# 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 .)

# Knots as ribbons

I’ve continued with my project to edit the 3D printed models my Fall 2014 Math 341 Introduction to Topology class made. Recently, I came across two of my favorite pieces. The first is a model that was designed by Emily Jaekle (’16) and is a ribbon version of the (3,5) torus knot.

This $$(5,3)$$ ribbon torus knot was designed entirely in Cinema4D. The curve was created using the Formula tool with parametrization $$x(t)=(2+\cos(5t))\cos(3t), y(t)=(2+\cos(5t))\sin(3t), z(t)=-\sin(5t)$$ for $$t\in[-\pi, \pi]$$. The trianglulated surface was created by first adding in a small rectangle, then using the SweepNurbs (without caps). The rectangle was also rotated 1800 degrees in the process. The small gap was fixed using the Bridge tool in Edge mode. This ribbon knot was originally printed in blue on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in black resin. You can find the model here on Thingiverse.

The second model was designed by Cathy Wang (’15) and is a ribbon version of the (3,2) trefoil knot with an amazing color scheme.

The entire model was designed in Cinema4D. The knot was created using the Formula tool with parametrization $$x(t)=(2+\cos(2t))\cos(3t), y(t)=(2+\cos(2t))\sin(3t), z(t)=-\sin(2t)$$ for $$t\in[-\pi,\pi]$$. The trianglulated surface was created by first adding in a small rectangle, then using the SweepNurbs (without caps). The width and height of the rectangle was adjusted so the band is not a constant size through the knot. The overlapping edges and small gaps were also fixed. Finally, the knot was colored with a beautiful rainbow-gradient. This ribbon knot was originally printed in rainbow colors on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in black resin. You can find the model here on Thingiverse.

# Knots, a Seifert Surface, and an Octopus.

I’ve recently been going back to the 3D printed models my students designed in my Math 341 Introduction to Topology class in Fall 2014. I know so much more now than I did then when I first started on this journey. So, I decided to go back to the designs, check over them, edit them (as necessary), then reprint them and publish them on Thingiverse.

The first two models I looked at where Candace Bethea’s (’15) $$6_2$$ knot and Hayley Archer-McClelland’s (’15) three interlocking trefoils. In a previous class using 3D printing, Professor Aaron Abrams contacted the developer of the program Seifert View, and arranged for the computed curves and surfaces to be able to be exported as a file. (This free software normally doesn’t allow you to do this.) We have since found this to be an invaluable tool for 3D printing knots and knots with Seifert surfaces.

The $$6_2$$ knot was first adjusted in, then exported from SeifertView, and then opened in Cinema4D. The surface of the knot needed fixing due to overlapping polygons. This was fixed by deleting overlapping polygons, then filling in the gaps using the Stitch and Sew tool in Edge mode. The knot was originally printed in orange on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in clear resin. You can find the model here on Thingiverse.

The three interlocking trefoils were designed entirely in Cinema4D. The parametric equations of a trefoil knot are $$x(t)= (2+\cos(3t))\cos(2t), y=(2+\cos(3t))\sin(2t), z=-\sin(3t)$$ for $$t\in[-\pi, \pi]$$. We first made the curve with the Formula tool for $$t\in[-3.14, 3.14]$$. We added a SweepNurb (with no caps) consisting of a circle with radius 0.2 cm around the curve. The choice of $$t$$ meant there was a small gap between the ends of the tube. We again sealed this gap using the Stitch and Sew tool in Edge mode. The knots were originally printed in pink, pale green and blue on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in clear resin. You can find the model here on Thingiverse.

Mithra Muthukrishnan (’16) used Seifert View to design a figure-8 knot and its Seifert surface. After tweaking in Seifert View, she exported the surface to Cinema4D. It ended up being a complex design process, since we wanted to color the knot and two sides of the surface with different colors. There, the surface was extruded in both directions creating the two sides of the Seifert surface. Finally, three copies of the surface were made. In one, all the surfaces were deleted leaving the knot. In another the knot and one side of the extrusion was deleted. In the final copy, the knot and the other side of the extrusion was deleted. This left three pieces which could each be given their own color before printing. The knot was colored red/pink, the different sides of the Seifert surface were colored white and yellow. The model was originally printed in color on the Projet-260 3D Systems printer. Later, I printed it on the MakerBot 2x printer in bright white. You can find the model here on Thingiverse.

The octopus model was designed by DanJoesph Quijada (’15) entirely in Cinema4D. The legs were made using parametric equations like $$x(t) = a\sqrt{2}t, y(t)=0, z(t) = b^2 cos^2(ct)e^{-dt^2}$$ for various constants $$a, b, c,$$ and $$d$$, and bounded time $$t$$. These curves were then thickened using a SweepNurbs. To make the octopus head, we first made a box, then extruded parts of the sides to alter the shape, then applied the Subdivision Surface tool to it. Finally we made minor adjustments, such as adding the eyes and changing the dimensions of the octopus to make it look more realistic. The model was originally printed in pink on the Projet-260 3D Systems printer. Later, I printed it on the MakerBot 2x printer in bright white, though I had some trouble with the legs. You can find the model here on Thingiverse.

