The 5 Platonic & 13 Archimedean Solids
Before we proceed with a 3D geometric explanation of the electron cloud, we need to be clear about the limitations of 3D space. It is an interesting fact that, just as 2D space is limited to only 2 types of regular 2D tessellation can fill a plain with just two colours, so there are only five regular solids. These ‘Platonic Solids’ have the characteristic of being formed from the same equal-sided shapes. Three of these, the Tetrahedron, Octahedron, and Icosahedron have triangular faces, whilst the Cube has square faces, and the Dodecahedron pentagonal.
From 5 Platonic Solids another set of semi-regular polyhedra, called the 13 Archimedean Solids can be derived. Aside from the Truncated Tetrahedron, the other 12 fall into two distinct categories. one based on the Octahedron and Cube, that exhibit octahedral symmetry, and another six derived from the Dodecahedron and Icosahedron, that exhibit icosahedral symmetry.
Three Geometric Processes
The 13 Archimedean Solids are produced from the 5 Platonic Solids through three geometric processes: truncation, explosion and twisting.
Truncation: Each side can be divided into two or three and the corner parts removed. Truncating a solid reduces its volume, i.e makes it smaller. Solids that can be created through truncation include, the Cuboctahedron, the Truncated Cube and the Truncated Octahedron, all of which are used in the construction of the Atomic geometry.
Explosion: By moving the faces of a solid away from its centre until the spaces between each face can be filled with other regular shapes we create an explodes solid. Examples of this include, The Rhomi-cuboctahedron, and the Great Rhombi-cuboctahedron, both of which appear in Atomic geometry.
Twisting: a square can transform into two triangular faces when it is twisted into a parallelogram. In this way the Rhombi-cuboctahedron can be transformed into a snub cube. Twisting differs from the other two processes as it is chiral, meaning it can occur in a left or right orientation.
Not only have we discovered that the sub-orbitals occupy spaces that can be defined by these polyhedra, we also suggest that it is these three geometric principles that are at work, transferring the energy of the electron instantaneously between shells, creating the quantisation effects found at the atomic scale.