Thus far, we have ascribed various polyhedra to each orbital type. Moreover, we have identified the appearance of other solids such as the Truncated Cube and Octahedron, and Snub Cube, who play a role within the atomic structure, which goes beyond this article.
Whilst a comparison of orbital types to fundamental solids provides a compelling model of the atom, in and of itself, it does not provide any kind of experimental evidence. In order to produce more support for the theory of Atomic Geometry, we should be able to tie in this geometric view with experimental data. As we are suggesting that the atom is a geometric construct is make sense that the sizes of each atom should follow a predictable pattern based on the polyhedra.
The discipline of ‘Geoquantum Mechanics‘, or ‘Geo Mechanics‘, is a field of study that examines the ratios created by the transformation of polyhedra through specific geometric processes. In the final part of this article, we will provide a very basic introduction to this fascinating area of study.
In, Mid, and Out Spheres
All solids exhibit an In-sphere, Mid-sphere, and Out-sphere, (or circumsphere). We prefer to use the words IN, OUT and MID to maintain clarity. Let us take the example of a Cube with a side length of 1. The diagonal of its side length will measure √2, and the distance between opposite corners will be √3. therefore, a sphere placed inside of the cube will have a diameter of 1, the IN-sphere. By increasing the size of the sphere, it will touch the centre of each side, producing a MID-sphere of √2. Finally the OUT sphere encompasses the Cube.
Sketching out the Atom
Armed with the knowledge of the In, Mid and Out sphere of a Cube, side length 1, we can begin to map the Geometry of the Atom.
The atomic radius of each type of atom have been determined experimentally through measurement to within a tollerance of 5 pico-meter (pm). Within the theoretical framework of quantum theory, two datasets have been produced that try to explain the variations in atomic sizes. We can make a comparison between these data sets and see how Atomic Geometry compares.
If we take a very rough overview is can be seen that almost every block exhibits points structure where a sequential series of elements share exactly the same radii. Putting aside the the P1 and P3 orbitals for a moment and turning attention to the other orbitals, we notice there seems to be a specific radius where each set levels off. P2 elements level off at around 100 pm, whereas P4, all the D-block elements level off at around 140pm. The first half of the F-block elements have radii of about 185 pm, falling to 175pm at around the halfway point.
If we imagine a cube with a side length of 200 then we find that each type of orbital radii roughly falls over the point where the IN, MID and OUT sphere are located. Whilst this is just a rough overview it a good starting point for scaling our atomic model the the values of the radii. Not only do the orbitals fall at around the correct radius, that also fall into the cube in the exact configuration we have allocated for each orbital type.
These values are only a rough guide, and in reality there is a tremendous variation in size for each type of atom. However, by using the principles of that transform Platonic into Archimedean Solids, we are able to generate a far more accurate match. So accurate in fact, that it supersedes all other existing predictions.
The Extended Jitterbug
We are not the first people to suggest that the atom follow a geometric structure. A similar idea of the atom was conceived of by Buckminster Fuller. He discovered that by collapsing the square faces of a Cuboctahedron into two triangles will create an Icosahedron. Continuing this process produces an Octahedron at the final stage. This transformation was termed the ‘Jitterbug’ and was suggested by Fuller to be in operation within the atomic structure.
Both the Octahedron and Cuboctahedron are 2 forms we have identified from the orbital geometries.
Within the theory of Geoquantum Mechanics we extend the jitterbug concept to include two new forms, the Snub Cube and Rhombic-Cuboctahedron. The ‘Extended Jitterbug’ contains five polyhedra, with the Cuboctahedron located at the centre of the set. With the exception of the Snub Cube, all of these geometries have been identified as having a particular relationship to the different orbital types. The pattern terminates with the Rhombic-Cuboctahedron, which is also formed by the arrangement of the hexagonal F-orbitals. What follows is a small introduction to Geoquantum Mechanics, that we believe is enough to prove that validity of Atomic Geometry.
The second important geometric transformation used in Geoquantum mechanics is ‘Truncation’. This is enacted by the division of the side length of a Platonic Solid into two or three. By dividing the side of a Cube or Octahedron in half, the Cuboctahedron is formed. By dividing the sides into three, the corners of each polyhedra can be removed, forming the truncated Cube or Octahedron.
We can place these objects in a row, with the Octahedron at one end and the Cube at the other. Comparing this to the ‘Extended Jitterbug’ both begin with an Octahedron (P-orbital). The truncated series collapses the Octahedron into a Cuboctahedron, whereas the Jitterbug expands into the Cuboctahedron. The first diminishes in size and the latter grows in size, meaning that the two Cuboctahedra exhibit side lengths that are double in size.
The final geometric concept utilised by Geoquantum mechanics is not a process, rather a combination of platonic duels scaled to a specific ratio. A compound solid is one where the mid-sphere of the dual pair is exactly the same size. The Star-Tetrahedron is an example of a compound of two Tetrahedra. The mid-sphere, in this case, contains an Octahedron. The Cube and Octahedron also form a perfect compound, and this time we find a Cuboctahedron occupying the central space. The last of the simple Platonic Compound involves the combination of a Dodecahedron and Icosahedron.
