Possibly new sequence related to dodecahedron
Franklin T. Adams-Watters
franktaw at netscape.net
Fri Dec 31 02:16:36 CET 2004
A pair of related, also finite, sequences: the number of "polypents" (connected sets of pentagons, up to symmetry) on an dodecahedron, and "polyiamonds" on an icosohedron.
These sequences can also be done for smaller Platonic solids, but are too simple to be interesting. This is especially so for the tetrahedron, where the sequence is:
1,1,1,1,1
For the square, we have:
1,1,1,2,2,1,1.
For the octahedron, the sequence is (I think):
1,1,1,1,3,3,3,1,1
"N. J. A. Sloane" <njas at research.att.com> wrote:
>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
>
--
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
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