Troublesome Triangles

This project was originally part of the activities for videoconferences led by Dr Ian Johnston of the Open University's Technology Faculty.  Ian is a practising engineer, and devised this game as a way of helping students to understand how certain physical systems work.  These are known as self-organising critical systems.  Such systems are self-organising, because they do not require an outside agency to make things happen - for example, avalanches are self-organising systems, no one tells the snow when to fall.  They are critical because something dramatic is likely to happen at any moment as the system re-organises itself, trying to achieve stability.

Why do this project?

Doing this project, particularly if you spend some time in discussion with the students, is a great way to introduce them to mathematical modelling.  It's also great fun!

If you are not an engineer ...

Engineers work in the real world, trying to solve problems by understanding more about the properties of complex systems.  Mathematical modelling is the process by which they do this.  Reality is often too complicated to be analysed directly, so mathematicians and engineers create models which they hope will be straight-forward enough for analysis, but will have enough of the properties of the real system to give useful predictions which can then be tested on the real system.
This project provides students with a mathematical model to investigate.  It models real world systems like avalanches.  Snow accumulates on a slope and at a certain point, the snow falls off the slope.  Sometimes it falls off slopes that don't appear to be very steep, while it stays on slopes which are much steeper.
Modelling avalanches is a way to understand what is going on, so that engineers can find ways to prevent them or at least ensure that their effects are less devastating than they might otherwise have been.  It is easy to see how a pile or sand or rice provides a model for an avalanche.  The Triangle Game is a human model in which each student is equivalent to a single flake of snow or crystal of sand or grain of rice.  
Walking round aimlessly is just a way of producing a random collection of 'flakes of snow' waiting to fall.  When the instruction is given to make triangles, this models the flakes of snow falling onto the slope, and settling into a stable position.  For some it happens easily, others will jostle about, moving others which were previously in a stable position, but now are not.  
The main effects to notice are:

• initially there is a lot of movement, which eventually begins to settle down
• it may happen that just as everyone thinks equilibrium has been reached, one person moves and everyone else is displaced

This is self-organising because each person moves independently of the others, with no overall control.  It is critical because one person moving can cause everyone else to move - an avalanche.
The background to this project.

Key questions:

• How quickly did the game settle down?  Did it always take the same number of moves to settle down?
• What makes it easy/difficult to make triangles?
• What difference does it make if there is a wall in the way?
• How do these things relate to a physical system like an avalanche?