Rotation Groups

TL;DR

This paper explores two types of rotation groups generated by edge rotations of a tetrahedron, demonstrating that under certain conditions, their orbits are dense in three-dimensional space.

Contribution

It introduces and compares stationary and peripatetic rotation groups, proving density of orbits in space for skew axes and infinite order rotations.

Findings

Both orbits are dense in 3 under specified conditions.

The distinction between stationary and peripatetic groups is clarified.

Infinite order rotations lead to dense orbits in 3.

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 $\Re^3$.