97-310 Charles Radin, Lorenzo Sadun

ROTATING RATIONALLY (565K, plain tex) Jun 3, 97

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Abstract. In 3-dimensional Euclidean space, let $A$ be a rotation by $2 \pi/p$ about a fixed axis, and let $B$ be a rotation by $2 \pi/q$ about a second axis that makes an angle of $2 \pi n/m$ with the first, where $p,q,n$ and $m$ are arbitrary positive integers. For each value of $n,m,p,q$, we find a presentation of the group $G_{n/m}(p,q)$ generated by $A$ and $B$. We show that there are never any unexpected relations between $A$ and $B$. Rather, all the relations between $A$ and $B$ are direct consequences of three simple identities involving rotations by $\pi$ and $\pi/2$.

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