An icosahedron is the largest of the five Platonic Solids (by face count). It captivated me when I was taking my first Computer Graphics course due to my use of it as a 20 sided die, or d20 as it is known, in my long-running D&D sessions with friends. Thus, I wanted to make a perfect representation of it in digital form.
Originally, I just wanted the shape itself, which required defining its vertices and faces, with the later requiring vertex-triplets and normal vectors. I was able to accomplish this in short order, and produced the following animations via OpenGL in C++:
Two years passed, and I wanted to use it again in a VR video game project. It imported into Unity just fine, but I remembered I didn't give it a proper UV map back in the day. Thus, I set about drawing one out on paper, using the existing and fixed real vertices to stitch it correctly onto the model:
After some fine tuning, I was able to map the UV vertices to their Real counterparts with some set arithmetic, and all looked well, except it seemed a dice texture I applied seemed mirrored.
It was at this point I read a bit more about the file format I was using, Wavefront OBJ, and realized that it's a right handed coordinate system; With how I had drawn the slices of the icosahedron to map UV vertices to Real vertices, I was using a left handed coordinate system. Without getting to more into what all that means, to fix it I simply had to redraw the slices, this time with the z-axis pointing down (not up), and redo my relations, and ...
After many off-by-one errors, it worked. To understand the full algorithm, I leave as an exercise to the reader to follow the carefully commented code, along with running the program for yourself.
