Ferromagnetism and conductivity are 2 key aspects in the generation of electrical energy. In this post, we examine the geometric foundations of the Brillouin zones, which are at the heart of this important phenomena.
D-orbital Geometry (review)
Atomic Geometry is a new theoretical model of the Atom, based on the geometry of the 4 types of orbital produced by the Schrödinger Equations. These come in 4 types, S, P, D, and F, that appears in sets in various shells of the electron cloud. As each shell completes, so a new Orbital type is added to complete the next. up to the 4th shell, where the F-orbitals form. After this, the number of orbitals in each shell starts to fall, so the 5th shell only contains a complete set of S, P and D-orbitals. By the 6th shell, only one set of S-orbitals and half a set of P-orbitals form before the unstable radioactive elements begin.
This means that there are a total of 3 stable D-orbital sets that appear on the Periodic table, in the 3rd, 4th and 5th shell of the atom. Note that the traditional depiction of the periodic table gives the impression that the D-orbital elements appear in the 4th, 5h, and 6th shell, however this is not actually the case.
From a total of 81 stable elements, only 3 exhibit ferromagnetic properties at room temperature. These all appear in the 1st D-orbital set. Out of these, Iron (26) exhibit by far the greatest ferromagnetic qualities, followed by, Cobalt (27), and Nickel (28). Antiferromagnetism is the exactly opposite, in that the these elements resist change in a magnetic field. This is most prevalent in the 2 preceding elements, Chromium (24) and Manganese (25). Technically, as these atoms have many unpaired D-orbital electrons, they should possess ferromagnetic qualities. Yet, compounds of these solids generally align in opposite directions, which candles out magnetic properties of these atoms.
Electron exhibit either an UP or DOWN quantum spin value. Both of these antiferromagnetic elements exhibit a full complement of UP spin D-orbital electrons. Once all the UP spin electron complete the D-orbital set, the DOWN spin electron begin to form, completing each orbital with 2 electrons. Therefore, the three magnetic elements each possess a mixture of filled and half-filled D-orbitals. However, both Copper, and Zinc, exhibit a full complement of D-orbital electrons, just as Chromium (24) and Manganese (25) exhibit a full set of half filled D-orbitals. This is termed the Aufbau Anomaly, which is never explained properly to most Chemistry Students, which is partly due to the fact that there is no solid scientific foundation that can offer a sensible reason for this occurrence.
Copper (29) and Zinc (30) also do not exhibit any ferromagnetic properties. However, Copper (29) is one of the best conductors of electricity. Only Sliver (47) is a better conductor, which appears in the 2nd D-orbital set. The other element that noted for its conductive capacity is Gold (79). All of these conductive element appear as the second to last D-orbital element, in group 11.
Finally, it is worth mentioning the apart from the 1st D-orbital element in the 1st set, all other elements exhibit a radius of either 1.4Å, or 1.35Å, which is traditionally ascribed to the ‘shielding’ by the D-orbital elements, which is supposed to explain this uniformity. However, this idea is presented without any kind of mathematical description that can validate the claim. According to the standard model, as the more protons and neutron are added to the nucleus, so the atomic radius should become reduced. Yet this is simply not the case. In fact, even D-orbitals in higher shells generally exhibit a radius between 1.35Å-1.45Å, despite a large increase in the number of nucleons between different sets. Similar uniform radii can also be identified within the P-orbital elements, for which current theory simple can not explain, and so fails to acknowledge.
Atomic Geometry is the first model of the atom that explains the properties and curious anomalies of these D-orbital elements, not though electromagnetic ‘shielding’, or similarly vague concepts. Instead, we find that the 2 types of radii, 1.4Å and 1.35Å are produced from the out-sphere of the Rhombic Cuboctahedron, Snub cube and their duals. As the number of protons and neutron increase, so the atom evolves through geometric principles. Chromium has 24 protons, which equates to the corners of the Rhombic Cuboctahedrons. Manganese has 55 nucleons, which produces a large Cuboctahedron. After this, the 26 protons of Iron provides a perfect match for the Dual of the Rhombic Cuboctahedron, the Deltoid Icositetrahedron, after which the radius drops from 1.4Å to 1.35Å for Cobalt, whose 27 protons can now form a Cube. This collapses the structure of the electron cloud, from the into the Snub Cube, which transition into its dual, the Pentagonal Icositetrahedron, for the final 2 elements Copper and Zinc.
