Rotation Groups

Abstract

A query, about the orbit $P\mathcal{W}$ in real 3-space of a point $P$ under an isometry group $\mathcal{W}$ generated by edge rotations of a tetrahedron, leads to contrasting notions, $\mathcal{W}$ versus $\mathcal{S}$, of "rotation group". The set R $=\{r_{{\mathsf{A}}_1},r_{{\mathsf{A}}_2}\}$ of rotations $r_{{\mathsf{A}}_i}$ about axes ${\mathsf{A}}_i$ generates two manifestations of an isometry group on ${\Re }^3$: (1). In the {\em stationary} group $\mathcal{S}:=\mathcal{S}$(R), all axes {\sf B} are fixed under a rotation $r_{\mathsf{A}}$ about {\sf A}. (2). In the {\em peripatetic} group $\mathcal{W}:=\mathcal{W}$(R), each $r_{\mathsf{A}}$ transforms every rotational axis $\mathsf{B}\ne \mathsf{A}$. {\bf Theorem.} \ If the line ${\mathsf{A}}_1$ is skew to ${\mathsf{A}}_2$, if each $r_{{\mathsf{A}}_i}$ is of infinite order, and if $P\in {\Re }^3$, then both of the orbits $P\mathcal{S}$ and $P\mathcal{W}$ are dense in ${\Re }^3$.

6 pages