cube and extruded the legs out some more. Once I applied the Subdivision Surface tool to it, I got a pair-of-pants with extra long legs. I again used the Boole Tool with cubes, to get two rings. These were extruded and finished as before.

These models are currently on display at the Taping Shape* exhibit at the Rueben H. Fleet Science Center in San Diego, California. The pair-of-pants and bent pair-of-pants surfaces can be found on Thingiverse:  http://www.thingiverse.com/thing:1279118 and http://www.thingiverse.com/thing:1298073.

*The Taping shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587).  (The InforMath Project is a partnership between San Diego State University and several museums at the Balboa Park, including the Rueben H. Fleet Science Center .)

Pair-of-Pants surfaces, the math

A pair-of-pants is a surface that looks exactly like a pair-of-pants that you wear. Technically, it is topologically equivalent to a sphere which has been punctured three times, or a disk which had been punctured twice (shown below). It is an orientable surface of genus two having three boundary components. They are useful objects in topology, in that they give a different decomposition of surfaces. We usually think of closed connected surfaces as spheres, where either handles or cross-caps have been added. More formally, recall the Classification of Surfaces Theorem: Any closed, connected surface is topologically equivalent to a sphere, a connected sum of tori, or a connected sum of projective planes.

It turns out that we can cut up just about any orientable closed surface into pairs of pants with simple closed curves. This is called a pants decomposition of a surface. Pants decompositions are not unique. For example, we can cut up a genus 2 surface (a sphere with two handles) in two different ways:

What happens in general?  Suppose our surface has $$g$$ handles, where $$g\geq 2$$. Then we can slice the surface with $$3g-3$$ “vertical” simple closed curves, which decomposes the surface into $$2g-2$$ pairs of pants. The genus 3 case is shown below and illustrates the general idea.

Since a pair-of-pants is a subset of a thrice punctured sphere, it also admits a hyperbolic structure. Alternatively, simply construct a hyperbolic pair-of-pants by gluing together two right angled hexagons (hyperbolic) along alternate edges. More generally, it is relatively straightforward to show that there exists a unique hyperbolic pair of pants with cuff lengths $$(l_1,l_2,l_3)$$ , for any  $$l_1,l_2,l_3>0$$. Here, cuff lengths refers to the lengths of the three boundary components. Even more can be said about hyperbolic surfaces and pants decompositions, but this will lead us too far astray.

A new challenge

In December, I was contacted by Professor Ricardo Nemirovsky from San Diego State University to design 3D printable surfaces for the  Taping Shape* exhibit at the Rueben H. Fleet Science Center in San Diego, California. The exhibit runs from January 30 through June 12, 2016.

The exhibit contains a structure made out of packing tape with
three interconnected regions: a torus, a topological
equivalent to Schwarz P surface, and a pair-of-pants
surface with the legs twisted. The structure is large enough for visitors to walk and crawl through. There are three “work tables” (one for each region), with materials, suggested activities, poster displays, etc. The 3D printed models will be a part of the work table and displays.

Ricardo requested I make pair-of-pants surfaces with caps that can be joined together in different ways, Schwarz P surfaces that can be joined together, and also a frame that allows the Schwarz P surface to be created as a soap film spanning the frame. The challenge was on!

In the following blog posts, I’ll explain a bit about the math behind the surfaces, and how we figured out how to build and print them.

*The Taping shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587).  (The InforMath Project is a partnership between San Diego State University and several museums at the Balboa Park, including the Rueben H. Fleet Science Center .)

Decorative knots

I’ve been back printing on the FormLabs 1+ liquid printer this week. I decided it was time to print out some more knots, this time for use as festive decorations. I printed out Laura Taalman’s Trefoil Managerie and used some red gift ribbon to create ornaments. Previously Laura had given me a torus with a (5,2) torus knot. I also turned this into a decoration.

I’ve also printed more snowflakes. Again from Laura Taalmans’ Snowflake Machine, the small snowflakes that have a circle so you can hang them up. Awesome.

Math is all around us…

The holiday’s are fast approaching, and I had a flash back to the movie Love Actually and the song “Love is all around you”. I feel the need to change the first word here (Math = Love?) , since many Christmas decorations show a remarkable amount of geometry to them.

I went to this wonderful tree lighting ceremony and cookie party at W&L and was completely distracted by both the amount of sugar available (cookies, cupcakes, brittle, candy, cider, hot chocolate, …) and the fabulous decorations. These included the stellated star (above) and the fabulous minimal surfaces created by stretching fabric between a table top and it’s legs. The red lighting just added extra pizzazz.

I’ve also seen a lot of knots out there in the real world. My favorite is this folded trefoil knot on the back of a truck. Exactly the kinds of knots I’ve been researching with my undergraduate students.

One of my friends I knit with has the best earrings – nice and knotty! Fun way to be festive at this time of year. (She’s a graphic designer, so was greatly amused at my need to take a picture of her earrings!) I believe that this earring is a true lover’s knot.

Snowflakes

I’ve a had blast the past week or so 3d printing some snowflakes. They are for the math department for decorations around the holidays.

I’ve been using Mathgrrl’s (aka Laura Taalman) snowflake ornaments (Thigiverse thing 195032). She’s since customized the design here: http://www.thingiverse.com/thing:570339.

I also had a blast printing some of her fractal snowflakes from the Snowflakerator snowflake machine found here: http://www.thingiverse.com/thing:1159436.

I even designed a few of my own found here, here and here.

Math at the Simon’s Center for Geometry and Physics

I had the very great privilege of being a co-organizer of a workshop held at the Simon’s Center for Geometry and Physics and NYU Stony Brook. This was the workshop on the Symplectic and Algebraic Geometry in the Statistical Physics of Polymers. It was my first time to this campus, and I had a blast with both the math at the workshop AND all the visualization of math in the environment.

My first hint that things were going to be special, was the fantastic Umbilic Torus sculpture found at the end of an avenue of trees between the center and the math department.

The sculpture is by Dr Helamun Ferguson, click  here to find a photo gallery showing the design and construction of the piece.

The sculpture consists of a space filling curve all over the surface of the sculpture. The sharp curve along the edges is a trefoil knot, winding three times around the central hole (the longitude on the torus) and twice around the sculpture the other way (the meridian on the torus).

The base of the sculpture is a large round granite disk with a 3 sided deltoid mirroring the 3-fold symmetry of the sculpture overhead. The base had to be left to settle for a year, and was greatly loved by the local skate-boarders!

The Simon’s Center itself is in a wonderful airy building, with mathematical themes blended seamlessly in the design. I kept finding treasures as the workshop went on. The most obvious, is the sandstone wall behind the stair case leading up to the cafe on the second floor. It is covered with small math motifs from knots, to physics, to finding the square root of 2.

Even the screens on the side of the first floor lounge are mathematical, with different tilings of the plane illustrated. Just love the artistry of the designs in them.

Models have arrived!

Our 3D printed math models have arrived in the W&L Mathematics Department ready for the Fall semester. They are in labeled clear plastic containers right over the biz hub in the math work room, so everyone can easily get to them.

We are looking forward to seeing what people do with them in class this coming semester. In particular, if we get any more requests for builds or rebuilds.