Geometry : Golden Rhombic and Edge Augmented Polyhedra

In a nutshell, here is what I want to present in this post:

• Modifying the look of the render and loading OBJ files on demand.
• Rebuilding the main Golden Rhombic Polyhedra from the base shapes and building newer models
• Finding a great piece of software that only runs on MacOS : Hedron
• Coming up with the concept of edge augmentation and applying it to convex polyhedra

• The viewer
• The Edge Augmented Polyhedra gallery
• The Rhombic Polyhedra gallery

Render

For some reason, I was not happy with the render anymore, so I decided to give it a (small) lift-up. I now use some Lambert material and some ambient and directional lights. I tested a lot of variations, including shadows but everything else was bad. I guess I am too used to path tracing renders now or my skill is too poor. I have also implemented the loading of OBJ files on demand. I don't think it will speed up anything though. Basically, during the initialisation of the viewer, when having lots of model, most of the time is spent on setting the GUI buttons for the models and almost none on the loading of the models data. The good point though is that I don't have to write a JS file with the geometry data which was a (working) very unelegant solution. This way, I won't have to duplicate the data for different viewers.

Golden rhombic polyhedra

I have already talked about rhombic polyhedra in a post about origami. This time though, I would like to go a bit deeper on the topic. As the name suggests, they are polyhedra made exclusively of rhombus. What defines a rhombus since its edges are all equal in length already is the ratio of its diagonals. When this value is the golden ratio, the rhombus is called a golden rhombus and a polyhedron made from golden rhombi only is a golden rhombic polyhedra. There exist only 5 of them which are convex, they are called the Golden Isozonohedron:

• the acute golden rhombohedron
• the obtuse golden rhombohedron
• the Bilinski dodecahedron
• the rhombic icosahedron
• the rhombic triacontahedron

When looking on the internet to see if anyone else has been trying to construct or study golden rhombic polyhedra, I stumbled across Hedron a MacOS piece of software that seems to let you assemble bigger polyhedra from basic shapes. I could not buy and test the software though since I don't have an Apple computer with me which is unfortunate. However, as a game, I have replicated the shape used as the logo since it is a rhombic golden polyhedron.

Edge-augmented polyhedra

All of the work described above started with some renewed interested on some work I had dropped a long time ago. To recall, after making the origami based on the rhombic polyhedron, I wanted (and I did) make the renders of these "augmented" rhombic polyhedra. To this end, I have to somehow figure out the position of the vertices. At first, I tried to find a close-form solution using Sympy and Maxima but since it was my first playing with these programs, I struggled a lot and ended it up discarding the project. I did find the coordinates for the model based on Bilinski dodecahedron though. On a second try, I came up with a more general using a numerical solver. I had tackled the problem and could do the renders but I was not satisfied yet. These two attempts happened years ago and I was not so happy with the results.

Recently, I started from scratch, stated the problem differently and used tricks that I learned along the way to solve for the point coordinates using Sympy. It works well on the platonic solids, I tried it on the Archimedean solid too and it could be generalized further but I am satisfied with the current implementation. I also tried a different manual approach which consists in establishing the correct topology and using Stella to figure out the correct position since I still have not implemented a reliable generic solver for regular faced polyhedra. The correct topology is easy to make. Take any polyhedra, split each face by introducing a new point at its center, then split each edge of the base mesh in two by introducing a new vertex and it is done. I call them edge-augmented polyhedra because it looks like a pyramid was added onto the edge like it happens on face-augmented polyhedra. With this operation, for instance, the edge-augmented cube is the same as the face-augmented rhombic dodecahedron. and the edge-augmented dodecahedron is the same as the face-augmented rhombic triacontahedron.