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# Simplifying Transformations

This problem follows on from Combining Transformations .

We shall concentrate on the transformations $I$, $R$ and $S$ in this question. Here they are again:

We can combine $R$, $S$, $R^{-1}$ and $S^{-1}$ in lots of different ways.

$S S R S R^{-1} S R S R^{-1}$ and $ S^{-1} R R S R S R R^{-1} S R^{-1}$ are two examples of transformations obtained like this.

In fact, there are infinitely many ways to combine them.

How many different transformations can you find made up from combinations of $R$, $S$ and their inverses?

(We shall count two transformations as the same if they have the same effect on all starting shapes.)

Can you be sure that you have found them all?

Convince yourself that $R S = S^3 R$.

In Combining Transformations you found simpler expressions for powers of $R$ and $S$.

Use these and $R S = S^3 R$ to simplify $S S R S R^{-1} S R S R^{-1}$ and $S^{-1}R R S R S R R^{-1} S R^{-1}$.

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Age 11 to 14

Challenge Level

This problem follows on from Combining Transformations .

We shall concentrate on the transformations $I$, $R$ and $S$ in this question. Here they are again:

We can combine $R$, $S$, $R^{-1}$ and $S^{-1}$ in lots of different ways.

$S S R S R^{-1} S R S R^{-1}$ and $ S^{-1} R R S R S R R^{-1} S R^{-1}$ are two examples of transformations obtained like this.

In fact, there are infinitely many ways to combine them.

How many different transformations can you find made up from combinations of $R$, $S$ and their inverses?

(We shall count two transformations as the same if they have the same effect on all starting shapes.)

Can you be sure that you have found them all?

Convince yourself that $R S = S^3 R$.

In Combining Transformations you found simpler expressions for powers of $R$ and $S$.

Use these and $R S = S^3 R$ to simplify $S S R S R^{-1} S R S R^{-1}$ and $S^{-1}R R S R S R R^{-1} S R^{-1}$.

You might like to look at the article Grouping
Transformations , which explains some of the mathematics behind
these problems.

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?