# 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.

# Torus Knots

I modeled the trefoil knot as two torus knots $$T(2,3)$$ and $$T(3,2)$$. The parametric equations for a $$T(p,q)$$ knot are $$x = \cos(pt)*(3+\cos(qt)), y=\sin(pt)*(3+\cos(qt))$$, and $$z=\sin(qt)$$. Here, $$p$$ is the number of times the knot winds around the longitude of a torus, and $$q$$ is the number of times the knot winds around the meridian of a torus.

Both models were printed on the FormLabs printer. I first made a small $$T(2,3)$$ knot with a label extruded out of the curve (as shown to the left). I used Cinema 4D to design the model by using the Formula Spline to draw the curve, the Sweep NURB to give the curve depth, and the Wrap Tool to wrap the text around the curve. I also used the Extrude Tool to give the equations depth and the Boole Tool to connect the equations to the curve. For both knots I had to make sure the ends of the knots overlapped correctly. Before printing the $$T(3,2)$$ knot, I had to change the range of $$t$$ to $$t=[0, 2\pi]$$ instead of $$t=[0, 5\pi]$$ (I initially used $$5\pi$$ to be sure that the curve closed).

The first $$T(2,3)$$ knot came out nicely, however the text was a little small. Using the subscript made the numbers too small, so I reprinted the knot used parentheses instead, as shown here. The $$T(3,2)$$ knot also looked great, as it was smooth and there were merely small nubs where the supports were, which could be removed with an exacto blade. We’ve discovered that the FormLabs printer makes smoother surfaces and finer curves than does the MakerBot, which is why it is ideal for printing knots.

You can find the torus knots on Thingiverse here T(2,3) and here T(3,2). Instructions on how to make torus knots in Cinema 4D can be found here. Professor Denne has also created another worksheet in Mathematica about Torus knots. It can be found here.

# Helicoids

My next project (after finally finishing all the quadratic surfaces) was to make a helicoid. I spend some time on Mathematica creating different helicoids by changing the parameters of the formula. The helicoid is pararametrized by $$x=u\cos(t), y=u\sin(t),$$ and $$z=u$$, where $$u\in[-1,1]$$ and $$t\in[0,2\pi]$$.

Professor Denne and I decided to print two of the ones I created to start (we may print more!). I exported the following Mathematica files and imported them into Cinema 4D.

Professor Denne used these files (as well as others) to create another worksheet in Mathematica. It can be found here

After making them the correct size I optimized the polygons and extruded them by 0.20 cm (a process I can now do very quickly after all my practice with the quadratic surfaces). I then printed each of them on the liquid printer and had fantastic results!

The .STL and .form files for both of these helicoids can be found on Thingiverse.

Later on, I made another helicoid, this one with $$u\in[0.25,1.25]$$ and $$t\in[0,2\pi]$$. This model can be found on Thingiverse here.

# Klein Bottle

Just a really short post to share our general excitement over having just about completed all of the objects from Multivariable Calculus. We just have a few more to print out. We will spend our remaining week(!) printing out some interesting topological objects – many of these directly from Thingiverse.

We printed one such object today. This is the Voronoi Klein Bottle from MadOverlord on Thingiverse. We printed this on the MakerBot 2X with a raft but no supports. After a moment’s thought one can see that the print succeeds (despite the short horizontal lines on the design) because the Voronoi cells are small enough. Interesting! The black filament also hides a few rough spots on the print.

The Klein bottle is named after Felix Klein (25 April 1849 – 22 June 1925), a German mathematician who saw many connections between Group Theory and Geometry. It is a one-sided surface and is a generalization of a Mobius strip. (In fact, it is topologically equivalent to two Mobius strips glued together along their boundaries.)

There are many fabulous descriptions of this topological object, one of my favorites is The Adventures of the Klein Bottle found on YouTube (from the wonderful folks at the Frei Universitat in Berlin).

# Walking down memory lane

In Fall of 2014 I taught Math 341 Introduction to Topology. As part of the class I had the students design and then print a topological object. For most students, this ended up being the highlight of the course.

We spent a week of class in the IQ center under the guidance of David Pfaff. He showed us how objects can be viewed in the stereo 3D lab and gave us a crash course in Cinema 4D. Students then let loose their imaginations and creativity. Many students chose to learn about knots and links, ribbon knots, and Seifert surfaces of knots and links. They produced some wonderful models. Other students chose to create objects with symmetry (like the 20 sided die), or the cube-like Cayley graph.

It turned out that getting the objects that could be 3d printed was hard work! Many objects had not been optimally made (for example with normal vectors pointing inwards). We were fortunate to have David Pfaff’s expertise in sorting out these errors. Eventually all the objects were printed using the IQ center’s ProJet 260. Some of them needed to be printed twice, as they broke when being removed from the printer. Many 3d printed math objects from this class and from Aaron Abrams first year seminar currently reside in the Mathematics Department.