As promised (in the previous post) I've now made Jmol scripts to show the atom/ring equivalence classes. I still think that the ring ones are more clear, but I suppose it depends on what aspect of the symmetry of the structure is needed. As an example:

Shown here is a C70 structure, with coloured circular plates at the centre of each face. It should be clear that there is an axis of symmetry running through the middle, from one blue plate to the other. Around the blue is a ring of green, and 5 rings in between.

The slight difficulty in all this was working out the ring equivalence classes. There is an existing CDK method to do this - in the SSSR ring finder - but it seems to give too many classes. The way I did it was to first find atom equivalence classes (or 'orbits') using signatures. Then each ring is a circular list of the orbit indices : which I'm going to call a 'ring code'. See this image for illustration:

These two rings (A and B) have the same ring code, written as the smallest concatenated string formed from their orbit indices. In other words, the signatures of each atom in the ring is converted to a number based on that signatures index in a list of all the signatures for all the atoms. Obviously other atom-equivalence class methods could be used to find the initial orbits; the rest of the procedure would be the same.

I do wonder if it would be quicker to just find the orbits of the dual of the embedding. However, that involves making that embedding first, so probably not...

Shown here is a C70 structure, with coloured circular plates at the centre of each face. It should be clear that there is an axis of symmetry running through the middle, from one blue plate to the other. Around the blue is a ring of green, and 5 rings in between.

The slight difficulty in all this was working out the ring equivalence classes. There is an existing CDK method to do this - in the SSSR ring finder - but it seems to give too many classes. The way I did it was to first find atom equivalence classes (or 'orbits') using signatures. Then each ring is a circular list of the orbit indices : which I'm going to call a 'ring code'. See this image for illustration:

These two rings (A and B) have the same ring code, written as the smallest concatenated string formed from their orbit indices. In other words, the signatures of each atom in the ring is converted to a number based on that signatures index in a list of all the signatures for all the atoms. Obviously other atom-equivalence class methods could be used to find the initial orbits; the rest of the procedure would be the same.

I do wonder if it would be quicker to just find the orbits of the dual of the embedding. However, that involves making that embedding first, so probably not...

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