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Joining octahedra

Posted 11 years ago
Dear members of Wolfram community!
I have one mathematical problem from an area of structural inorganic chemistry. Short introduction: anions [TiF6]2- are simple octahedral units, that can associate with each other forming more complex anionic units by sharing vertexes or edges of octahedra. Thus, in the case of two octahedra there are two variants of connection - by sharing one vertex and by sharing one edge (because of symmetry, vertexes and edges of single octahedron are indistinguishable).

Now, let's consider the case of sharing vertexes only. Three octahedra can be joined by three variants: three octahedra in a line, three octahedra forming right angle and three octahedra forming a planar ring (circular combinations are also possible). 
In the case of four octahedra additional dimension appears (3D) and if I have calculated right, there are 10 variants of their combination (by sharing vertexes only).

On the basis of this information I try to develop an algorithm, by which one can calculate the number of possible combinations of n octahedra (for the beginning by only vertex sharing), and I need your help to do this.

Thank you in advance,
POSTED BY: Igor Shlyapnikov
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Dear Sean!
Thank you for your reply. Unfortunately, since I am not a mathematician, I can't define the problem correctly, i.e. to formalize it. And this is the problem.
POSTED BY: Igor Shlyapnikov
I'm not good at combinatorics or group theory, but since no one has responded yet, I'll take a guess.

I wouldn't start by thinking of this as a programming problem. I would first try to formally answer the problem for the case of size two using Burnside's Lemma.

If you can do that, then you've formalized the problem's definition using group theory which should be useful. 
POSTED BY: Sean Clarke
Since a 3-member ring is allowed, I understood the problem to be like growth of a freely-jointed polymer chain, where the octahedra may be linked by both axial and edge connections.  This seems to rule out simple symmetry operations to count the possible structures. 
POSTED BY: Fred Doty
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