Crystallization in the Geocubic Model

by Tom Gilmore
Copyright 2017
All graphics by Tom Gilmore
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Part I - Cubic Coordinates

Cartesian Quantum Coordinates

Einstein’s Relativity deals with the concept of individual viewpoints, and how they relate to fixed space.  Each viewpoint sees a different perspective of objects in space.  To represent this mathematically, he asserts that each viewpoint can be represented by Cartesian coordinates centered on that viewpoint, and that all viewpoints (special relativities) can be coordinated to a general relativity. In his book "The Meaning of Relativity" 5th Edition, Einstein laboriously proves that the Cartesian coordinate system must be based on unit-measure, and rejects the concept of points of zero-dimension.  He wrote on page 15, "think of a Cartesian system of co-ordinates as a cubical framework formed of rods each of unit length".

This concept of rods is diagrammed below-left as an 8-cube cube.  To the right, red dots are placed at the centers of the 8 cubes and connected by red unit-rods, forming a single cube of the same unit size as the 8 cubes, but at the center of the 8-cube cube.  This illustrates how Einstein’s rods actually connect unit-cube centers.

The Cartesian coordinates should represent cubes, not points.


Quantum Crystal Facets

A crystal lattice is a repeating connected pattern.  A given lattice type is assigned a “coordination number” (shown in parentheses after the name of the lattice, for example Body Centered Cubic(8)).  The coordination number of a lattice is the count of adjacent atoms to any given atom.

There are 26 crystal facets, and there is a correlation between the 3 main lattice coordination numbers and the aspects of a cube. 

Lattice                           Cube

Body Centered Cubic (8)         8 Corners 
Face Centered Cubic (12)      12 Edges
              Basic Cubic (6)           6 faces

The illustration above shows cube aspects of corner, edge and face as indicated by colored dots.

The cube aspects are actually quantum positions of a subdivided 3x3x3 cube, as indicated in the graphic above by subdividing the cube but keeping the dots in-place.  These dots locate the 26 outer cubes of the 3x3x3 cube (1 cube is enclosed at the center). 

The graphic below-left shows a solid form of the 26 crystal facets, and to the right how the facets relate to the 26 outer cubes of the 27-cube cube. 

Quantum Lattices

A “unit-cell” is the smallest number of atoms that display the basic lattice pattern.

The “Body-Centered (8)” Cubic Lattice

The lattice called “body centered” is better termed “corner connected”.  This can be seen from the Geocubic Model below, where a central cube is surrounded by 8 cubes attached at the corners.   

The graphic above depicts the unit-cell of the b.c.c.(8) lattice, which fits within a subdivided 3x3x3 cube.  (All of the lattice unit-cells fit within a subdivided 3x3x3 cube).

Cubic Source of the Tetrahedral Lattice

The tetrahedral(4) lattice has 2 chiral (mirrored) unit-cells.  The Geocubic Model reveals the chirality is due to a “halving” of the b.c.c.(8) lattice.

A tetragon has 4 equal triangular faces, as diagramed below to the left (in mirrored forms)  The middle illustrations show a cube added to an outline of the tetragon and the vertices are marked with black dots.  These dots indicate the 2 chiral forms of the tetrahedral lattice.  To the right, the Geocubic Model of the unit-cell is shown, where a central cube is surrounded by 4 cubes located at the corners corresponding to the black dots.

The “Face-Centered (12)” Cubic Lattice

The face-centered crystal is shown to the left below.  The mid-edge correlation to the crystal is demonstrated to the right below, where the 9 frontal points of the 12 mid-edge points are connected by lines that outline the solid crystal.  The “face centered” is better termed “edge connected”.

In the Geocubic Model, the f.c.c.(12) unit-cell is a central cube surrounded by 12 cubes attached to the 12 edges (13 total atoms).  This is diagrammed below, with the outline of the f.c.c (12) crystal wrapped on the unit-cell.

As with all lattice unit-cells, the f.c.c.(12) unit-cell is within a subdivided 3x3x3 cube.

Cubic Source of the Hexagonal Lattice

The hexagonal(6) lattice has chiral (mirrored) unit-cells, and the Geocubic Model reveals the chirality is due to a “halving” of the f.c.c.(12) lattice.  For the hexagonal lattice, 6 of the 12 edges are utilized, and the other 6 are utilized for the mirrored form.

Part II - The Crystal Linking Force

Bias Planes

In the Geocubic Model, each Proton is encased in a spherical force field that is compressed into a unit-cube.  The atomic number is the number of Protons (and thus spheres) of an Element.  The compressed spheres must arrange according to a general symmetry law that dictates that the arrangement must be symmetrical in 2-dimensions, but not in all 3-dimensions.  Elements that cannot meet these symmetry requirements must bond with other Elements to achieve symmetry (see article on atomic bonding).  Every non-inert Element with a valid arrangement will have a ‘Bias’ of one or more internal force planes.  These Bias planes are a crystal force that attracts and aligns other atoms along that plane and links them together.  However, crystal linkages are subject to fracturing apart.

Bias planes are energized by the Protons at the centers of spheres that are exactly aligned on the plane surface.  Protons are isolated at the centers of the spheres, and Electrons are spinning on the force surface of the spheres, attracted by the Coulomb force but unable to penetrate the surface.   Neutrons are unaffected by the sphere.  The function of the Neutron in the Nuclide is to induce transmutation of the Elements.  See the article on Isotopes for detail. 

