3D Printing in Clay

This week has been an exciting one here at Pitzer College. I’m currently co-teaching a class on “Mathematics and 3D Printing” with our Ceramics Professor, Tim Berg. Tim is also teaching a class on mold-making that I’ve been auditing when I can. For the benefit of both classes, Tim invited Bryan Czibesz, from SUNY New Paltz, to come lead a workshop in which we built a 3D printer that creates objects out of clay. This is particularly timely; Bryan had just come from the first Clay Fab Lab at the National Council for Education in the Ceramic Arts conference. An article about this meeting just appeared (featuring Bryan) on the Shapeways blog and a similar one in 3DPrint.com.

Bryan has plans up on Thingiverse for the printer we built. In advance of his visit, I printed some of the components in ABS plastic on our Flashforge Creator, and Tim cut the wood pieces on a CNC router. One of the features of this particular 3d-printer design is that all of the parts are easily made or acquired. Here are some of the parts laid out before assembling:


Bryan did a fantastic job involving the students in the actual construction of the machine.


And here’s the completed machine….

The circular collar you see hanging in the center holds a tube of wet clay that looks like an upside-down ketchup bottle. Inside that tube is a plunger, and the top end is connected to an air compressor. When you turn on the compressor, the air pushes the plunger down, and squeezes clay out of the tip.

Unlike a plastic extrusion printer, once you turn the compressor on it continuously squirts the build material (clay). In my initial experiments, I’ve found that this significantly constrains the geometry of printable designs. Essentially, each horizontal slice of the model being constructed must be a deformed circle. To relax this constraint, Bryan is working on a way to have a computer control an on/off valve for the clay,  but that wasn’t ready to implement with our machine.

With all that said, the machine is still capable of producing some amazing designs. Here are a few images of pieces Bryan had brought with him that were made on a similar machine.

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You’ll notice in these images that there’s a lot more than just 3D printing involved. Each design is both hand-glazed and kiln-fired. Some of Bryan’s designs are assembled from multiple printed pieces, and some combine printed and hand-built components. Clearly, 3D printing can be a valuable new tool in the arsenal of the traditional ceramic artist, but to produce designs like these, knowledge of traditional techniques is still necessary.

In future posts I will share my own designs made with our new printer (I’ve got a lot of ideas!). In the meantime, here’s a video of our new machine in action.

3D Printing in Clay

The Islamic Cube

Many years ago my wife and I co-taught a class on the Mathematics of Tilings. We have a lot of Islamic tiling patterns around our house, and I think we were excited about the class because it gave us the chance to learn more about the general theory of tilings, as well as about how these particular kinds of tilings can be constructed.

Around that time I bought her a gift of some small jewelry. I thought I’d be cute and make a little box for it out of paper. So I searched the internet for a nice pattern, and came up with this image, created by Craig S. Kaplan with his wonderful applet Taprats:


You can read more about how this program works in an article Craig wrote for a Bridges conference, here.

To make the box, I simply used Taprats to make an image similar to the one above, printed it, cut, and folded, and viola!IMG_5518

I’m pretty sure she liked the box better than the gift that was inside, or it wouldn’t have survived in our house for this long.

A few years after this I started getting into 3D printing and more artistic work. I did a woven coffee sleeve (I’ll write a post about that sometime!), and after seeing it my wife suggested I make the pattern on the Islamic box. The idea was to interpret the printed lines as literal curves in space that weave through each other as in the image.

Taprats was helpful once again in generating the raw curves that the above tiling is based on. These come from a pattern inside an octagon, 12-gon, and an odd bow-tie shaped piece, as shown here.

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Once I had these curves in Rhino, I could manually manipulate them to make the correct over/under pattern. (In a future post I’ll describe a program I wrote later to do this automatically, to create models of alternating knots.)

