This shows Catalan Solids and their Archimedean Duals. Specificaly, it is about constructing these polyhedra with vZome. This project started with some queries and discussions about how to build them. Catalans can be built from midpoints of edges of their Archimedean duals as follows; These midpoints can be shown to be equidistant from a center point, which is to say, they lie on an indicated sphere. Struts can be constructed that are orthogonal to edges at midpoints and also orthogonal to radii, i.e. tangents to the sphere. This is most easily done with Cross Product tool. Intersections of these struts define vertices of the polyhedron. This series of models follows the sequence listed in the linked article, with Archimedean Duals as open frames.

3D designs created in vZome. Use your mouse or touch to interact.

Rhombic Dodecahedron with Cuboctahedron frame
Triacontahedron with Icosidodecahedron frame
Deltoidal Hexecontahedron with Rhombicosidodecahedron
Deltoidal Icositetrahedron with Rhombicuboctahedron

Note that the two following models are chiral. Only one of the two possible versions (left or right handed) of each are shown.

Pentagonal Icositetrahedron with Snub Cube
Pentagonal Hexecontahedron with Snub Dodecahedron
Triakis Octahedron with Truncated Cube

A continuation of this series can be seen at Here