Atomic geometry is the only model of the atom that can accurately predict the atomic radii, the reasons for the Aufbau anomalies, and for the various properties inherent in the different types of elements. This article expands on the Atomic geometry model, and examines the reason for magnetic and conductive nature of different compound and materials. Before reading on, we recommend going through our articles that examine the D-orbital geometries in greater detail, upon which this article is based.
D-orbital Geometry - Part 1
Today, permanent magnets are a common place occurrence. From simple fridge magnets, to hard disk storage, (although storage systems are moving onto solid state devices (SSD) that are not magnetic in nature), and, most importantly the generation of electricity, magnets have played a key role in the development of modern day technology.
Due to its strong magnetic tendencies, Iron (26) is the most recognised magnetic substance on the periodic table. This property of iron is attributed to the fact that, when complied in a solid, (lattice structure), the electron spins of the atoms tend to align in the same direction. When a magnetic filed is applied to these kinds of element, the poles tend to align themselves to an even greater degree.
Often, solids are comprised of more than 1 type of atom, each possessing a different strength of field, due to their size or electron configuration. If the spin of each atom still aligns in the same direction, then the solid still produces a ferromagnetic property.
Antiferromagnetic elements exhibit properties whereby the electron spins of each atom in the lattice appear in alternating directions. The most prominent example of this kind of behaviour is exhibited by Chromium (24). Elements of this nature specifically resist forming electromagnetic fields, not because they are not affected by the externally applied magnetic field, but because the influence causes more of the atoms to align in alternating polarities.
The final type of configuration is the found when the atoms do not align in either the same or opposite directions. This is called a ‘spin glass’, which is often formed by mixing ferromagnetic and antiferromagnetic elements to produce small areas where some atoms are aligned in the same direction, whereas other align in opposite directions.
At cooler temperatures, the magnetic orientations of the elements remains fixed. However, when heated above their Cure Temperature, the spins of various atoms becomes free, and can be reorientated. By heating a substance and applying a strong magnetic field, permanent magnets can be created, which hold their magnetic properties once cooled.
Material that are not ferromagnetic or antiferromagnetic fall into two main categories. If an atom exhibits any orbitals that are unpaired, i.e. they exhibit orbitals that a filled by only 1 electron, the substance is paramagnetic, and will be weakly attracted in a magnetic field. If the orbitals are completely filled, such as Copper (29) or Zinc (30), the substance is diamagnetic, and is repelled by a magnetic field.
The most commonly recognised Antiferromagnetic atom is Chromium (24), followed by Manganese (25). According to present theory, elements with unpaired electrons should exhibit ferromagnetic qualities. Yet, both of these atoms have an electron configuration which consists of a full complement of half filled D-orbitals. When we compare these atoms to Iron (26), the most ferromagnetic element, we find they all exhibit the same radius (1.4Å), and even form the same body Centred lattice formation in a compound. Chromium even exhibits the same number of neutrons (30), as Iron. Why then should these elements exhibit exactly the opposite ferromagnetic behaviour?
Traditional theory tends to offer quite vague explanations, which suggests the reason is due to the fact that the atom will ‘fall’ into the easiest possible configurations. Yet, this is quite an unsatisfactory explanation. The theory of Atomic Geometry offers a far more convincing answer. The reason is simply due to the ‘shape’ of the atom.
The 24 protons of Chromium form the geometry of the Rhombic Cuboctahedron. This shape can be combined with a Cube and Tetrahedron to form a Cubic lattice structure. Manganese exhibits the form of a cuboctahedron, which can also form a lattice when combined with Octahedra.
Electrons can only exhibit 2 types of quantum state, with either an up or down spin. This produces the effect that the atom is orientated with a north-south pole. In the 4D interpretation of the electron field, quantum spin is ascribed to the rotation of a 4D object on its w-axis, which in turn produces a rotation of the solid in quantised steps around a central axis. In the case of the Rhombic Cuboctahedron and cuboctahedron, this midsection is formed of an octahedron, and hexagonal plane respectively. As one atom rotates in a clockwise direction, so the others rotate in an anticlockwise direction. This is very easily visualised by the mechanics of a rotating set of cogs.
