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.

Leaves

Rarely have I enjoyed the first frost as must as this year.

It’s time to look back, and the theme leaves suggests that I list the books that I found memorable this year.

In the English language, there were several books I really liked. Zero K by Don DeLillo, and The Buried Giant by Kazuo Ishiguro. DeLillo’s book is atypical for him, the humor of his earlier books has disappeared, and the discussion of the acceptability of death reminded me of classic greek theater. Ishiguro has been writing against being compartmentalized for a while, and his Buried Giant is no exception. I must admit that he has tricked me with this book: I thought it was an easy read, but only later realized that I have forgotten crucial parts, very much akin to the forgetting that is happening in the book itself. It’s a daunting book.

I haven’t read as many French books from this year as I would have wished, mainly because I will probably forever play catch up with previous years. The one outstanding book though is the completely devastating Chanson douce by Leïla Slimani. The book begins with the death of two children, killed by their trusted baby sitter. While we learn more about it, we have to reconsider what makes a life worth living. This book has won the Prix Goncourt this year. While I don’t trust book prizes blindly, they sometimes get it right.

Then there are the books in German. For me, the clear winner is Am Rand, by the Austrian writer Hans Platzgumer. Once more this year we hear about a life, and its end. This appears to be this year’s theme in literature: Ways of dealing with death. This sounds morbid, but the point is that while the protagonists approach death one way or the other, we learn about how they deal with life, and in all the books above there is a lot to learn.

Incidentally, by favorite book this year is not about death at all but rather about a desperate attempt to grasp life. J.M Coetzee’s The Schooldays of Jesus is the sequel to his The Childhood of Jesus, and it is pretty clear that there is more to come. We follow two immigrants (a man and a boy) in a nameless, kafkaesque country. The man is willing to accept his new life, while the young boy questions everything, creating meaning in a senseless world.

Treescapes (Red River Gorge State Park III)

Even more than the curious Natural Bridge and the Rock Garden, a highlight of Red River Gorge State Park is the view of distant treescapes.

Compared to Indiana, the vegetation of Kentucky has deeper reds and more pronounced greens in late Fall.

Living with daily views like these would be paradise for me. Admittedly, I like complicated things.

Rock Garden Trail (Red River Gorge State Park II)

I am a big fan of rock/tree formations, and thus the Rock Garden Trail in Red River Gorge State Park was a must for me.

The rock garden is, expectedly, very different from designer Japanese landscapes. Because everything is here by chance, all depends on the viewer.

It is hard to say what I like about these landscapes. They are clearly not everybody’s taste.

Maybe they vaguely suggest archaic temples to me — either unfinished, or in utter destruction. The trees serve as columns, while the rocks are material for more permanent structures, to be used or in slower decay.

The Nihilist’s Bridge (Red River Gorge State Park I)

The Natural Bridge in Red River Gorge State Park (Kentucky!) doesn’t bridge anything, like most natural bridges.

One can walk under and across and thus use both functions of a bridge, contemplating the pointlessness of the act in this case. It helps that the bridge is more or less on top of a mountain.

That there are trees on top is giving hope, though.