### 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.

### How many isomers of C4H11N are there?

One of the most popular queries that lands people at this blog is about the isomers of C4H11N - which I suspect may be some kind of organic chemistry question on student homework. In any case, this post will describe how to find all members of a small space like this by hand rather than using software.

Firstly, lets connect all the hydrogens to the heavy atoms (C and N, in this case). For example:

Now eleven hydrogens can be distributed among these five heavy atoms in various ways. In fact this is the problem of partitioning a number into a list of other numbers which I've talked about before. These partitions and (possible) fragment lists are shown here:

One thing to notice is that all partitions have to have 5 parts - even if one of those parts is 0. That's not strictly a partition anymore, but never mind. The other important point is that some of the partitions lead to multiple fragment lists - [3, 3, 2, 2, 1] could have a CH+NH2 or an NH+CH2.

The final step is to connect u…

### Generating Dungeons With BSP Trees or Sliceable Rectangles

So, I admit that the original reason for looking at sliceable rectangles was because of this gaming stackoverflow question about generating dungeon maps. The approach described there uses something called a binary split partition tree (BSP Tree) that's usually used in the context of 3D - notably in the rendering engine of the game Doom. Here is a BSP tree, as an example:

In the image, we have a sliced rectangle on the left, with the final rectangles labelled with letters (A-E) and the slices with numbers (1-4). The corresponding tree is on the right, with the slices as internal nodes labelled with 'h' for horizontal and 'v' for vertical. Naturally, only the leaves correspond to rectangles, and each internal node has two children - it's a binary tree.

So what is the connection between such trees and the sliceable dual graphs? Well, the rectangles are related in exactly the expected way:

Here, the same BSP tree is on the left (without some labels), and the slicea…

### Listing Degree Restricted Trees

Although stack overflow is generally just an endless source of questions on the lines of "HALP plz give CODES!? ... NOT homeWORK!! - don't close :(" occasionally you get more interesting ones. For example this one that asks about degree-restricted trees. Also there's some stuff about vertex labelling, but I think I've slightly missed something there.

In any case, lets look at the simpler problem : listing non-isomorphic trees with max degree 3. It's a nice small example of a general approach that I've been thinking about. The idea is to:
Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }For each d_i in D, connect the degrees in all possible ways that make trees.Filter out duplicates within each set generated by some d_i. Hmm. Sure would be nice to have maths formatting on blogger....

Anyway, look at this example for partitioning 12 into 7 parts:

At the top are the partitions, in the middle the trees (colored by degree) …