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Avalanches and sandpiles

Engineers solve problems in the real world using mathematics and modelling. "Modelling" means looking at a real world phenomenon, and using mathematics to try to say what is happening, and to make predictions which can then be tested against the real events. The type of problems we are looking at here are known as self-organising critical state problems.

But what does this mean? Many problems engineers study have straightforward answers, like how much force will it take to break a chair. You can test a chair, and find numbers to give you the answer you want. This type of problem is known and predictable. There are also problems which are neither known nor predictable - for instance, we know there are cosmic rays in space that penetrate the earth's atmosphere. They might zap a satellite so it no longer works properly. However we don't know when this might happen and we have no way of predicting it.

There is a third class of problems which comes in between these. These are 'sort of known', and 'sort of predictable', and this is the type of problem we are considering here. We can't tell when any one thing will happen, and we can't work out any one solution, but we can work out general patterns. These phenomena have big events every so often, and little events more often. We don't know exactly when something will occur or how big it will be, but we can work out roughly how often we might expect an event of a particular size.

Avalanches are an example of these self-organising critical systems. However they don't always do what you might expect. Snow falling on a quite gentle slope may cause an avalanche, where snow falling on a much steeper slope stays where it is. So we can see there is a lot for us to find out about avalanches if we are to prevent disasters. Another example is lightning. Lightning is unpredictable also. But in both cases we can work out the frequency of big and small events even though we cannot know exactly when to expect an event.

Imagine a single flake of snow falling on a pile of snow on a hillside. What happens? Well, the snow might hit the pile causing the whole lot to fall down the hillside. This is an avalanche. Or the snowflake might stick to the pile without anything else happening, or it might stick to the pile, and the pile roll slowly down the hill.

But let's suppose this snowflake causes ten flakes to fall. Each of them might also cause ten flakes to fall, so a hundred will fall. What happens if they all cause ten flakes to fall, and they all cause ten flakes to fall, and they all cause ... Well, I am sure you can imagine the result! Eventually we would have a big, dramatic avalanche.

We can see what happens in a small avalanche by modelling it with rice. If you pour rice a bit at a time through a funnel (so that you control the rate at which the rice falls), a small heap will form. As you add more rice, the heap will get bigger. Notice the angle the side of the heap forms with the horizontal surface you are pouring it onto. This is known as the angle of repose (repose meaning rest), and it is the angle the rice likes to form with the horizontal. You can experiment to see if it is always the same.

As you add more rice, the pile becomes pointed at the top. Then adding one or two more grains will be enough to cause a collapse - try it for yourself, if you haven't already. You will see that there are three things that might happen: nothing much (the rice just sticks to the pile), a few grains might roll down the slope, or there might be a big fall (an avalanche). We can observe small events - just a few grains -medium events and large events - the whole side of the pile of rice slips. Small events are note very interesting. Large events are very dramatic, and model what happens when snow engulfs a whole village.

Why are these known as self-organising critical states?

Well, they are self-organising because no one tells the rice or the snow what to do. The rice decides what sort of pile to form, the angle with the horizontal and the shape, just as snow on a hill side does. There is no external organiser telling it what to do. All the patterns we observe come from the system itself. This is different from looking at a block of flats, and working out how it stays up. If an engineer wants to know how a block of flats stays up, she or he will have to look at the plans or ask the person who designed it. It might be that the outside walls are holding the block up, or there might be a concrete bit up the middle. Whatever it is, someone outside the block of flats planned it to be that way. In a self-organising system, however, we can see exactly what is going on, and no one is involved in doing any planning or designing of the system.

A critical state means that something dramatic is about to happen all the time. The pile of rice or snow is critical because it is always just about to collapse. So a self- organising critical state means something which constantly reorganises itself so that something dramatic is just about to happen.

But where's the maths?

By now, you might be wondering what this has to do with maths, however. Engineers need to know what will happen if they want to build railways or put up a TV transmitter in an area where there is a lot of snow, because they don't want them to swept away by an avalanche. The engineer has to know how often to expect an avalanche, what conditions cause an avalanche, and how you might try to stop an avalanche from starting in the first place.

Engineers have to ask mathematicians or be mathematicians to find out the answers to these questions. Mathematicians create models which they hope correspond reasonably well with the real thing.  But a model is much easier to study and to use to make predictions than reality.

This is what you will all do in the project work linked to this page. The projects appear quite different, but they are all self-organising critical systems, so you will find similar things from them all.

A human self-organising critical system

Try this short activity in which you all become the grains of rice or flakes of snow in a self-organising critical system.

Choose two people to be observers. They don't take part, but watch what is happening, and report back at the end. Everyone else chooses two others without saying who they are. Then everyone walks around aimlessly.

When the teacher says "Stop!" everyone tries to get into an equilateral triangle with the other two people they have chosen. This might happen quickly, or it might take quite a long time. Any one person moving can cause almost everyone else to move sometimes.