The Carbon Bias

Carbon has a unique Bias among the Elements, with 2 diagonal interior planes crossing along the cube diagonal line (as illustrated above).  The Bias of an Element can rotate in the 3 dimensions of the cube.  In the case of Carbon (the only Element with this Bias) there are 12 spatial orientations.

The Carbon Ring

The Crystal Linking Force aligns Bias planes, creating a continuous connected plane between atoms, linking them.  The crystal force rotates the internal planes to align them by rotating the internal spheres while retaining the basic sphere arrangement.

In order for a crystal Carbon ring to form, 3 Bias plane orientations must be involved, and must uniquely pair in 3 crossing pairs.  Using a, b, and c for the 3 planes, the 3 crossing pairs must be (a,b), (a,c), and (b,c).

The three spatial orientations utilized in depicting the Carbon-Ring diagrammed below are each used twice in the ring. 

As shown above right, the Carbon Ring is within a subdivided 3x3x3 cube.


The Carbon Ring is the unit-cell of the graphite lattice, which extends on the diagonal plane (a section shown below).

In graphite, multiple layers of lattice planes lay side by side, only held together by static forces, so they can slide on each other, making graphite an excellent dry lubricant.  It is known that the layers always offset to one of 3 different orientations called "A, B, and C".  This childish designation arose because academia has not realized the hexagonal lattice is cubic, and that the 3 offsets are along the 3-dimensional axis of X (right/left), Y (up/down), or Z (front/back).

In the photo below, cubic wood blocks are marked with the intersections of the Carbon Bias planes with the cube edges and faces.  The cubes are stacked according to the alignment of the Bias planes.  Only the front layer of cubes are marked, the supporting cubes are unmarked.  A pattern of upright and inverted triangles is revealed by the markings.  The upright triangles result from the hexagonal Carbon Rings, but the inverted triangles are merely the spaces between the upright triangles.  In the center a hexagon is formed from 6 triangles (3 upright and 3 inverted).  If the cubes were extended on the slanted plane, then a pattern of interlaced hexagons would appear.

Those wood blocks marked are all marked identically and are merely rotated into the 3 different orientations that align the internal Bias planes according to the crystal force.

The Sugar Crystal

The 2 chiral forms of sugar are named for the direction light is refracted by the crystal.  The chiral forms of sugar are termed dextrose (right-turning), and laevulose or (left-turning).  Sugar is a crystal (a combination of crystal forces and atomic bonds (6 linked carbohydrate molecules)).  The Carbon Ring arrangement of 6 Carbon atoms is retained in the sugar molecule (a clathrated Oxygen atom is at the center) but a different crystal alignment links them, due to the atomic bonds (which alter the number of spheres in the cube).

Each diagonal plane has an energy of 56 kcal.

Carbon Ring – 12 planes of 56 kcal = 672 kcal
Dextrose Bias – 9 planes of 56 kcal = 504 kcal

Refer to the article on the Carbon Cycle for detail on the sugar crystal/molecule.

Part III – Chirality

The Chiral Carbon Ring

The mirrored form of the Carbon Ring has an opposite “slant”.  In the graphic below the 3 spatial orientations of the 2 chiral forms are shown to the left.  The “right-facing” forms are shown below-upper, with the slant plane in orange.  The chiral “left-facing” form is shown below-bottom, with the slant plane shown in blue.  Both sets of 3 forms are the same except for the slant.  As illustrated below to the right (in purple), the chiral form of the Carbon Ring utilizes the six mid-edge quantum locations not utilized by the other ring.


Due to the viewpoint of the cubes from the right, the slant-left plane is seen from the edge.

An Alternate (Quantum) Coordinate Notation

The conventional notation used in crystallography uses a unitary directional measure along the 3 (x,y,z) axis coordinates (see below-left), with the numbers 1, -1, and zero (where the -1 is shown as a 1 with a bar over it).  Generally, the crystal facet locators are depicted on the vertices of 8 cubes subdividing a cube, with the 27 vertices labeled (see below-right).


In the quantum reality, these “points” are actually center points of the 27 sub-cubes of a 3x3x3 cube, but even without recognizing that the coordinates are cubes (quantum coordinates), it is strange that a numbering system became established since it is better replaced by labeling the 3 numeric axis coordinates by the following descriptive words.

Right/Left       (x-axis)
Up/Down        (y-axis)
Front/Back      (z-axis)


The 3x3x3 cube above is labeled using the initial letters of the descriptive words, for example:
RUF is Right, Up, and Front

Corners require all 3 axis words.
Edges only require 2 axis words, as the third coordinate is zero
Faces only require 1 axis word, as the other 2 coordinates are zero.

Quantum Rotation of the Carbon Ring

There are only 2 chiral forms of the Carbon Ring (in the Geocubic Model called the “slant”).  The slant is assigned to the x-axis of right and left.  Each chiral slant has 4 quantum rotational positions for a total of 8 orientations (and the reason there are 8 orientations is that there are 8 corners of the cube). 

The 4 slant-right rotations are diagrammed below.
The rotations can be labeled with the alternate quantum notation.



In the diagrams above, the 3 Bias forms involved in each rotation are shown.  The “controlling” rotational form is at the right, preceded by the 2 associated forms that supply the slant of the ring.  Notice that the 2 planes of the controlling Bias form are repeated separately in the 2 associated Bias forms that share the slanting (dark) plane.

The 4 slant-left rotations are


(Due to the standard Right,Up,Front perspective of the cubes, the slant-left plane appears as a diagonal line)

Note that the 8 rotational depictions utilize 3x8=24 Bias forms, and that the 12 different Bias forms are utilized twice each (12x2=24).

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