On the former page You can already see illustrations of different aspects of Close packing of spheres and their relation to the creating universe.
This page will deepen a little more the understanding of the basic concept.
There is actually only one single platonic solid,
The Tetrahedron is the basis for the cube and the octahedron, which are two tetra compounds. and for the Icosa- and Dodedecahedron as five cube - ten tetra compounds.
The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size.
adding those same tetrahedra forms the stella octangula, which re ensembles the cube.
The vertices of a cube can be found in any regular geometric close packing, whereby simple cubic refers to packings where only the vertex positions are filled.
simple cubic packing of spheres touching at six points fills a maximum of 52 % of space.
Centered cubic close packing of spheres is the geometry of the space-motion-time compound as described before.
The difference to the simple cubic packing is that here, at the centre of the cube, an additional sphere is added.
Instead of six contact points for the simple cubic packing, we now achieve eight.
The result is a lattice of flattened bi-pyramids as shown in the image to the left.
Six of these bi-pyramids join together to form the cubic tessellating Catalan solid called Rhombic Dodecahedron.
This geometry is a stellation between the cubic lattice of simple packing and the octahedral lattice formed by the cube's centering spheres.
The most compact regular form of cubic packing is face centered packing.
Instead of the centre, it has lattice points on the faces of the cube. This constellation results in a 3D lattice formed by two different volumes and two different geometries:
A grid formed of neighboring Tetrahedra and Octahedra which are non tessellating geometries on their own, but in a combination of two tetrahedra per one octahedron they combine to form Trigonal Trapezohedra, which are in fact tessellating geometries.
The image to the left shows the tetra-octa lattice.
Body-centered packing allows the spheres to touch at fourteen contact point.
Basic geometries of the four alchemical elements from the three cubic packing principles.
The Four essential solids that can tessellate (or "pack") 3-dimensional space:
- The Cube
- Trigonal Trapezohedron (one octa- plus two tetrahedra)
- Rhombic Dodecahedron
- Truncated Octahedron
Body centered cubic packing - Body centered cubic packing - Face centered cubic packing
Rhombic Dodecahedra Truncated Octahedra Vector Equilibrium
(as trigonal trapezohedra compound)
The image shows face centered cubic packing principles and how tessellation is achieved by combining octahedra and tetrahedra, and how the cub octahedon or Vector Equilibrium relates to the
Trigonal Trapezohedra grid.
The proportion is Two tetra- for any octahedron, but in volume the relationship is the opposite, because the two tetrahedra combined have only about half the volume of the octahedron.
In consequence, despite the cub octa having equilibrium in reference to traversing vectors, it lacks equilibrium in vertex distance. It has either square or triangular faces.
The Trigonal Trapezohedra grid forms Star-Tetrahedra geometry inside every single cube of the space filling lattice.
As mentioned before, all tessellation is based on cubes.
The tetrahedra from the Trapezohedra lattice basically accentuate all the diagonals inside of the wave field (as shown in yellow).
The Octahedra, on the other hand, accentuate the three dimensional planes of the cube (as shown in red color).
concave (yellow) and concave lens projection convex lens projection
convex (turquoise) lenses
Projection and Reflexion in the Cube-Sphere
as naturally occurring in "Bubbelogy"
(expanding spheres in body centred close packing)
1. refraction on six concave lenses projects and expands spheres into planes.
2. mirrored spheres of neighboring cubes are becoming projected towards centre of the initial cube, forming three dimensional planes at 90º angle.
3. mirror effect caused by the two intersecting concave lens projections.
4. convex lenses form by means of sphere intersection at eight faces of the cube's inscribed truncated octahedron. Convex lenses project the centring sphere toward the cube's vertices.
5. six concave lense-projections at the cube's faces and eight convex projections through diagonals.
6. along the seven symmetry axes 20 projections take place. Here the twelve concave- plus eight convex projections.
Please refer to my album "The Cube, the Sphere ..." to find out more about what's in a cube ...
Schematic representation of the reflection-projection sequence relative to one pair of concave lenses (out of the total of three pairs), forming one single dimensional plane, horizontally centering the sphere.
Centering planes in cube-spheres are magnetic South.
The centering sun or nucleus is the magnetic North pole.
The four pairs of convex lenses are the position of magnetic East and West.
Inverse diagonal and edge length proportion of the regular Octahedron, coherent in face centered packing,
(in combination with tetrahedra in relation 2:1. Forming the Trigonal Trapezohedron ),
and the square Bipyramyd as repetitive tesselating cell for body centered packing.
( six bipyramids forming the rhombic dodecahedron ).