The main effects you might notice are these. There is a lot of movement at the beginning, then it gets quieter. Just when you think that it's all over, someone takes a small step and the whole thing becomes unstable again. This is a simple example of a self-organising critical system. It is self-organising because no one else is telling you where to go - each of you decides for yourself where to go. It is critical, because the position of every single person is important. If one person moves, somebody else must move unless you have someone who has not been chosen by anyone else.

Think about what it's like trying to form the triangles. What difference would it make if there was a wall in the way? A wall will cut off some positions, so some people might have to move much further. This will then have a knock-on effects on others.

Another self-organising critical system

In the Elasticated Cups project, you join disposable cups together with thin elastic, or elastic bands knotted together. If you put some rice in the cups, they will be more stable. Join four or five cups in a row, then move one of the end cups in small steps, maybe 1 or 2 cm at a time and see what happens. To start with, the only cup that moves will be the end cup you are moving. This is like what happens when the snow sticks on the side of the hill. Then you might get a small amount of movement in the next cup. If you look at the elastic bands you will see that they are getting quite stretched. The cups still aren't moving much, but there is a lot of energy stored in the bands. If you carry on, soon all the cups will move. This is like an avalanche.

You can do this with 10, 20, 30 ... cups, or you could arrange the cups in a square. If you do this, you will see strange and wonderful things happen. You can even arrange the cups in equilateral triangles, like the Triangle Game. As you see, all the projects are basically about the same thing - but using different physical systems.

Mathematical modelling

To model what is happening in your project, you must be able to describe it. Maths is about making accurate models of the world. It isn't enough just to say "Oh, that was fun!" or "That was less fun." or "Something happened." or "Not very much happened." You need to be able to make this kind of thing quantitative - to be able to put quantities on what happens. The very simplest way to do this is to call events small, medium and large.

The first thing you need to do is to decide what are small, medium and large events. Think very carefully about what each of these mean in the context of your project. The next thing you need to do, is to see how often these events happen. Then graph your results, say, for 50 movements or spoonfuls of peas. See how many result in a small, medium or large event.

This is the beginning. You're then on your way to being able to measure a self-organising critical system. Nobody else is doing exactly your experiment. Your results will be completely unpredictable and different from everyone else's. You are doing genuine research, finding results which no one in the world could duplicate or predict. A single flake of snow has an effect which would not have happened without that flake of snow. If you move in the Triangle Game, or add a pea to your pile, or pull you cup a few centimetres, whatever happens is completely unpredictable and would not have happened if you hadn't done that. So this is all your own.

Once you've begun to get an idea of how often different events happen, the really exciting thing is to see how you can change things. Each of the project notes gives you ideas on how to do this, but you will have ideas of your own by that stage.

Developing the models

One technique engineers use is called pinning. In the Triangle Game suppose some of you couldn't move, or had your shoe laces tied together, so you could only move very slowly. What effect do you think this would have? Most people think it would make the whole thing much quicker. But would it really? Those people moving into position might have helped others move into position. How many people have to stand still to have a really big effect on the whole group?

The next thing you might do is to have a few people deliberately disrupt things. Suppose some people were told "Always take a step in the wrong direction." What would that do?

Similarly, you can change the behaviour of the rice. Suppose you added some peas. Peas are round and smoothish - what effect would that have? Suppose you added some raisins? Raisins are sticky and "glue" to each other, so they're not as keen to fall down as the rice. But does this mean you'll get less happening, or more happening? If they're not as keen to fall down, then it's harder to get an avalanche started, but if lots and lots are piled up, sticking together, maybe the avalanches will be bigger when they do happen. 

What happens if some of the cups are easier to move than others? More rice in a cup means it's harder for that cup to move. What happens if some are empty, or if some are more full than others? If you make some areas on the table sitcky, so that some cups occasionally get stuck for a bit, what effect does that have?

In the Triangle Game you can do something similar - you could mark areas on the floor where you have to move very slowly.

Planning and testing your own mathematical model

There are three stages you should go through in your project work.

First you should decide how you're going to describe the system you're looking at. You will need to discuss this - everyone's opinion is important, there is no single right answer to this. This is real maths which in the real world mathematicians and engineers care about and argue about all the time. Your results are just as interesting and just as good as those of people who do this for a living.

Secondly, you will need to describe what your system is doing. Be able to tell other people what happens in a way everyone can understand and relate to. 

Thirdly, don't stop too soon! Think of ways to modify what you've done so far. Try to find out ways of stopping avalanches happening, or ways of making the avalanches bigger or more frequent or more exciting. Do whatever you want, but try to make it different.

You might like to investigate how these things are done in real life. Do an internet search to find out how real avalanches are stopped. See what sorts of things engineers actually do - find out if they're the similar to the things you suggested. What they do is not always at all obvious. Just holding the snow in the right place is not necessarily the right thing to do, just as holding some of the cups in one place is not necessarily the right thing to do.