In all of the examples above, we see that the intersection each solid is located at the halfway point on its sides. Previously we identified the same mathematical process to define the position of each orbital type. The division of a side into two equal parts.
We can combine the three previous geometric concepts, the Jitterbug, Truncation, and Compounds, to produce the ‘Geoquantum Blueprint’ that we can use to map the atomic structure.
The compound of the Cube and Octahedron, share the same mid-sphere, which each defines a Cuboctahedron at its center. Both the Octahedron and Cube can have a portion of their corners removed to produce their truncated versions.
The Octahedron can explode through the Jitterbug process, to create a larger Icosahedron and Cuboctahedron (double in size), and then on to create the Snub Cube and Rhombi-Cuboctahedron.
The Cuboctahedron in the centre of the cube/octahedron compound can also expand and shrink through the same jitterbug process. As the Cuboctahedron collapses into a smaller octahedron, so new compounds can be formed, firstly from the Icosahedron/dodecahedron pair and then another Octahedron/Cube. This can also be truncated and als contains a smaller Cuboctahedron as its centre. Again it can expand and contact in size, through the extended jitterbug process.
Each of these polyhedra has an IN, MID, and OUT sphere, most of which exhibit different sizes. However, some of these spheres do fall in exactly the same space. The mid-sphere of the Cube/Octahedron compound, which also form the out-sphere of the Cuboctahedron, for example.
This Geo-mechanical scaling is what we believe structures, the not just the radii of the atomic shells, but more incredibly, the variation of atomic radii for every element on the periodic table.
Predicting Atomic Radii
When we think of an atom, we often imagine that the size must be increasing in incremental steps as more electrons begin to fill each shell. However, this is far from the truth. The atomic radius seems to vary widely for each type of atom. Additionally the different sizes seem not to follow any kind of predictable pattern. Sometime we see noble gases like Helium (2) and Neon (10) expand in size, whereas others, such a Argon (18), contract in size. At the same time we find certain element exhibiting the exact same radius. Why?
Using the Geoquantum Mechanical model we are able to explain this large variation, whilst maintaining a logical geometrical process. In this way we can produce a curvature that almost exactly matches the experimentally measured values for all the atomic radii on the periodic table.
Currently, there are two sets of data that are used to try and calculate the atomic radii. The first is the Bohr Radius, which is a theoretical value based on an infinitely dense atomic nucleus. The second is the Van da Waal radius, which considers the atom as a hard shelled sphere.
However, for the majority of elements we see that the predicted values do not even come close to the data extrapolated by experimental results, that are accurate within a tolerance of about 5 picometers.
The table below shows the results generated for each atomic radii for the first 18 elements. The blue line provides the results generated by experimental data. the dotted green line might be hard to see, as it is so close to the measured data, whereas the red dotted line depicted the data derived from the Bohr radius, and the Yellow line from the Van da Waal radius. Clearly Geoquantum mechanics is a more accurate representation of experimental results.
All of these radii are derived from the Geoquantum Blueprint. Two Cube/Octahedron Compounds are scaled through the extended jitterbug, so that the fist exhibits a mid-sphere radius of 50pm and the second a radius 100pm
In the graph above the green sections produce radii based on the Cube, Truncated Cube, Snub Cube, or Rhombi-Cuboctahedron, the red section is based solely on the Octahedron, and the blue section is based on the Icosahedron/dodecahedron compound.
From this we can see that the first 3 element expand through a Cube and then an Octahedron, after which the 4th element collapses back onto a truncated version of the cube. The atoms diminish in size, until 50pm, which is the mid-sphere of the first compound. Then Neon (10) grows dramatically in size, filling the pentagonal mid-sphere of the Icosahedron. The 10 electrons of this noble gas, are a perfect fit for the rotation midsection of an icosahedron.
As more electrons are added, the radii decreases through an Icosahedron, compound with a Dodecahedron, before collapsing into the Snub Cube. The atom continues to decrease in size, levelling out at element number 15-17. This occurs in the mid-sphere of the second Cube/Octahedron compound, which also happens to contain a third polyhedra, the Cuboctahedron. Having fill each of these forms that occupy the same space, the atom suddenly collapses in size by a factor of √2, as it falls into the in-sphere of the Cube or onto the out-sphere of an octahedron nested inside. this completed octahedra now forms the 3rd noble gas. Argon (18).
This explains why Argon (18) has a radius of around 71pm, whereas Helium (2) has a much larger radius of 120pm. It all follows a logical geometric progression. In this way we are able, not just to construct the most accurate model of the Atomic radii so far produced by any atomic theory. We are also able to show the logic that produces it.
The implication it that the space around the atom is geometric, and that is why the electrons are held in such stable ‘orbits’. This insight sheds a completely new light on the functioning of the atom, which validates the theory of Atomic Geometry, and introduces another new filed of science to the world. Geoquantum Mechanics.