Rotation Groups

Abstract

A query, about the orbit P W in real 3-space of a point P under an isometry group W generated by edge rotations of a tetrahedron, leads to contrasting notions, W versus S, of "rotation group". The set R = {r A 1 , r A 2 } of rotations r A i about axes Ai generates two manifestations of an isometry group on ℜ 3 :(1). In the stationary group S := S(R), all axes B are fixed under a rotation r A about A.(2). In the peripatetic group W := W(R), each r A transforms every rotational axis B = A.Theorem. If the line A1 is skew to A2, if each r A i is of infinite order, and if P ∈ ℜ 3 , then both of the orbits P S and P W are dense in ℜ 3 .