Honeycombs and Macrocyles

Small rings in chemistry are usually laid out as polygons; 5-membered rings as a pentagon, 6-membered as a hexagon, and so on. Once you get beyond about 9-10, this tends to look a little nasty. Or at least, unconventional.

So for 'macrocycles', it makes sense to make a less circular, and more wavy outline. Or, to be more exact, there is an inner cycle and several outer cycles, like this:

To make it clearer, I have used a chemical-like structure with oxygens in the inner ring. These crown ethers are a particularly clear-cut case, as the ethylene linkages force the particular geometry of the drawing. However, it is not so obvious for other sizes of rings - what possible arrangements are there?

Well, it occurred to me today that there is a simple formula for these macrocycle drawings. For a ring of size n with an inner ring of size i and outer rings or size o, you have to have n = (i * o) - i. The formula can be rearranged, but the idea is that you add up all the outer rings and remove the inner one.

So here are some crude representations of such cycles, for n in {9, 12, 14}. Below each cycle is the list of outer cycles. All of these examples are regular, in that they have the same number of vertices in the outer cycle. It is possible - although less desirable - to have different numbers of vertices in the outer cycles.

The version on the left is somewhat uglier than the 'puffed out' one on the left, but on some sense they are the same drawing. They are both (4, 6, 4, 6) in the notation of the previous image. Note that 24 = 4 + 6 + 4 + 6 - 4, which suggests that the previous formula can be generalised a bit to : n = (x0, ..., xr) - r. Each x in the equation is a ring size, and it is the sum of these minus the number of rings.

This immediately suggested a way to make examples - use partitions again! In other words, partitions of n + r for r in the range (3, floor(n / 3)). This produces a whole lot of horrible drawings, such as (3, 3, 9), but it does work. The code is here.