Distinct Hamiltonian cycles of the icosahedron and dodecahedron I am seeking a listing of the distinct Hamiltonian cycles following the edges of the icosahedron and the dodecahedron.  By distinct I mean they are not congruent by some symmetry
of the icosahedron or dodecahedron (respectively). So they do not make the same sequence of angular turns. For example (as Gerhard corrected me in the comments), there is just one distinct Hamiltonian cycle on the cube.
Hamiltonian cycles of the Platonic solids are all over the web, but I am not finding a definitive list of the number and a description of each.  Thanks to anyone who can point me in the right direction!
 A: There is a listing of a hamilton cycle count of 2560 at this website: http://mathworld.wolfram.com/IcosahedralGraph.html.
A: There are 17 Hamiltonian Circuits on an icosahedron which have a unique shape. You have to find a description of the circuit that codes the this shape. I coded the succession of  turnings that the circuit takes at every vertex. On an icosahedron the circuit has one of the four possible continuations, say A, B, C, D. You find  a description of the Hamiltonian cycles as lists of ABCD. Then you can easily find the unique ones (remember that each vertex can be the beginning one).
I programmed this in Mathematica. 
Later I found, that A. Sainte-Lague published this already in 1937 in a book called: Avec des Nombres et des Lignes.
By the way: The tetrahedron, the cube and the dodecahedron have one Hamiltonian Circuit (HC), the octahedron has two and the icosahedron  17 !
Lei Willems
A: bit late to answer here but been looking into this myself. I found a presentation showing that if one is interested in the unique sequences of the magnitude of the angle (i.e. viewing A=C and B=D in the answer by user93483), then for the platonic solids we find:
Tetra = 1 HC, Cube = 1 HC, Dodeca = 1 HC, Octa = 2 HC, Icosa = 11 HC
This results in a total of 16 topologically distinct Hamiltonian cycles for the platonic solids.
Link to presentation:
http://www.cs.berkeley.edu/~sequin/TALKS/Banff05_HamSymm.ppt
