Duals of Polyhedra

thingiverse

OpenSCAD can generate various polyhedra based on sets of points. If these points represent an Archimedean solid, the output is a Catalan solid; for Thomson problem solutions, it's the convex hull dual. When points are face centers equidistant to the origin, the polyhedron itself is formed. Multiple input point sets create compound polyhedra or material shared by multiple solids. However, directly generating Archimedean polyhedra isn't possible due to unequal distances from the origin. Instead, treating each set of faces separately and combining them can achieve this. Sample scad files are provided for a cuboctahedron from face centers, a rhombic dodecahedron from cuboctahedron vertices, a compound of both, and common material between them. Note that compound_cuboctahedron_dual.stl requires external support during printing. The method used in the program involves two spheres - one arbitrary and a unit sphere containing the points to be considered. A vector is drawn from the origin through a point on the unit sphere, creating a plane normal to the vector at a specific distance from the origin (inside the arbitrary sphere). All material beyond this plane's outer side is removed. This process repeats until all points are treated. The program's simplicity and ability to let openSCAD generate triangles are appreciated.