I saw this dodecahedron with an Islamic-inspired pattern designed by Taj Ragoo. As soon as I saw it, I knew I had to make one. I studied the pattern, wrote some Python, and made myself a PDF. I cut it out, folded it, glued it together, and now I have one of my own:

I love that this elegantly combines two pure geometric forms: the Platonic dodecahedron (12 uniform pentagons), and an Islamic pattern using five-pointed stars.

Looking closely, details emerge:

Each face has ten small stars in a ring. I’ve lightened them a bit in the front face here. At the center of each face is a ten-pointed star (highlighted in red), made of two overlaid five-pointed stars.

The real genius of the pattern is at the corners. I’ve highlighted one in blue. It’s a star made of the same parts as the central ten-pointed star, but there are only nine points. It works because three pentagons lying flat touching at a point occupy 324 degrees, leaving a 36-degree gap.

When the dodecahedron is folded together, the gap is closed. 36 degrees is exactly one-tenth of a complete 360-degree circle, so exactly one point of the ten-pointed star is missing, leaving a perfect nine-pointed star using the same shapes, spread over the corners of three pentagons. Beautiful!

If this appeals to you, follow Taj on Instagram: he’s got more Platonic/Islamic mashups to enjoy. The paper versions are just prototypes of the final versions he makes in wood.

Of course, you can get my PDF and make one for yourself:

The Python code to draw the net isn’t great: it has no real parallels to the structure of each face. It’s a lot of math and line drawing to get things in the right places. My ideal would be to have a toolset that used a tile-placing abstraction, to be able to do more interesting designs. Some day.

It was a joy to work on this though. It was a slow process of studying the original, working out the math, then mulling over coding approaches. The code was developed in small steps over weeks. Then printing initial versions, marking them up, working out the tab structure. Some copies were colored to understand how the lines flowed across the whole dodecahedron. It was good to be working in both the mental and physical worlds: