Pentakis dodecahedron Himmeli

My Himmelis are getting bigger as I learn and develop my technique, and for this one I also decided to use a better camera setup. This himmeli is approximately a pentakis dodecahedron; the true Catalan solid has slightly varying edge lengths, but I'd already cut my straws to equally long pieces for more regular polyhedra, so this will have to do.

At some point I wondered if this approximation closes up nicely, but by the time I'd finished a hemisphere, things were looking good. For one thing, you can regard the structure as a kind of stellated dodecahedron; while the stellations make the detailed structure bumpy, it doesn't change the base dodecahedron that keeps it all together.

I managed to finish this with a single thread, splicing in more length as I went on. Due to the non-Eulerian graph structure, this meant backtracking on a few edges — basically once per every odd-valued vertex.

I used a simple recipe to keep track of the structure: (a) The neighbours of each 6-valent vertex are alternately 5- and 6-valent, and (b) each 5-valent vertex has only 6-valent neighbours. This turns out to be enough, but you can also check in the end that you get 12 5-valent and 20 6-valent vertices. (This is the dual of the buckyball, which has 12 pentagonal and 20 hexagonal faces.)

#himmeli #strawart #pentakisdodecahedron #catalansolid #polyhedron #geometricart #algorithmicart #algorist #mathart #laskutaide