Rotation Groups

Abstract

A query, about the orbit $P{\cal W}$ in real 3-space of a point $P$ under an isometry group ${\cal W}$ generated by edge rotations of a tetrahedron, leads to contrasting notions, ${\cal W}$ versus ${\cal S}$, of "rotation group". The set R $=\{r_{{\sf A}_1},r_{{\sf A}_2}\}$ of rotations $r_{{\sf A} _i}$ about axes ${\sf A}_i$ generates two manifestations of an isometry group on $\Re^3$: (1). In the {\em stationary} group ${\cal S:=S}$(R), all axes {\sf B} are fixed under a rotation $r_{\sf A}$ about {\sf A}. (2). In the {\em peripatetic} group ${\cal W:=W}$(R), each $r_{\sf A}$ transforms every rotational axis ${\sf B\not=A}$. {\bf Theorem.} \ If the line ${\sf A}_1$ is skew to ${\sf A}_2$, if each $r_{{\sf A}_i}$ is of infinite order, and if $P\in\Re^3$, then both of the orbits $P{\cal S}$ and $P{\cal W}$ are dense in $\Re^3$.