# As Much As Possible

## Tesselation

Say you had a table top that you wanted to cover with bits of paper.

In this context the constraint as much as possible (AMAP) means 'cover as much table as you can with the least paper'

If you start out with the output of a good paper shredder as a paper source, the irregularity of the paper makes covering the whole space almost impossible. There are always gaps between the bits. To fill the gaps you need tinier bits, which make even smaller gaps. Before long you are getting REALLY tired of placing bits of paper.

But say you started with a bunch of identical paper triangles. Then you can fill the space completely with no gaps and no overlaps. That is the area of the paper in the tiles is equal to the area of the tabletop. I started thinking about 'as many as possible' years ago when I worked with tilings after reading Grunbaum and Shephard's amazing *Tilings and Patterns*. That book is basically an index of all possible 2d tilings and the patterns they make. We set up a situation with two constraints; identical bits, and as much coverage as possible And we find that there are surprisingly few shapes that can satisfy the constraints.

For instance, among regular geometric figures, only those with two, three, or four sides can form tiles.

MC Escher famously explored how to make very quirky tiles.

One of the things that are apparent about tilings is that you see a pattern when the state of "as much as possible" (AMAP) is realized.

Roger Penrose invented a couple of particularly interesting tiles derived from pentagons. These can tile a plane in two distinct sort of ways. One way is quite regular once you see how one part looks you can predict how other parts of the same tiling will look.

The other sort of tiling is aperiodic you can't predict exactly the positioning of tiles in the tiling until you actually build the tiling to that location. This is challenging enough to make an amusing passtime. So with Penrose Tiles we have two different ways of getting as many tiles as possible on the plane.

The interesting thing is that they are incompatible. The edges of the two tilings won't line up perfectly producing a crack.

(This picture shows the two sorts of penrose tilings and the crack.)

When I was first studying tilings I was a lithographer making original prints. You'd draw on an aluminum plate and make as many prints as you want, and then use a 'graining machine' to scour the image off the plate so it could be reused. This machine was essentially a big tray full of ball bearings that was shaken in a circular motion by a motor.

I used to love using that machine. I'd tilt the tray up so that all the bearings rolled to one end and put in the plate and let the bearings roll back over. They'd all be scattered at random at first, but soon a hexagonal pattern would emerge because that's the most efficient packing for circles on a plane.

The system would start off in a a chaotic state, but each shake pack more and more bearings onto the bottom layer until it was in a highly ordered state. A state that gets as many as possible bearings into the bottom layer.

The size of the bearings didn't go evenly into the size of the tray, so it was impossible for bearings to cover the whole surface. The resulting gaps would self organize into long cracks through large regions of bearings dancing a hexagonal pattern. And these cracks were shifting dynamically over time as space was found for more and more bearings below.

One perspective we can take on biology is that it's a system that includes as much of the material of the earth within itself as possible. It's regularly shaken, like the graining machine, and each shake gives the system the chance to change in a way that lets it incorporate even more material.

I've observed the same with fruit flies. They like to settle as closely together as they can under certain conditions. Given the constraints of the fly shape and AMAP they organize themselves into a characteristic pattern.

And of course honeycombs provide another example AMAP larvae in an area. We have seen how systems where the AMAP constraint works often self-organize into structures that show patterns. Are these patterns 'real' or are they just apparent to people? How would we decide?

Perhaps one way of thinking about this to see that the physical situation is very similar whether the pattern is there or not.The 'tiles' are still all physically present if the pattern is messed up. So if we could find some sort of physical effect that depends on the presence of some sort of pattern would that tempt us into saying that patterns are real and not just apparent.

What do you think?