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 .)
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.
Our summer build team attended the centennial MAA MathFest Conference in Washington DC. We had a wonderful time attending many talks, the exhibits and the evening entertainment. (Some of our favorites included the talks by Erik Demaine, Noam Elkies, and the Cirque de Mathematiques.)
Emily and Ryan gave a wonderful (and well attended) talk about their summer work. I spoke about our work in the What Can a Mathematician Do with a 3D printer? session organized by the inspirational Laura Taalman and Ed Aboufadel (below left). Everyone who brought printed objects got to display them at the front of the room (below right).
Laura had her MakerBot mini printing before the session started. A small collection of her models was placed in front of it. Our models were right in front of Jason Cantarella’s 3D printed calculus robot Cy. They were very well received by everyone present.
Here are some of the wonderful models by Christopher Hanusa from Queens College CUNY (left), and Lila Roberts from Clayton State University (right).
Laura spoke about how she designed and printed the Catalan Wireframe Polyhedra, shown below. We were even lucky enough to each be given one by her! I’ve come away from the session with many good ideas of using 3D printing in the classroom, as well as designing new math models.
One last post about the summer printing. We did end up using the ProJet 260 (gypsum) powder printer. We printed two objects – the solid Strange Bowl, and the Tumor Model. Both had colors added with help from Dave Pfaff. (There is a complicated color bit map involved.) The tumor model’s colors roughly correspond to the distance from the center of the model. While these models are beautiful, they are not as robust as some of our other models. They won’t go into general circulation, but instead will be in our display case.
Our summer research project has officially ended. Emily and Ryan have been phenomenal. Together, we’ve designed and 3D printed over 46 math models during the past 9 weeks. Given that our first week was spent working on the math and learning computer programs, we’ve averaged a little over 6 models a week. Phew!
We also 3D printed models that other folks designed, which means we’ve well over 55 different models in total.
We are all currently at the 2015 MAA MathFest conference in Washington DC. Emily and Ryan will be talking about their work in a student research session, and I will be discussing our work in a session on What can a mathematician do with a 3D printer? organized by the inspirational Laura Taalman and Edward Aboufadel.
Before I left to come to MathFest, I had W&L photographer Kevin Remington take some stills of just a few of our models in a professional light box. The results are fantastic. Many of these photos appear in Thingiverse, as well as my web page.
These photos show: a few of the quadratics surfaces we designed and printed; the strange bowl family; some of our ”sliced” volumes; and the part of a helicoid, the “Bulge Head” solid, and Voronoi Klein Bottle.
The neat thing that we’ve been doing in the past couple of weeks is to use the FormLabs Form 1+ liquid resin printer. It is just so cool!
The first objects we printed were the strange bowls (shells, washers and smooth). Previously we tried to print them on the MakerBot 2X, but the sheer number of supports meant the print was not a great success. However, the FormLabs printed them beautifully. We all loved watching the bowls slowly come out of the liquid resin.
We next printed was the Bulge-Head solid. It is one of our favorites!
Finally, we had great success printing parametric curves and other surfaces with the liquid resin printer.
I then made a series of models of parametric curves. The first was a model of a spiral that increases in diameter as it travels along the \(z\)-axis. The curve comes from Section 10.7 in Stewart’s Essential Calculus. The curve is defined by the equations \(x=t*\cos(t), y=t*\sin(t)\), and \(z=t\). I designed the model in Cinema 4D 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 them with the curve. The print failed a few of times due to a tangled filament and a jammed extruder, but it worked after the fourth try. The result was that the equations looked messy and the letter \(t\) is hard to make out in places. Also, the MakerBot did not include supports for the last rotation, which caused the print to be messy towards the vertex of the spiral. We remedied these issues by printing another version of the spiral on the Formlabs printer, where we imprinted the equations into the object. The print came out much better as the equations were neater as was the end of the spiral. This model can be found on Thingiverse.
I used the FormLabs printer to create a model of a space curve from Stewart’s Essential Calculus (Section 10.7, exercise 18). The first challenge was to draw the object in Cinema 4D without a self-intersection (3D printers do not accept intersecting geometry). Professor Denne suggested that I make two half-curves that intersect, then make a Boolean out of them. The suggestion worked, so I was then able to put text onto it. It was tricky to figure out how to get the equations onto the curve, but I decided to put them on top of the lower ring of the model. They turned out well, as did the curve, which was very smooth and with minimal deformation due to its supports. You can find this model on Thingiverse here.
My next print was of a helix on the Formlabs printer. I first printed a Black one with a radius of 2 mm, but it turned out to be very small and frail, and the equations were hardly legible. I then fixed these issues by making the radius 4 mm, but the equations are again hard to read because of the white resin. This model can be found on Thingiverse here.
Our experience has taught us that equations are easiest to read on the FormLabs prints in grey. You can find instructions on how to use Cinema 4D to add equations to parametrized curves here.
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.
In order to make a monkey saddle I created the surface in Mathematica. I then exported it as a .wrl file and imported it into Cinema 4D. Once it was in Cinema 4D, like all surfaces, I made it the correct size, optimized the polygons and extruded them by 0.20 cm.
I then added an equation to the surface by punching it all the way through. This time with the equation I used Arial as the font instead of Times New Roman to hopefully avoid the issues we had with the formula when printing the hyperbolic paraboloid.
I printed the monkey saddle using the liquid printer and the formula in Arial font ended up looking great. The model can be found on Thingiverse here.
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.