Duals of Polyhedra by pmoews

Last crawled date: 3 years, 1 month ago

Here is a simple openSCAD program which outputs the "dual" of a set of points on a unit sphere. If the points are the vertices of an Archimedean solid the output is a Catalan solid. If the points are a solution to the Thomson problem, (see thing:22663, Thomson Problem Polyhedra), the output is the convex hull dual. If the points are the centers of the faces of a polyhedron all equidistant from the origin, the output
is the polyhedron.
More than one set of points can be used as input so as to output compound polyhedra or material common to two or more polyhedra. For example the Archimedean polyhedra cannot be output directly as their faces are not equidistant from the origin. However the Archimedean solids can be generated by treating each set of faces separately and combining them.
Some sample scad files and their outputs are provided. They are:
A program to generate a cuboctahedron from its face centers:
cuboctahedron_to_self.scad
A program to generate a rhombic dodecahedron from the vertices of the cuboctahedron:
cuboctahedron_to_dual.scad
A program to generate the cuboctahedron - rhombic dodecahedron compound:
compound_cuboctahedron_dual.scad
A program to generate the material common to a cuboctahedron and a rhombic dodecahedron:
common_cuboctahedron_dual.scad
The image at the upper left shows the four polyhedra. Note how the vertices of the cuboctahedron protrude through the centers of the faces of the rhomic dodecahedron in the compound polyhedron.