## Stellating the Icosidodecahedron in Black and White

It is also this time of the year to chase away the dark hours by making presents. As in previous years,
we will make a stellation out of paper without glue. This year, we are going to stellate the Icosidodecahedron, one of the fancier Archimedean solids.

The stellation is quite simple, it is also a compound of the dodecahedron and icosahedron. The simpler compounds of a Platonic solid with its dual are also doable, see the post from two years ago.

To make it, we will need 20 triangles and 12 pentagons, so printing and cutting two of the templates below will do. I suggest to print four templates in two different colors and to make two models.

Then we start by sliding five triangles into one pentagon like so:

Next another five triangles:

Now we have finished one half of the model. This already would make a nice dome for the backyard.

You can make another half and try to attach them, but I think it is easier to just keep going.
This next step is a little tricky, because to prevent the polygons from falling out, it is best to add a ring of alternating pentagons and triangles. When done, it looks like this:

The last two steps (add five more triangles and one more pentagon) are then pretty clear, but still tricky because you have to insert the new polygons in four or five slits essentially at once.

## 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.

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

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.

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.

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:

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.

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.

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.

Below is the inside of the completed third layer.

Two more to go. Layer 4 is easy:

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