Math cure for piles

Alan Turing developed a mathematical model for the patterns you get on stripey and spotty animals. Suppose each bit of the skin is either light or dark. Bits will tend to be light or dark if all their neighbours are light or dark: bits on the edge, with half light and half dark neighbours can change to preserve the average lightness or darkness in the region. You can simulate this with discrete cellular automata, but there are also continuous differential equations with the same properties.

There are some simple predictions. The regions of light and dark will tend to have smooth edges. If the light-to-dark ratio is close to 0.5 then you will get lines: if it isn't, then you will tend to get spots. However, where the animal is strongly curved, like the tail of a leopard, you will get stripes that run around the tail. Spots are pretty stable, but stripes tend to straighten up until the run around the animal, and incomplete stripes tend to grow out. This last prediction was verified on angelfish only in the last few years.

Okay, now let's try the same thing on the inside of the animal. Suppose in the colon your regions are not light and dark, but convex and concave. This would normally give you a set of ridges running around the colon, but isolated blobs between the ridges could be stable. However, unlike the coloured regions we had before, these regions of concavity and convexity are changing the local geometry too. If one of these blobs was wide enough, then it could grow ridges of its own on the surface; to maintain the balance of concave and convex, the neck of the blob would grow, and you have a polyp, poor you.

If you cut off the head of polyp, you may still have the local imbalance of concavity and convexity in the bit at the base. Once the thing has healed, the most likely thing it will do is grow right back again. The right solution would seem to be to pinch up a ridge between the polyp and the nearest ridge. Instead of a line and a dot, you will have a line with a little dead-end line growing off it. When the two regions have joined, take off the clip doing the pinching. The dead-end line should shrink back into the original ridge, and never want to grow back.

Well, so goes the theory anyhow...