The difference however between the 2D rotation of the cog model and the 4D interpretation of the electron field, is that as the 4D object rotates on its W-axis, it momentarily ceases to exist in 3D. Only as it completes a full rotation does it pass back into the 3rd Dimension, rendering ‘frames of time’. This rotational speed of governed by the speed of light with unifies the 4D rotation throughout a material, and produces the quantised effects of electron spin. This view expresses a major diversion from the standard notion of time, which is normally considered as a continuous function, as suggested by the tradition notion of space-time in physical space. This view explains the quantised nature of the electron cloud, and the capacity of electrons to only exhibit either an up or down spin, and their ability to ‘magically’ jump from one shell to another, something which the standard model still cannot explain.
Furthermore, we find that when Chromium is mixed with an impurity, such as oxygen, its ferromagnetic properties suddenly manifest. In simple terms, we can ascribe the counter rotating cogs with oxygen atoms, which in turn aligns all the Chromium atoms in the same spin orientation.
A similar solution can be applied to Manganese, whose magnetic properties are transformed by the addition of other elements. The increases the distance between atoms, which is the accepts view of quantum mechanics to explain this transformation. However, as the theory tends to ignore the geometry of the atom itself, the expression is formulated through more complex explanations, as it tried to maintain the idea that the electron is a particle, rather than a 4D field. As with all things, we find that the most simple answer is usually the best, which in this case only requires that we view the atom as a 4th dimension entity.
The Deltoid Icositetrahedron and Ferromagnetism
Iron (26) is the most ferromagnetic element on the periodic table. This nature is traditionally ascribed to the fact that it has many unfilled electron pairs. Yet, this is also true of many of the other D-orbital elements that do not exhibit ferromagnetic properties. For example, Ruthenium (44) and Osmium (76) exhibit exactly the same number of unpaired D-orbital electrons but do not exhibit ferromagnetic qualities. Therefore, such explanations of ferromagnetism are completely unfounded, and illogical.
The theory of Atomic Geometry ascribes the 26 Protons of Iron to the Deltoid Icositetrahedron, which is the Dual of the Rhombic Cuboctahedron. This form is created by defining the centre of each face, which become the corners of the dual. Whereas the Rhombic Cuboctahedron is formed of 2 types of face, square and triangle, the deltoid Icositetrahedron is formed of only one type of ‘kite’ shaped face.
The construction of these Dual can be viewed from the perspective of an Octahedron and Cube. In the Rhombic Cuboctahedral model of the D-orbital elements, the Cube with a side length of √2 can be nested inside, with is corners defining the centre of each triangular face. The Octahedron will have its corners centred on the square faces on the x, y and z axis, (dark green in the image above). As the Deltoid Icositetrahedron ‘swaps’ places with its dual, so these corners push through the centre of each face, which increases the size of the Octahedron and Cube. As the out-sphere of the Iron atom remains consistent at around 1.4Å, the Octahedron will now exhibit the same size as the out-sphere, which is just under √2. The Cube with its corners nested in the centre of the triangular faces of the Rhombic Cuboctahedron get reduced in size slightly, which bring it into unity with the larger octahedron, forming a compound. This is easier to visualise when viewed as a 2D projection.
In the image above, we can see the square side of the cube, seems to fit almost exactly inside the red square of the Octahedron. This produces a compound of the 2 solids, which, when the corners are connected, form the Rhombic Dodecahedron, which is the template for the 4D hypercube. Shortly, we will examine the Brillouin Zones of various elements, which for the Body Centred Cubic (BCC) lattice also happen to be the Rhombic Dodecahedron.
The second geometric reason for the exceptionally strong ferromagnetic qualities of Iron (26), is found in the nature of rotation. Within the D-orbital set, 1 of the 5 orbitals exhibits a torus form, with two orbital lobes in the north-south orientation. The Rhombic Cuboctahedron it the only Archimedean Solid that exhibits an octagonal mid-section, that can be rotated. This geometry matches that of the torus orbital, which is one of the reasons the Rhombic Cuboctahedron appears in the set.
