Quaternions and conjugate system of tetrahedra (Q1803514)

Let \(| A|=A_ 0A_ 1A_ 2A_ 3\) be a tetrahedron in a three dimensional projective space. Given a point \(P\) of the tetrahedron not on the faces of \(| A|\) the lines \(A_ iP\) meet the face of \(| A|\) opposite \(A_ i\) at the point \(T_ i\). The two tetrahedra \(| A|\) and \(| T|\) are called perspective with \(P\) as the center of perspectivity. Two tetrahedra \(| B|\) and \(| C|\) are called conjugate with respect to \(| A|\) if the polars of the vertices of one with respect to \(| A|\) form the faces of the other one and vice versa. The authors show that for a given point \(P\) (not on the faces of \(| A|)\) a conjugate pair of tetrahedra with \(P\) as a vertex of one of them can be obtained using quaternions. Thus further properties of such systems of tetrahedra can be studied.