# some technical informations

### regular polyhedra

### groups of isometries

### semi-regular polyhedra

__The snub cube and the snub dodecahedron__ may be obtained from the two regular polyhedra: first perform a similitude on each of its faces (center of the similitude on the center of the face), then truncations to create the chains of triangular faces around the transformed faces.

cube : 16° 28' and 0,4376 - dodecahedron : 13° 6' and 0,5621
__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.

**Rotations in space ** belong to the necessary tools to describe polyhedra. Mathematica defines a rotation using Euler's angles, but it is more convenient to use the matrix of a rotation of angle *α* around an axis defined with an __unit__ vector {u,v,w}:

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) |