The Deltoid Icositetrahedron radically changes the toroidal nature found in the Rhombic Cuboctahedron. The midsection flattens into a single plane. This means the Deltoid can be rotated in two halves. A similar behaviour can also be attributed to the Octahedron and Cuboctahedron, these exhibit a mid-plane of a square and a hexagon respectively. The Deltoid Icositetrahedron maintains its octagonal shape, which is now rotated by 22.5°, compared to the octagon at the centre of the Rhombic Cuboctahedron.
The rotation of the central octagonal plain of the Deltoid Icositetrahedron changes the fundamental structure of the BCC lattice. The flat faces of the Rhombic Cuboctahedra touch each other so that as one turns clockwise, so the adjacent atoms will rotate in the opposite direction, causing the spins to align in opposite directions, forming the antiferromagnetic properties of the lattice. For Deltoid Icositetrahedra, the corner points of each octagon touch, which produces more space between each atom. This means, unlike the cog analogy, the atoms can rotate in the same direction, which unifies their spin orientation producing the ferromagnetic properties of the lattice.
Furthermore, the Deltoid Icositetrahedron also exhibits a single mid-plane in the vertical direction, which means that a similar rotation process can be applied to the cells above and below. When complied into a cubic arrangement, each atom becomes spaced apart by a 4 pointed star, which give each atom the ability to unify the spins over a much wider area.
Iron plays a central role in the Haemoglobin, the biological mechanism that is used to hold oxygen atoms in red blood cells. When we examine the active site of the molecule, we find 4 nitrogen atoms, inside which a single Iron atom is suspended within a square. When we examine this geometric configuration, we find that the projection of the Deltoid Icositetrahedron can be superimposed over the active site, in various scales and orientations. Nitrogen, exhibits 7 neutrons and 7 protons, which when multiplied by 4 total 56 nucleons, which is the same number as found in the Iron atom itself. This correlation is not presently recognised in the field of biology, and begins to offer a new geometric perspective as to how this mechanism, which is essential for all red-blooded animals, can produce such an intricate atomic mechanise.
As noted previously, Chromium (24), exhibits the same number of protons as the corners of the Rhombic Cuboctahedron. Chromium (25) has 55 nucleons,, and so falls into a larger Cuboctahedral form. The next element, Iron, has 26 electrons and protons, which fall into corner points of the Deltoid Icositetrahedron. This is the final boundary before the radius drops to 1.35Å, as the D-orbital structure collapses into the Snub Cube.
Snub cube Magnetic Order
After Iron (26), the remaining D-orbital elements in the 1st set all express the same radius of 1.35Å. This shift is represented geometrically as the collapse of the Rhombic Cuboctahedron into the Snub Cube, as part of the Extended Jitterbug transformation. Just like the Rhombic Cuboctahedron, the Snub Cube is formed of 24 corners. The key difference is that the diagonal across 12 of the square faces on the Rhombic Cuboctahedron collapses into 2 triangles. With a side of 1, the diagonal of the square is reduced by a ratio of √2:1. This is achieved by a 22.5° rotation of the Rhombic Cuboctahedrons faces found on the x, y, and z, axis. This reduces the out-sphere of the Rhombic Cuboctahedron from 1.4Å to 1.35Å, which is the difference between Iron, and the rest of the D-orbital elements that follow it.
The Snub Cube is a Chiral Geometry, which means the square faces can be rotated either clockwise or anticlockwise to produce 2 different versions of the same form. When placed in a honeycomb, as in the previous example of the Rhombic Cuboctahedron, a clockwise rotation on one from will produce an anticlockwise rotation in those directly adjacent. As each solid get reduced in size, so gaps form in between, which allow each atom to rotate in the same direction. I.e. the spins of each align in the same direction.
After Iron (26), the only other elements that are ferromagnetic at room temperature are Cobalt (27), and Nickel (28), although to a much lesser degree. In the theory of Atomic Geometry, the 27 protons of Cobalt for a Cubic arrangement as the geometry transitions into the Snub Cube for Nickel (28).
