Charles Radin, Lorenzo Sadun
ROTATING RATIONALLY
(565K, plain tex)
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 $\pi/2$.