Study of Parametric Dodecaherdon in OpenSCAD

thingiverse

I've been obsessed with dodecahedrons for quite some time now, and lately, I've been diving deep into trigonometry, so I thought I'd give creating one a shot. While at it, I figured I might as well draw numbers on it to help someone build a 200mm D-12 die. I'm struggling to figure out how to solve for dihedral angles, especially when dealing with regular polyhedrons like the dodecahedron, but also when combining any number of angles that close a vertex of three or more planes to form a closed mesh. Someone must have cracked this code using trigonometry, linear algebra, or whatever it is that Buckminster Fuller types use to generate geodesics and such. Clearly, I need to head back to school - my skills are clearly marketable! How does a building contractor join a 1:1 pitch roof to a 1:2 pitch roof that meets at a 120-degree angle? Can you enlighten me?! I did learn that the dihedral angle of a dodecahedron is the arc-cosine of negative one over the square root of five... but why, how, when, and who came up with this formula?! Anyway, here's a parametric Customizer I wrote in OpenSCAD and a few STL examples, including a 200mm D-12 twelve-sided die - enjoy!