The Snub Cube offers a wide variety of possible lattice configurations. When positions corner to corner, the shape between each solid resembles that of the Deltoid Icositetrahedron lattice, but exhibits two kinds of side length. When viewed from a 2D projection, as each rotates, these alternate on the x and y-axis. As there are two different side lengths in the projection, the solid can also be orientated to produce a ‘rhombic’ shape or an off-set start, (bottom left below). If each form appears in the lattice with the same chiral rotation, then it also produces a similar type of shape in the spacing of the lattice. Finally, the Snub Cubes can be squished together so that their flat faces touch. Again, this can form in 2 types of orientations, which depends on whether the square or triangular faces are placed up against one another.
in 3 dimensions, there is only one recognised space filling structure that can be formed that includes the Snub Cube. The Alternated Cantitruncated Cubic Honeycomb contains Snub Cubes in alternating chiral symmetries. The Gaps in between are filled with Tetrahedra and Icosahedra that exhibit two types of side lengths, 1 and 1.04277. This makes it a ‘near-miss’ regular polyhedral structure.
Presently, the notion of the Snub Cube is not apparent within the literature of modern Chemistry. However, a class of atomic lattice, called Perovskite Crystals, does exhibit a similar rotational phenomenon. These are formed of Octahedral configurations, normally formed of smaller atoms, which are interspersed with other atoms with a larger radius. Just like the squares on the faces of the Snub Cube, these Octahedra can be rotated to produce a wide variety of lattice types. This interesting class of materials is able to produce substances with numerous unexpected qualities. These adaptive composites are often able to change their electrical or magnetic behaviour, when placed in a magnetic or electric field, which includes, Colossal magnetoresistance, ferroelectricity, superconductivity, and Charge Ordering, to name by a few, even through the atoms that comprise these compound do not exhibit these properties independently. These crystal structures have a wide range of modern applications, including, electrical sensors, high-temperature superconductors, and fuel cells.
The reason the Snubification aspect of the elements is not presently recognised appears to be due to the fact that Chemistry tends only to observe the structures of these material, rather than the invisible geometry with allows these structures to be manipulated. The theory of atomic Geometry suggests that within these crystal lattices it is the Snub Cube that is at work, that produces the rotation of the lattice structure, and is responsible for these exotic effects.
Magnetic domains and the Tribonncci constant
As previously, explained, the capacity of a material to exhibit magnetic properties is due to the alignment of the quantum spins of large groups of atoms. These are called ‘magnetic domains’. A magnetic material will have a number of domains where the clusters of atoms all align. When an external magnetic field is applied, some of these unify and orientate themselves in the same direction, which increases the magnetic effect. If one domain is larger than the other, then it will tend to influence the smaller, so both domains now align in the similar direction. However, if each domain is roughly the same size, then the magnetic effect candles out. This is why even Iron, the most ferromagnetic material, can be demagnetised. This video offers a great overview of the traditional theory of magnetism, from the atom, to the magnetic domain. Note that the term ‘magnetic moment’ means the same thing as quantum spin, which can only take 1 of 2 values, up, or down.
Whilst the traditional view of magnetism provides a surface level explanation for the magnetic properties of different elements, it fails in its application across the entire periodic table. Furthermore, it cannot explain the exact mechanise that forms these magnetic domains in the first place. Additionally, when a magnetic material is heated past is Curie Temperature, these domains become disordered, and the material loses it magnetic properties. Yet when cooled, the domain will re-establish themselves. So what is the principle that can allow this to happen?
The theory of Atomic Geometry offers a clear solution. However, to explain this, we first need to examine the properties found in the Snub Cube. What makes this solid unique amongst the Archimedean Solids is that contained within its structure is the Tribonacci Constant. This is derived from a similar idea to the more popular Fibonacci numbers, which are formed by added together the two previous numbers in the series, starting with 0 +1=1. The Tribonacci numbers are similar but, add together the previous 3 terms of the series, stating with 0+0+1=1, then, 0+1+1=2, 1+1+4=4 and so on.
Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
Tribonacci Numbers: 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274…
Interestingly, we can see that both series contain the number 13, found in the nature of the Cuboctahedron, and is also half the number of Protons found in Iron. This is followed by 24 in the Tribonacci series, which s the number of corners of the Rhombic Cuboctahedron. Also, it is interesting to note that element 43 is strangely radioactive within the series of stable elements, after which element 44 begins a second block of stable elements. This means there are only 81 stable elements on the periodic table. Both of these appear in sequence in the Tribonacci numbers.
The Golden Ratio (Φ) is approximated by dividing adjacent pairs of Fibonacci numbers. The higher the number pair chosen, the closer the result will be to Φ. The same can be said of the Tribonacci Numbers. Therefore, an approximation of the Tribonacci Constant (η) can be found by dividing 24 by 13, or 81/44.
The unique property of the golden ratio is that, when squared, the result will be exactly 1 greater than Φ. This satisfies the equation:
Φ² – Φ – 1 = 0
The Tribonacci Constant (η) is similar but for the cubic version of this equation, therefore:
η³- η² – η – 1 = 0
The golden ratio appears most prominently in the geometry of a pentagon. In 3D, it is found in the mid-section of the Icosahedron. The Rhombic Cuboctahedron on the other hand Exhibits the Silver Ratio, √2±1, created though its Octagonal mid-section. As √2 is the diagonal of a square with a side of 1, it naturally appears in the mid-plane of an Octahedron. A Cuboctahedron, has a radius that is the same as its side-length and so expresses the ratio 1:2 across its Diameter.
Each of these forms appears in the Extended Jitterbug, and all have one thing in common. They can be rotated to produce what is known as a pseudo version of the solid. Whilst the Rhombic Cuboctahedron and Icosahedron have a mid-section, the Cuboctahedron and Octahedron have central plains. The odd one out in the sequence seems to be the Snub Cube. Due to the 22.5° rotation of its square faces, it loses its mid-section, and central plain. However, it does encode the Tribonacci Constant within its structure. When considering the Extended Jitterbug, we can see that each form exhibits a particular quality. Either a fundamental geometric ratio, metallic mean, or the Tribonacci constant. Viewed in this way, the Extended Jitterbug begins to express a deeper, hidden relationship between these fundamental geometric ratios.
The Tribonacci constant has a value of 1.83928… It can be constructed from a circle with a radius of 1 placed inside a square. A right-angled triangle, is rotated away from the vertical axis, by a certain degree, When the distance from the right-angle to the vertical matches the distance from the horizontal, the resulting hypotenuse will be equal to the Tribonacci Constant.
Whilst the equation which generates the exact value for ξ is rather complicated, involving the cubic roots of 3√33±19, which has no solution in terms of compass construction, a close approximation can be found by √8-1, and √1.5 ×1.5. By placing an octagon inside the large square, which is then repeating in just one quarter, we can see that the right angle corner just about touches the edge of the square place inside the octagon in one of the small corners. As the base of the triangle passes, the exterior of the square, it nearly touches the corner of the large octagon in placed in the circle, (bottom left above). In this way, we can intuitively understand the Tribonacci constant exhibits a type of fractal construct, similar to Φ.
In fact, the Tribonacci Constant does manifest a Rauzy fractal. This rough looking shape remarkable nests into sets of 3 to create an exact replica of the original piece. Each unit is the same shape, and is proportional to each other in such a way that is satisfies the Tribonacci Series. Each unit is able to tile a flat surface with 3 sizes which are an exact replica of each other. This video provides a great overview of the Tribonacci Constant and the Rauzy Fractal.
Unlike the Fibonacci Numbers, the Tribonacci’s have never been identified as appearing in nature. However, the tessellation of the Rauzy fractal bears a striking resemblance to the formation of Magnetic domains within a ferromagnetic material. When we examine a close up cross-section of such a material, we find the domains are formed in apparently random clusters. Yet, when the Rauzy fractal is superimposed over the domains, we find that there are large portions of coherence.
Whilst this is far from conclusive, it does point to an interesting correlation between the ferromagnetic elements, Cobalt (27), and Nickel (28), which also exhibit magnetic properties, and assume the form of a Snub Cube, with a radius of 1.35Å. But what about Iron? In Atomic Geometry, this element has a radius of 1.4Å, and is ascribed the form of the Deltoid Icositetrahedron, which appear to bear no relation to the Tribonacci constant.
