Hi Vladimir
I like your Beautiful Universe. It is like a Bohr model of space. I think you might be interested in quaternions, which are mathematical extensions of complex numbers. They are associated with spin in quantum mechanics.
A quaternion can be thought of as four numbers, any one of which may be taken as a reference for the other three. This conceptual link between the numbers leads to a picture of the quaternion as a tetrahedron.
Your face-centered cube model first struck me as odd since I might have expected simple hexagonal shapes, or the occupation of alternate nodes of a regular grid. The first possibility arose mainly from other investigations, and the second possibility was based on the three dimensional model of a quaternion I had been considering which places the tetrahedron representing the quaternion on four corners of a cube.
But then on reflection it became clear that the face-centered cube has alternate nodes occupied, and so could very well be related to a 3D structure for a model quaternion. In the attached diagram, the quaternion is shown in a 1/8 segment of a face-centered cube, ie 1/2 x 1/2 x 1/2 of a FCC.
Mathematically these cubes are assumed to be cyclic like circles or toruses, but that could also correspond to a grid of identical cubes in which case nodes could be placed on other cubes leading to different geometrical configurations of a quaternion, including the condition where the nodes are in the same plane in which case the quaternion bounds no volume, only area.
If you are interested, download John Baez The Octonions to see where I get this idea, which I readily admit could invite criticism as a gross misinterpretation of the mathematics as physics. First the octonions: they are an extension of the quaternions consisting of eight numbers, double again the size of a quaternion. As Baez puts it, "the octonions are the crazy old uncle nobody lets out of the attic".
Ignore the math. There are three diagrams on page 7. At the top is a circle with three nodes labeled i, j, k. These nodes correspond to three numbers of a quaternion with the reference number being hidden.
The second diagram on the page shows a triangular transformation table for the octonion (again not showing the reference number) with an inscribed circle with three nodes e1, e2, e4 on its circumference. I take these to refer to the nodes i, j, k above.
The third diagram on the page shows a mapping of the eight numbers in an octonion onto the eight corners of a cube. The reference number is now included and labeled with the number '1'. Notice that the reference number and e1, e2, and e4 connect diagonally across the faces of the cube, and occupy alternating nodes. I think that this could be a picture of a quaternion in space, but I doubt if that was intended.
I would have overlooked this possibility without the diagrams. You have obviously realized the importance of graphics in your essay and website.
Quaternions and octonions are the basis of modern mathematical physics and I would not be surprised to find them in your model.
Best wishes and I hope your essay does well.
ColinAttachment #1: quaternion3D.pdf