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Exploring Tangles: A Mathematical Activity with Moves and Patterns

Engage in a collaborative activity with four participants (A, B, C, and D) using two ropes to create and untangle intricate mathematical configurations known as "tangles." Follow specific moves: the twist (B swaps with C) and the turn (everyone passes their end clockwise) to develop a unique number represented by each tangle. Investigate how these operations can relate to rational numbers and the mysterious concept of infinity. This exercise, inspired by mathematician John Conway, challenges participants to create and untangle numbers while learning about deeper mathematical concepts through group interaction.

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Exploring Tangles: A Mathematical Activity with Moves and Patterns

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  1. www.carom-maths.co.uk Activity 2-4: Tangles

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

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

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

  5. 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. represents the number 0. 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.’

  6. 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?

  7. 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.

  8. There is much useful Tangles material on the Nrich site at the links below. http://nrich.maths.org/5776 Link 1 http://nrich.maths.org/5777 Link 2 Link 3 http://nrich.maths.org/5899 http://nrich.maths.org/5681 Link 4

  9. 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.

  10. 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...

  11. With thanks to: VinayKathotia John Conway Nrich. Carom is written by Jonny Griffiths, hello@jonny-griffiths.net

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