Maths and music

Answers to Task 3b: combining the Symmetries of the Square

aaa;  aaaa;  aaaaa;  bb;  ea;  ae;  eb;  be

1. aaa means rotate through 90 degrees clockwise three times, which is equivalent to a rotation through 270 degrees clockwise (or a rotation through 90 degrees anticlockwise).
2. aaaa means rotate through 90 degrees clockwise four times, which is equivalent to a rotation through 360 degrees clockwise. This takes the square back to its starting position, and so gives the identity, e.
3. aaaaa means rotate through 90 degrees clockwise five times, which is equivalent to a rotation through 360 degrees clockwise, plus another 90 degrees clockwise. It is therefore the same as a.
4. bb means reflect in the diagonal axis from top left to bottom right twice, which takes the square back to its starting position, so gives e. (Any rotation repeated twice gives e: they are 'self-inverse', because they are their own inverses).
5. ea means do nothing, then rotate through 90 degrees clockwise, which is the same as just doing the rotation, and is therefore equivalent to a.
6. ae means rotate through 90 degrees clockwise, then do nothing, so is also equivalent to just a. (Any symmetry combined with e is equivalent to the symmetry alone. This property is what defines an identity.)
7. eb is equal to just b.
8. be is equal to just b also.

Show bc = a  and  dc = aa:

1. bc means first reflect the square in the diagonal axis from top left to bottom right, then reflect in the vertical axis. The effect of this can be seen in this diagram:



This is equivalent to rotating the square through 90 degrees, or a.

2. dc means reflect first in a horizontal axis, then in a vertical axis. The effect of this can be seen in this diagram:



This is equivalent to rotating the square through 180 degrees, or aa.

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