# Museum Magic

Recently, I had a wonderful two days visiting Ricardo Nemirovsky and his team in San Diego. I started by visiting the wonderful Fleet Science Center and being the mathematician for their Meet the Mathematician event on the Sunday afternoon. (I even got my own poster – wow!) Ashanti Davis met me and showed me around the Taping Shape* exhibit for which I had designed the 3D-printed mathematical models.

These photos don’t really capture how marvelous it was to walk inside the topological shapes. How often does a mathematician get to explore the interior of a torus, or walk down the leg of a pair of pants?!? The lighting kept changing color as well, adding to the experience.

I was able to see my 3D-printed models in action; anything from little kids throwing them around, to big kids and grandparents building more complex topological shapes.

The folks at the museum had set up a tank with soapy water, and the frame for the Schwarz P surface could be lowered into and out of the soapy water by folks using a wheel. The frame worked perfectly, beautifully showing the Schwarz P surface.

I spent my hours in the museum talking with people who stopped by about the math of the Taping Shape exhibit and of the 3D printed models. Using zome tools and another tank of soapy water, I was able to demonstrate just a few of the many different shapes that soap film (a.k.a. minimal surfaces) take. The kids experimented with their own zome tool shapes. They created some bizarre models giving interesting soap films. Many of the soap films we created showed the classic angles where three or more soap films joined together. (Math in action really does work!) It was a wet and fun time, and my hands ended up very, very clean. Thanks go to Ashanti for setting things up and keeping me company during much of the afternoon.

A number of my designs had been 3D-printed into giant sized models, which was great to see. On the final day of my visit I was able to meet with the entire Informath team. I was also able to hang out with Bohdan Rhodehamel and see his lab. He was responsible for 3D-printing and then assembling the models for the Taping Shape exhibit. I wrapped up my trip by giving the math department’s colloquium on Mathematics and 3D Printing at San Diego State University.

*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 .)

# 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 .)