# Possibly new sequence related to dodecahedron

N. J. A. Sloane njas at research.att.com
Tue Dec 28 17:26:10 CET 2004

The following is an interesting question raised by
Wouter Meeussen <wouter.meeussen at pandora.be> on Dec 27 2004:

He said:
>> Based on the symmetry point-group, how many 'different' sets of 1, 2, .., 10 vertices
>> exist on a dodecahedron?
>>
>> If I count different norms of sums of 1, 2, .., 10 unit vectors, I find:
>> 1,5,12,22,34,50,65,78,78,86
>> but there might be 'different' sets that accidentaly sum to the same resultant's norm.
>> So the given integers are a lower bound.

We can rephrase the question as follows.

Let G be the full icosahedral group, of order 120.
Let v_1, ..., v_20 be the vertices of the dodecahedron.
Let S(n) be the set of vectors

v_{i_1} + v_{i_2} + ... + v_{i_n}

where 1 <= i_1 <= i_2 <= ... <= i_n <= 20.
Then what is s(n), the number of orbits of G on S(n)?
(so s(1) = 1, s(2) = 6, ...)

Presumably this is different from A039742:
...
%S A039742 1,6,50,475,4881,52835,593382,6849415,80757819,968400940,11773656517,
%T A039742 144791296055,1797935761182
%N A039742 Lattice animals in the fcc lattice (12 nearest
neighbors), connected rhombic dodecahedra, edge-connected cubes.
...

Repeat for icosahedron, etc.!

NJAS