Thickening the resulting curves turned out to be one of the biggest challenges. Each strand was to have a rectangular cross-section, but these cross-sections need to stay horizontal with respect to the plane of the pattern. There are also issues where strands make sharp angles. There was no built-in way to get Rhino to do this! so … I wrote a Grasshopper script to do it. (If anyone out there would like it, let me know!)  Here’s the result of running that.

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The next challenge was to make a cube. The obvious thing would be to take six copies of the above square, and put them on the faces of the cube. However, I had a lot of trouble connecting them that way. Instead I ended up bending the square along it’s diagonal, and putting together 12 of them so that each bent diagonal became an edge of the cube:

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Finally, I was ready to print! I uploaded to Shapeways and ordered a cheap plastic prototype, before ordering it in bronze, as shown below. You can order your own in plastic, bronze or brass here.

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The Islamic Cube

3D Printing for the Visually Impaired

This project was neither Math nor Art, but it called on a lot of the skills I developed doing both, so I thought it appropriate to write about here.

Last June I was contacted by our Office of Student Affairs about a student who would be joining us in the Fall. The student is visually impaired, and they asked if I could create a relief map of the campus to help guide her.

After a few conversations, I realized there were a few important design constraints:

  1. The map had to be small enough in all three dimensions to fit in a backpack. In particular, that meant fairly shallow relief.
  2. It had to be large enough that the individual features could be easily detected by touch.
  3. The map should only include relevant details of the campus, or it would get too cluttered: only buildings, paths, and roads were necessary.
  4. Somehow this project shouldn’t take a lot of MY time. They weren’t paying me anything, and no one knew how useful the finished product was going to be to the student.

The last criterion was the most important. I could spend hours individually modeling each building, but that really wasn’t necessary.  So I had to think about other solutions…

I started by asking our Office of Communications for a simple line drawing of the campus, including only the relevant features. Here’s what they gave me:vectormap

Next, I was able to import this into Rhino3D, and automatically extract the outlines of each feature. It only took about an hour or two to raise each outline to a different height in the z-direction, and make them 3-dimensional. I made the heights correspond to the type of object, rather than any kind of representation of reality. So all buildings were one height, roads another, paths a third, etc. I was hoping that would provide enough tactile information to distinguish between them. With much more time I could have added textures (e.g. make grassy areas rough), but that didn’t seem necessary at this point. Here’s the finished digital model:


I realized the buildings were going to need labels. I thought about adding braille on top of each building, but that was going to take way too much time. After some internet searching, I discovered this braille label maker:

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So I ordered it, and showed the people at the Office of Communications how to use it to manually add labels. I’m sure that was labor intensive, but I didn’t end up having to deal with it. It’d be worth incorporating braille in the digital model if we were going to make these things on a regular basis, but as I said above, it wasn’t clear how useful the finished product was going to be, or if and when the college would admit another visually impaired student.

The finished map was printed at Shapeways.com. Here it is! If you look closely you can see braille labels on a few of the buildings. The whole thing is 10″ by 11″, and about one-third of an inch high, so it slips easily in a backpack between books.


I was recently told that it got used by the student extensively. It helped guide her around to the point she memorized the paths to take, which was exactly its purpose. Success!

3D Printing for the Visually Impaired

A Plurality of Polyhedra

A little over a year ago (February, 2015) I was contacted by Los Angles artist Clare Graham about making some models. He had become interested in the illustrations of polyhedra in the 1509 book De Divina Proportione, by Luca Pacioli. What’s significant about these  illustrations is that they were done from woodcuts by Leonardo da Vinci. Many of them represented the first depictions of polyhedra which allowed one to easily see their internal structure.

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Da Vinci made 61 illustrations for the book. About half of these are solid polyhedra, and half are the more open, skeletal designs like the one in the image above. He also included three solids that are not polyhedra at all: a sphere, cone, and cylinder.

After a few months of going back and forth with Clare about decisions on size, material, finish, etc, I set to work reproducing these. Each one took anywhere between 30 seconds and 3 or 4 hours to design.


