Two electrons define a line, three a plane. This plane may have any orientation in space. Therefore, to define three-dimensional isotropic space, four electrons are required. Since under the supposition that the nucleus, consisting of different kinds of particles, does not contribute to the definition of three-dimensional isotropic space, in case of the atomic shell electrons also at least four electrons are required for granting highest symmetry. In the state of highest symmetry, the electrons are arranged to form a tetrahedron. Each two of the four electrons at the corners of the tetrahedron may be viewed as having opposite spin and sitting on one of two orthogonal edges. Thus, in addition to the spin quantum number s, for full classification, a new quantum number, which shall be called the topical quantum number c , is required. A visualization of a pair of electron pairs with orthogonal spin axis is presented in Electron's 3D space and spin (Java applet).
Since the tetrahedron is a self related Platonic body and the Platonic spheres are of the same symmetry as the Platonic bodies, the electronic charges may be considered to be spread out evenly over the surface of the Tetrahedral Sphere, where each electron covers one of the four spherical triangles. By this tranformation, the role of the topical quantum number is not changed.If spin and topical quantum number cause the fourfold degeneracy of the lowest state in the PSE, one might infer, that this degeneracy will extend to the higher states and cause the Occupation of States accordingly.
In summary, the fourfold degeneracy appears to be a consequence of the inherent degree of freedom a particle with spin experiences in isotropic space. The resulting Double Shell Periodic System images directly the sequence of occupation of electronic states.