IV.3.3 The Octahedron
The Octahedron has eight triangular faces, six vertices, and twelve edges. A regular Octahedron consists of eight equilateral triangles, with four of those triangles meeting at each vertex. It is in fact the only one of the regular polyhedra to have an even number of faces meeting at a single vertex. Various minerals have been found in the form of octahedral crystals, including diamond, alum and fluorite. Interestingly, there are two different vertex angles: at any vertex, the angle between adjacent edges is sixty degrees, whereas the angle between opposite edges is ninety degrees.
Figure IV.6: The Octahedron, eight triangular faces, six vertices, and twelve edges, and its the dual of the Hexahedron.
As we mentioned above, Plato suggested that the Octahedron corresponds to the element Air, since its minuscule components are so smooth that one can barely feel it. This goes well in accordance with the complex-geometry of time as we noted above, since the Octahedron is the dual of Hexahedron and they both form two mutually orthogonal arrows of time which are the folds of super-symmetry, as we explained in Chapter II.