I saw the following figure in a Chemical Engineering article 'Catalysis on faceted noble-metal nanocrystals: both shape and size matter', Shuifen Xie, Sang-Il Choi, Xiaohu Xia and Younan Xia, Current Opinion in Chemical Engineering 2013, 2:142-150.
It represents a "continuous" family of polyhedra as weighted combinations of a cube, an octahedron and a rhombic dodecahedron (in fact, they consider discrete steps, in the spirit of cristallography).
Beyond the interest of seeing mathematics, and in particular 3D geometry, used in an applied context, I haven't been able to find mathematical works that consider this representation or an analogous one (spherical representation, or a more abstract one, using continuous groups ?).
My question is thus: has somebody ever seen any mathematical work connected to this representation ?
Edit (January 16th, 2018): I have found a complete study (though it only deals with convex polyhedra) in the very interesting book by Peter R. Cromwell named "Polyhedra" (Cambridge ed. 1997) ; I will not give details, but I advise strongly to look at it, with for ex. the following self-speaking topological representation in two figures and one table. (note the central rôle of n°5 = icosahedron).
This picture, centered on the great rhombicosidodecahedron, is extracted (page 15) from an article on the work of Haresh Lalvani in a 1980 issue of this interesting magazine.
Related :
Wythoffian operation in (https://en.wikipedia.org/wiki/Archimedean_solid)
(http://blog.physicsworld.com/2012/07/26/diy-build-a-particle-kit/)
(https://en.wikipedia.org/wiki/Waterman_polyhedron)
(https://www.redbubble.com/fr/people/alliweasley/works/15927513-polyhedral-relations)
(https://en.wikipedia.org/wiki/Uniform_4-polytope#History_of_discovery)