Icosahedral Symmetry on a Sphere

thingiverse

This OpenSCAD code covers a sphere with an icosahedral pattern, similar to "thing:39424 Octahedral Symmetry on a Sphere". The Platonic solids featuring five-fold rotation axes are the dodecahedron and the icosahedron, both sharing six 5-fold axes, ten 3-fold axes, and fifteen 2-fold axes. By mapping their vertices to a unit sphere, one can locate planar patterns with three or five-fold rotation axes and create an icosahedral symmetry sphere. Rosettes, consisting of a central circle and three or five equally spaced rays, were used to illustrate the method; see images on the left. The 3-ray rosettes are centered at dodecahedron mapped vertices while the 5-ray ones are at icosahedron mapped vertices. Both cases result in patterns with icosahedral symmetry. The code produces stl files for icosahedrons and dodecahedrons mapped to a sphere; see upper left image. Additionally, gen_rosettes.scad generates dxf rosette files while gen_icosahedral_sphere.scad creates stl files. Printing instructions are provided for the small and large stl files. More complex patterns, like a 20-ray rosette at icosahedral vertices, can lead to intriguing results; gen_rosettes.scad offers various rosettes for experimentation purposes.