Here’s how I did it. First, I installed the “Polyhedra” plug-in for Rhino3D. Most of the polyhedra I needed had already been programmed by the author of this plug-in, so all I had to do was type in the name of the shape I wanted. There was a little work to do to translate the latin names to the modern nomenclature, but most of that was fairly easy to guess. To create the open designs, I simply used the “extract wireframe” command to get the edges of each polyhedron, and then the “pipe” command to thicken them.

The solids were trickier. Each model was printed at Shapeways.com. Since they charge by volume, a large solid piece would be extremely expensive. So I made each one hollow with a removable tip that could be glued in later.


To get a tight fit for gluing, I chamfered the edges of each glue-in piece. It took a few hours of playing to figure out an efficient way to do this, but eventually I came up with a method that took under 5 minutes per model. The tricky part was getting the exact same chamfer on both parts to be glued.

A few of the da Vinci designs were not in the “Polyhedra” plug-in. Most of those were stellated versions of some of the polyhedra I did have access to, so I wrote a custom Grasshopper script to stellate any input polyhedron. If anyone out there is interested in that, I’d be happy to share it.

After the models arrived in a big box, I glued all of the solids together with super-glue. The Shapeways “White, Strong, and Flexible” plastic is notorious for picking up smudge
IMG_5107marks, so I thought they should also be sealed. After hours of internet searching I discovered “Pledge Floor Care Finish”, which is basically a clear acrylic sealer. This worked OK, but the next time I do something like this I’m going to try a penetrating stone tile sealer. (I just used one brand on some Saltillo tile I installed in my home, and was really impressed!) Here they are arranged on some plastic in my office, ready to be sprayed with sealer.

Finally, I handed the finished models over to Clare, who has them arranged in a  beautiful glass case in his gallery in Eagle Rock. (Apologies for the crappy quality iPhone 4S picture!)




All of these models can be purchased here at Shapeways. There’s a modest $10 mark-up fee that would go to me if you buy one there, but if you have just a little knowledge of Rhino (or any CAD package) you could also make them yourself in very little time and save that.

A Plurality of Polyhedra

Apollonian Gaskets with Grasshopper

My son and I were recently watching Vi Hart’s wonderful “Doodling in Math Class” Video on Infinity Elephants. Around 1:30 in the video she starts talking about how to draw an Apollonian Gasket in a triangle:

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This got me wondering about how to code this object. It’s not a new challenge. Lot’s of people have done it. You can even search Shapeways to find some amazing 3-D printed, 3-dimensional versions. But I couldn’t find much info on how its done. I did find an old discussion here where they were wondering how to do the same thing. (Nothing really useful there, except some nice references to hyperbolic Geometry-specifically a little paper by David Dumas-with a dead link to a program written by Curt McMullen.)

I’ll use Python when I have to, but my preferred method of construction is with the visual programming language Grasshopper, a plug-in for the CAD program Rhinoceros3D.  Clearly, the object I was trying to replicate is fractal, which means recursion is going to be unavoidable. That’s easy enough in Python, but it sucks in Grasshopper. The only way I know to do recursion in Grasshopper is with a 3rd party component. I think there are multiple options. The one I use is called “Hoopsnake.” It does the job, but for this kind of thing it’s extremely slooowwww. That’s OK, because to replicate the drawing in Vi’s video, I’m only going to do about 5 iterations.

It took a bit of experimenting, but eventually I got it down to a surprisingly small Grasshopper definition. The key was to use the CircleTanTanTan component, which automatically finds the circle that is tangent to three different input curves. For the Grasshopper savvy out there, here’s my definition:

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Here’s what it produces….

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…exactly what I wanted!!

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Once I had that, it wasn’t hard to make all kinds of designs. Here are some Apollonian Spheres.




I even experimented with an Apollonian Pocket Watch Design!

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Apollonian Gaskets with Grasshopper