Monthly Archives: February 2016

Constructing a Schwarz P surface

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The challenge: to construct a 3D printed Schwarz P surface piece for the Taping Shape*  exhibit at the Rueben H. Fleet Science Center, which could be joined to others to create a finite part of a Schwarz P surface. I’m not the first to do this,  Ken Brakke has already used his Surface Evolver program to create a beautiful and truly superior Schwarz P surface found on Shapeways.

With limited time before the exhibit, could we create a reasonable approximation of the Schwarz P surface using Cinema4D?Schwarz-smooth-1 We (Dave Pfaff and I) started by finding the minimal surface for a 4-gon with corners at the vertices of a regular octahedron. We then extended the resulting surface by 180 degree rotations about the straight boundary lines. This created a surface, but it was not quite right.  We needed to cheat a bit and make the 4-gon surface closer to a quarter circle in the middle. (The actual Schwarz P surface is not circular there, but is close.)

Schwarz-smooth-3After using the Close Polygon tool on the 4-gon, we used the Subdivide command for the 4-gon, then moved vertices closer to the circle. We subdivided again, moved vertices closer to the circle again and repeated the process. We then rotated 12 copies of the 4-gon unit around various edges to get the figure to the left.

Schwarz-smooth-4We then arranged 6 of these units in space, and added in a cube. We used the Boole command to cut out a cubical Schwarz P unit. I then extruded the surface, and added magnet holes as described previously in this post:

Joining models with magnets

Schwarz-smoothI made two sizes of models: 6cmx6cmx6cm and 10cmx10cmx10cm. We printed the models on the uPrint SE printer. They printed just wonderfully. The one small flaw in the design is that there is a slightly raised line in the place where we moved vertices to the circular arc. However, the model has many strengths: aside from the line it is quite smooth, and you can almost (but not quite) see the 4-gons. Given the time restriction before the exhibition, we decided to leave the model as is.

Schwarz-Bohdan-kids1To the left is some of the Schwarz P surface models printed for the Taping Shape exhibit.

You can find the files for the model, and instructions on how to place the magnets, here on Thingiverse.

*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 .)

Schwarz P surface – the math

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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 found here.

SchwarzP-lattice 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.)

SchwarzPIt’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

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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.pants-ringAfter 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

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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.

pants-cube   pants-full

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.

pants-extrudeThe 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.

pair-pants-capsOnce 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.  pair-pants-ringsI 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.

IMG_4012Finally, 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

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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. pair-pantsWe 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:

pants-decomp-2pants-decomp-1

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.

Pair-pants-2Since 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

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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 .)