Stellated Triacontahedron

If you have mastered the Slidables from last year and had enough of the past gloomy posts, you are ready for this one.

Let’s begin with the rhombic triacontahedron, a zonohedron with 30 golden rhombi as faces. There are two types of vertices, 12 with valency 5, and 20 with valency 3. In the image below, the faces are colored with five colors, one of which is transparent.

Triacontahedron

The coloring is made a bit more explicit in the map of this polyhedron below.

Triagraph 01

We are going to make a paper model of one of the 358,833,072 stellations of it. This number comes from George Hart’s highly inspiring Virtual Polyhedra.

Stellatriacontahedron

In a stellation, one replaces each face of the original polyhedron by another polygon in the same plane, making sure that the result is still a polyhedron, possibly with self intersections.

Newface 01

In our case, each golden (or rather, gray) rhombus becomes a non convex 8-gon. The picture above serves as a template. You will need 30 of them, cut along the dark black edges. The slits will allow you to assemble the stellation without glue. Print 6 of each of the five colors:

DSC 8159

Now assemble five of them, one of each color, around a vertex. Note that there are different ways to put two together, make sure that the original golden rhombi always have acute vertices meeting acute vertices. This produces the first layer.

DSC 8161

The next layer of five templates takes care of the 3-valent vertices of the first layer. Here the coloring starts to play a role.

DSC 8162

The third layer is the trickiest, because you have to add 10 templates, making vertices of valency 5 again. The next image shows how to pick the colors to maintain consistency.

DSC 8166

Below is the inside of the completed third layer.

DSC 8168

Two more to go. Layer 4 is easy:

DSC 8170

The last layer is again a bit tricky again, but just because it gets tight. Here is my finished model. It is quite stable.

DSC 8174

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