Math movie: How to play a Rubik’s Cube like a piano

By Murray Bourne, 09 Dec 2015

We use group theory every day, without ever thinking about it. Here's a simple example.

The integers form a group under the operation of addition.

4 + − 4 = 0

In this case, 0 is the identity and − 4 is the inverse. This example also demonstrates closure (4 and − 4 are integers, and when we add them, we also get an integer, 0.)

Another property of a group is that it is associative. That is, if we have more than one operation, it doesn't matter what order we do it in. For example:

(4 + − 4) + 3 = 4 + (− 4 + 3) = 3

Similarly, the positive rational numbers form a group under multiplication. If we start with the number 7, we can find its inverse, 1/7 and the identity is 1. Thus we have:

1/7 xx 7 = 1

Similarly, each of 1/7, 7 and 1 are rational numbers, so the group is closed.

Many kinds of groups

Group theory covers many different areas, including permutations, matrices, transformations, topology and so on.

Applications include number theory, chemistry, physics, mechanics and music.

Group theory and music

Notes on a piano are periodic. That is, the 12 notes starting from the lower end of the piano are repeated 7 times to form the rest of the keyboard.

A, A#, B, C, C#, D, D#, E, F, F#, G, G#

A lot of music makes use of what is called the cycle of 5ths, where the chord structure follows notes which are spaced 5 notes apart, like this:

B E A D G C F A# D# G# C# F#

This forms a cyclic group structure which can also be periodic.

And this brings me to today's math movie. It's a TED-Ed offering by Michael Staff and outlines the connections between Rubik's Cube and music.

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