Let us now reveal a new geometric secret, that has remained undiscovered by geometricians. In the Atomic Geometry model, the radius of 1.4Å is produces from a Rhombic Cuboctahedron with a side of 1. When this transforms into its Dual, is creates 2 different side lengths. The longer side (A) has a value of 1.1589, whereas the shorter (B) will be 0.8964. by dividing side A by side B we can identify the ratio between the 2, which is produce by the equation:
A * ( 4 + √2 ) / 7
When scaled to the values ascribed to the Deltoid Icositetrahedron in the Atomic Geometry Model, this result in a value that in incredibly close to the Tribonnaci constant divided by √2.
(A ÷ B) ≈ (η ÷ √2)
In fact, the exact ratio formed from this equation is √8-1, which, as noted previously, appropriates the Tribonacci constant. As 1:√2 is the ratio of the side of a square to its diagonal, we can see that the Tribonacci constant is express in the diagonal of the lattice, when viewed in 2D, which in turn produces the large magnetic domains unique to the Iron atom. This proposal is the first that finds an expression for the Tribonacci constant in nature, and opens new areas of investigation into the mechanisms that form the magnetic domains within a lattice.
The Geometric rules of Paramagnetic and diamagnetic Elements
Whilst Iron (26), Cobalt (27), and Nickel (28) are ferromagnetic, the next elements on the periodic table, Copper (29) and Zinc (30) are not. These elements are instead Diamagnetic, which means they are repelled when an external magnetic field is applied. Like paramagnetic materials, which exhibit exactly the opposite, and are attracted in a magnetic field, this phenomenon is extremely weak and difficult to detect. Present theory suggests that the reason for this diamagnetic behaviour is due to the fact that all the orbital shells a filled with both Up and DOWN electron spins. However, on closer examination of the elements that are Diamagnetic, this again turns out not to be true. For example, the majority of P-orbital elements are Diamagnetic, whether they exhibit half filled electron orbitals or not. Helium (1) has a half filled S-orbital and is also Diamagnetic. Once again, as with the ferromagnetic elements, we find that the theories that are commonly purported as being true, do not stand up under closer examination.
Atomic geometry offer a much more accurate generalisation. P-orbitals are ascribed to the Octahedron, whereas D and F-orbitals a given the geometry of a Cube and Cuboctahedrons. Therefore, as a rule of thumb, if an element is formed of P-orbitals, (an Octahedron), it will be Diamagnetic, whereas the Cube and cuboctahedron produce elements that are Paramagnetic (or Ferromagnetic). The exception are the D-orbitals that exhibit a full complement of election pairs. Whilst there are still exceptions, such as Oxygen (8), whose 16 neutrons for a hypercube configuration, (i.e a cube), and Aluminium, (13), whose 13 protons form a Cuboctahedron, the rule provides an exact match to the observed nature of elements.
Therefore, the theory of Atomic Geometry not only offers a much cleaner explanation for the ferromagnetic qualities of Iron, Cobalt and Nickel, it also produces a far more accurate prediction for the paramagnetic and diamagnetic properties of all stable elements on the periodic table, and can even provide a geometric explanation for those elements that seem to contradict this rule.
Blochs Theorm VS the Drudle Model
Copper (29) has a full complement of D-orbital electrons, yet only has 1 S-orbital electron in its outer valance shell. It is often taught that this is the reason for Coppers excellent conductive properties. The D-orbitals form bonds with each other, which then allows the single electron to travel freely over the through the material. However, like many of the superficial explanations produced by the mainstream science, this model has been proven to be incorrect. There are plenty of example of other D-orbital elements, such as Chromium (24), that also exhibit only a single electron in the outer shell, which do not exhibit any particular conductive properties. Within the 2nd D-orbital shell, we find that elements 41, 42, 44, and 45 all exhibit a single S-orbital electron in their outermost shell, but without exhibiting any kind of conductive properties.
