tori found in 3D-space. While they are still 2D surfaces, they are embedded in the 3-sphere. The 3-sphere can be stereographically projected onto the whole Euclidean 3D-space, and these tori are then seen as the usual tori of revolution. It can be seen that a point specified by undergoing a rotation with the - and -planes invariant will remain on the torus specified by . The trajectory of a point can be written as a function of time as and stereographically projected onto its associated torus, as in the figures below. In these figures, the initial point is taken to be , i.e. on the Clifford torus. In Fig. 1, two simple rotation trajectories are shown in black, while a left and a right isoclinic trajectory is shown in red and blue respectively. In Fig. 2, a general rotation in which and is shown, while in Fig. 3, a general rotation in which and is shown. Generating 4D rotation matrices Four-dimensional rotations can be derived from Rodrigues' rotation formula and the Cayley formula. Let be a 4 × 4 skew-symmetric matrix. The skew-symmetric matrix can be uniquely decomposed as into two skew-symmetric matrices and satisfying the properties , and , where and are the eigenvalues of . Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices and by Rodrigues' rotation formula and the Cayley formula. Let be a 4 × 4 nonzero skew-symmetric matrix with the set of eigenvalues Then can be decomposed as where and are skew-symmetric matrices satisfying the properties Moreover, the skew-symmetric matrices and are uniquely obtained as and Then, is a rotation matrix in , which is generated by Rodrigues' rotation formula, with the set of eigenvalues Also, is a rotation matrix in , which is generated by Cayley's rotation formula, such that the set of eigenvalues of is, The generating rotation matrix can be classified with respect to the values and as follows: If and or vice versa, then the formulae generate simple rotations; If and are nonzero and , then the formulae generate double rotations; If and are nonzero and , then the formulae generate isoclinic rotations. See also Laplace–Runge–Lenz vector Lorentz group Orthogonal group Orthogonal matrix Plane of rotation Poincaré group Quaternions and spatial rotation Notes References Bibliography L. van Elfrinkhof: Eene eigenschap van de orthogonale substitutie van de vierde orde. Handelingen van het 6e Nederlandsch Natuurkundig en Geneeskundig Congres,