some technical informations

regular polyhedra

infos 1

groups of isometries

infos 2

semi-regular polyhedra

infos 3
infos 4 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.
Catalan calculated the similitude's angles and ratios, but the values concerning the dodecahedron seem to be wrong...
  cube: 16° 28' 7"  and  0,437593
  dodecahedron: 22° 5' 18"  and  0,649979   (13° and 0.56 are more correct experimental approximate values)

Coordinates of the vertices of the rhombic semi regular polyhedra:

dodecahedron:   (±1, ±1, ±1)    8 points
(0, 0, ±3/2)  cyclically permuted      6 points
triacontahedron:  (±φ, ±φ, ±φ)    8 points
(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}: matrix
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)


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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects September 1999
updated 17-07-2007