ABSTRACT
To study quaternions, the authors can use two orthogonal planes in 4-dimensions to visualize simultaneously all 4-dimensions. One key fact about the pair of planes representation for quaternions is that the plane in 4-dimensions containing the vector corresponding to the origin O in 3-dimensions is special: algebraically this plane is isomorphic to the complex plane. The corresponding fundamental fact about the four planes representation for dual quaternions is that the plane in 8-dimensions containing the vector corresponding to the origin O in 3-dimensions is also special: algebraically this plane is isomorphic to the plane of dual real numbers. The authors can use this model of four mutually orthogonal planes to visualize two important effects in 8-dimensions: the effect of sandwiching with the isometries that represent rotations.
