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We consider the geometrical figure, Metatron's Cube. How is this figure derived?
Starting with a Cube, we flatten it along one of its diagonals. (We're talking about diagonals through the centre, not ones that are across the faces.)
We obtain the following:
We join up some of the vertices, adding new lines:
Now, we add in some guide-points. These won't be in the finished figure, but they allow us to divide each of the outermost sides in accordance with the golden ratio (click here for the "Further Cube Geometry in Nature" article, wherein is explained the golden mean and its Cubic basis).
We take the newly created points, and connect them as follows, adding also connecting lines to previously established intersections on the figure:
And, removing the guide-points, we draw some final lines in the centre.
Now, from the completed Metatron's Cube, we can derive two tetrahedrons:
As we can see, it reflects the relationship between the tetrahedron and the cube.
We also have the other four platonic solids: cube, octahedron, dodecahedron and icosahedron:
A platonic solid is defined as a polyhedron having faces that are all identical regular polygons. All of its faces are equal-sided polygons. In addition, we may require that its vertices all be the same distance from the centre—or, that with faces substituted for vertices and vice versa, the solid produced would also be a platonic solid. This type of manipulation, substituting faces for vertices, is known as creating the "dual" of the polyhedron.
So the tetrahedron has 4 faces and 4 vertices. The two tetrahedra are each other's duals. The Cube, however, has 6 faces and 8 vertices, while the octahedron has 8 faces and 6 vertices. The Cube and octahedron are mutual duals. The dodecahedron, however, has 12 faces and 20 vertices, while the icosahedron has 20 faces and 12 vertices. The dodecahedron and icosahedron are mutual duals.
It is observable that many naturally occurring crystals have structures conforming to particular platonic solids. Thus, although they are not overtly Cube-shaped, we nonetheless infer a Cubic influence.