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Cycles on Lattices

So, first a small correction to the last post : a 'honeycomb' layout is a cycle on a hexagonal lattice, while the other layouts there are not on-lattice. I've renamed them 'flower' layouts, as I'm not sure if they have a name - they're not at all new, I just haven't looked! To be clear:

these are two different layouts of the same 12-cycle. As it happens, the honeycomb layout is also a [5, 5, 5]-flower layout with straight edges on the first edge of the outer cycle. However, not every honeycomb is a flower.

There might be many ways to do it, but one way to make honeycomb layouts is to find cycles on a hexagonal lattice. Assuming, of course, that you have such a lattice already - it's not terribly hard to make one, but connecting it up properly is a bit fiddly. Luckily, it turned out that making the dual of a triangular lattice is slightly easier. To see how this works, consider the three possible (regular) planar lattices:

The grey dots are actually the dual lattice of the black lattice. Click-for-bigger to see the lines connecting the dual points. Of course, a square lattice (on the left) is self-dual, while triangular and hexagonal lattices are duals of each other.

Fine, so given such a hex-lattice, how to get cycles of the right size? Well, the simplest way might be to find all cycles in a grid of some size, and just take ones of whatever size. This ... works - but not very well. Here are the 10-cycles from the hex-grid above:

clearly they are all the same! In fact, this is one of the smaller 'families' (orbits) of cycles. There are 78 cycles of size 12, and 2,082 of size 22. Here is a table:
Clearly, this is not the right approach.


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