Maths and Music

Go to Symmetries of the Square (including Tasks 3a and 3b)
Go to Group Theory
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Session 3: Dancing with Mathematics

The content of this session is drawn from Chapter 2 "Dancing with Mathematics" from Mathematics Galore! by Chris Budd and Chris Sangwin (published by Oxford University Press, 2001, ISBN no. 0 19 850770 4)

In the popular imagination, the mention of folk dancing conjures up many images. Brownies shuffling round an old church hall, while Brown Owl plays the piano and Tawny Owl sings encouragement. Morris dancers, decked out in flowers and ribbons and only slightly drunk, hitting each other on the head with large sticks. Mad Scotsmen, hurtling across the mountains, jumping on swords, and doing strange things with a haggis. This all seems rather remote from the maths we were doing in the earlier sessions. So why should mathematicians be at all interested in folk dancing? One possible explanation is that to the general public, mathematicians and folk dancers have one thing in common: they are eccentric, mad, and generally detached from reality!

But there is a much more significant connection between mathematics and folk dancing: both are concerned with patterns. To be precise we can think of a folk dance as the performance of a series of simple motions according to a set of rules. The purpose of these motions may seem strange at first but when combined together they can produce some wonderfully complex dances involving every girl or boy in the room. Indeed some of the best dances involve very complex patterns which marvellously simplify at the end of the dance so that you always end up with the same partner as the one you started with. If you watched such a dance from a bird's eye view, you would see each dancer trace out an intricate path carefully interwoven with those of the other dancers. There in a nutshell is the basis for nearly all of the well-known traditional English, Scottish, and American dances, such as Lucky Seven, Dashing White Sergeant, Nottingham Swing, Dorset Four Hand Reel, most square dances, and, of course, the Hokey-Cokey (always a good dance to end an evening with, even if the mathematics behind it is a little simplistic).

This demonstrates the real power of good mathematics, that is the ability to transfer ideas rapidly from one area to a quite different one.

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Symmetries of the square

You are going to start by investigating the symmetries of the square.

Task 3a

Have a go at Task 3a
Answers to Task 3a

To save having to keep drawing pictures, we need a notation to describe these. Let's make a the rotation through 90 degrees clockwise, b the reflection in the diagonal from top left to bottom right, and c the reflection about the vertical axis. e is usually used for the symmetry where you do nothing (it’s known as the identity).

Symbol	Operation	Before	After
a	Rotation through 90 degrees clockwise about centre
b	Reflection in diagonal from top left to bottom right
c	Reflection in the vertical axis
e	Do nothing, or get back to where you started (identity)

Now let’s see what happens when we combine symmetries.

Discussion

• Try doing a followed by b - we'll call that ab.
• Now try doing it the other way round, that is, b followed by a, or ba.
• What do you notice?
• Take another pair of symmetries, and try combining them, changing which one goes first.
• What do you notice?
• Which symmetries does the order not matter for?

For ab, you should get the reflection about the horizontal axis (we could call this d).

Then when you do ba, you should find that this gives you c, the reflection in the vertical axis.

So unlike adding and multiplying numbers, it does matter which way round you combine symmetries. Depending on what other combination of symmetries you chose to try out, you may find it made a difference, which order you did them, or it may not. If you took a pair of rotations, for instance, it won't make a difference.

Task 3b

Now have a go at Task 3b.
Answers to Task 3b

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Group Theory

The set (or collection) of all the symmetries of a square is known as a group. A group is a set of operations which satisfies the following four properties:

1. The identity, e, is a member of the set.
2. When any two members of the set are combined, you get another member of the set.
3. If a is any member of the set, then so is the inverse of a (the element which reverses what a does, eg. rotation through 90 degrees clockwise is reversed by rotation through 90 degrees anticlockwise - or we could call that rotation through 270 degrees clockwise).
4. If you combine any three elements in the order a then b then c, you get the same result if you combine them in the order b then c then a.

Group theory is not something you normally meet at school. Usually people meet it for the first time when they study maths at university. However it is a very beautiful part of mathematics, and is also used by physicists studying elementary particles, and chemists studying the structure of crystals. It is also used by people who enjoy folk dancing, bell ringing, and knitting patterned jumpers (although they might not know it!).

A lot of the early work in group theory was done by a French mathematician named Evariste Galois (1811-1832). If you look at his dates, you will see that he only lived to the age of 20. He didn’t have a particularly good mathematical career at school, as he did not do the work people asked him to do, and his teachers regarded him as mediocre, or at best, eccentric. He knew he had it in him to do original mathematical research however. But when he submitted papers on what became the new topic of group theory, people lost them or did not understand what he meant. Then he got involved in a duel over a woman. The night before, he wrote a letter to his friends, in which he put down everything he could about group theory. This was just as well, as he was killed as a result of injuries he suffered in the duel. His work was soon seen for what it was: a very significant contribution to mathematics.

Additional notes on group theory 
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Back to dancing

You are now going to translate the abstract ideas of group theory into dancing! When we looked at the symmetries of the square, we identified the corners of the square with the letters A, B, C and D. Imagine instead you have four dancers called Andrew, Bryony, Chris and Daphne. They don’t have to stand in a square, but instead we will think of them as standing in a line:

A  B  C  D

Now remind yourself what the symmetry of the square we labelled b was:

  which becomes

This was the reflection in the diagonal from the top left to the bottom right. If our dancers change their positions like this:

A  B  C  D       A  C  B  D

that is the same thing – B and C swap places, A and D stay where they are. We can call this dance step an 'inner twiddle', and it is important to note that, mathematically, it is exactly the same as the reflection of the square, b, even if it doesn't look quite the same. When you are dancing, you can hold hands or brush shoulders – different dances do different things.

Task 3c

Now have a go at Task 3c.
Answers to Task 3c

This set of moves is called a ‘Hay’ in English folk dancing, a ‘Foursome Reel’ in Scottish dancing, and a ‘Reel of four’ in square dancing, but the mathematics is just the same.

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