Activity 2-4: Tangles

You need four people for this, A, B, C and D, and two ropes, starting like this:

You are allowed to make two moves. Move 1: B swaps with C, with C’s end going under B’s (a TWIST).

Move 2: everyone passes their end one person clockwise (a TURN). A Move 1 followed by a Move 2 looks like this:

As you mix up a number of Moves 1 and 2, you can develop quite a tangle in the middle. There is a way to interpret what is going on. Each tangle represents a number, and Moves 1 and 2 give a new number from the old. Move 1 represents ‘Add 1 to your number’, while Move 2 represents ‘Take -1 over your number.’ represents the number 0.

Task: using these rules, create the tangle representing 2/5. is one possible path. Task: half of you create a tangle-number, then pass it to the other half to untangle. Can they say what your number was? Note: what tangle represents infinity?

Task: given a rational number p/q, can you give an algorithm for creating it? Tangles were the idea of John Conway, English, (1937- ) a mathematician of great originality who has spent much of his working life at Cambridge and Princeton.

There is much useful Tangles material on the Nrich site at the links below Link 1 Link 2 Link 3 Link 4

There is an object called the modular group that is very important in advanced mathematics. Take the set of 2 x 2 matrices with a, b, c, d integers so that ad - bc = 1 (the determinant is 1). This set together with matrix multiplication forms the modular group.

It can be shown that the modular group can be generated by two transformations: These are exactly the transformations we have met in our tangle exercise. Coincidence? Who knows...

Carom is written by Jonny Griffiths, With thanks to: Vinay Kathotia John Conway Nrich.