Coordinates of the vertices of the rhombic semi regular polyhedra:
|dodecahedron:||(±1, ±1, ±1) 8 points and (0, 0, ±3/2) cyclically permuted 6 points|
|triacontahedron:||(±φ, ±φ, ±φ) 8 points and (0, ±1, ±(1+φ)) cyclically permuted 24 points|
Among the vertices of the rhombic dodecahedron we recognize those of a cube and those of a regular octahedron.
Among the vertices of the rhombic triacontahedron we recognize those of a regular dodecahedron (thus also those of a cube) and those of a regular icosahedron.
|reference:||"Mémoire sur la théorie des polyèdres" by M.E.Catalan (Journal of the Imperial Polytechnic School - folio XLI - 1865)
"Improve, on some important point, the geometric theory of the polyhedra." (in French)
(great mathematics prize of the Sciences Academy - Paris 1863)
||convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects||September 1999 |