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Simplifying Transformations

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Jannis Ahlers (Long Bay Primary) found 8 transformations:
"The answer is 8. I found this by finding all the possible positions the shape could end in by only using R, S and there inverses."

The 8 possible transformations are:

$I$, $S$, $S^2$, $S^3=S^{-1}$, $R$, $R S=S^{-1}R$, $R S^2=S^2R$, $ R S^3=S R$.

There are eight transformations made up only of $R$, $S$ and their inverses. Neat way to see this: draw the eight that you think exist, then note that applying $R$ or $S$ to any of them gives another of them, so we can't `escape' from these eight. The simplest expressions for the eight are:

$I$, $S$, $S^2$, $S^3=S^{-1}$, $R$, $R S=S^{-1}R$, $R S^2=S^2R$, $ R S^3=S R$.

Notice that $R S R^{-1}=S^{-1}$. (Of course, $R^{-1}=R$, so $R S R=S^{-1}$, and this can also be written as $S R=R S^{-1}$.)

So the two expressions simplified are:

$S S R S R^{-1} S R S R^{-1} = S S(R S R^{-1})S(R S R^{-1})= S S S^{-1}S S^{-1} = S$

and

$S^{-1}R R S R S R R^{-1} S R^{-1} = S^{-1}(R R)S R S(R R^{-1})S R^{-1} = S^{-1} S R S S R^{-1}=(S^{-1}S)R S S R^{-1}=R S(S R)=R S R S^{-1}= (R S R)S^{-1}=S^{-1}S^{-1}=S^{-2}=S^2$.