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.

Screen Shot 2016-03-22 at 8.37.05 PM

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.

Screen Shot 2016-03-22 at 8.41.20 PM

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:

Screen Shot 2016-03-22 at 8.43.20 PM Screen Shot 2016-03-22 at 8.45.20 PM

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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