You might think that conductivity occurs due to a combination of the fully paired D-orbital electrons, with the single outer S-orbital. After all, the same D-orbital configuration applies to Gold, and Silver that also exhibit strong conductive properties. However, the 4th most conductive element is Aluminium (13) which is not even a D-orbital element, and has 3 Valance electrons in its outmost shell. Whilst Gold, Silver, and Copper are Diamagnetic, the next most conductive elements are Paramagnetic, which therefore excludes this nature as the reason for conductivity. Furthermore, Cobalt (27), which is ferromagnetic and yet is more conductive than Zinc (30). Clearly, the conducive nature of these elements has little to do with the number of paired electrons, of the face that only one S-orbital appears in the valance shell.
One of the main problems with the common interpretation of the notions of electricity, is due to the insistence of trying to develop a model that is formulated around the idea that the electron is a particle, which in the 4D view simply is not the case. In fact, it has been proven that electricity does not ‘flow through the wire‘ as is commonly believed. Instead, it is the magnetic field that setups up the electric field. For this reason, electricity can be transmitted wirelessly, or be transmitted to an adjacent bulb prior to the current that flows through the wire. The wire then acts as a guide for the electrical circuit that also produces a ‘secondary’ magnetic field, which is stronger than the first. Yet, the fact remains that electricity is ‘transported’ due to the magnetic field, which forms when a change in the position of a particle. In 4D aether theory, this particle does not exist, moreover it is an electron wave. The wave only model of the electron is not recognised, as at the turn of the century, the particle notion of light was established as a solution to the Photoelectric effect and Ultraviolet Catastrophe. However, thorough our investigation of these experiments and the mathematical result that calculated the results, we have identified a flaw in the reasoning, which we cover in great detail in our articles on 4D Aether.
The 4D Aether
A more correct interpretation of the functioning of electricity comes from the examination of lattice structure formed by large numbers of atoms in a material. When combined, a secondary wave forms in the structure, which produces the flow of electricity. This video explains this nature in much greater detail, and clearly denotes the difference between a theoretical model, such as the electron particle, which is easy for us to conceptualise, and the reality of the physics that governs the formation of electromagnetic waves within a circuit.
The Drude Model of the that is continuously used to describe the flow of electricity has been disproven long ago. The only reason it is still taught today is because of its simplicity. But electron behave as waves. If this were not the case, the modern era of computing would simply not exist. Transistors, whilst often modelled by the particle electron, do not actually work by producing ‘holes’ in doped material. They work through Quantum Tunnelling, which when viewed from the particle perspective seems like a mysterious and bizarre quantum effect. Yet, once the electrons are viewed as a wave phenomenon, which can penetrate a material through evanescent coupling, the magic and mystery is simply replaced by common-sense logic. Concepts such a drift velocity of the electrons are in fact only theoretical in nature. No one has ever measured an electron moving, in the same way that no one has ever established the radius of an electron. Instead, the only model that makes sense of all the electromagnetic phenomena that we have observed is the wave model, which once the 4D Aether is re-introduced, that which removes the need for a particle of light, returning logic and normality to the quantum enigmas that plague the standard model.
Electron waves and the Tribonnaci Constant
Now that we have clarified the nature of conductivity from the wave perspective of the electron, we can examine the reasons for the conductive and magnetic nature of various elements. After Nickel (28) the next Element Copper (29) exhibits a full set of paired D-orbital electrons. Aside from the transition from ferromagnetic to diamagnetic, Copper also exhibits an extremely conductive nature. In the theory of Atomic geometry, Copper is assigned the geometry of the Pentagonal Icositetrahedron, which is the duel of the Snub Cube. Just like the transformation of Antiferromagnetic Chromium into ferromagnetic Iron sees the transformation of the Rhombic Cuboctahedron into its Dual, so the same process produces a transformation from the ferromagnetic element, Nickel into the Diamagnetic, yet conducive, Copper atom.
NOTE: The Pentagonal Icositetrahedron exhibits 38 corners. This happens to be the midpoint between Copper (29) and Silver (47), (i.e. 38 – 9 = 29, and 38 + 9 = 47). Whilst Silver is slightly more conductive than Copper, with values a difference of 0.63×106/cm Ω to 0.596 x106/cm Ω
Just like the Snub Cube, this form also exhibits the Tribonacci constant, which appears as the difference between its longer and shorter side length. The exact formula is a / (s+1), where ‘a’ is the longer edge length and s is the Tribonacci constant minus 1, divided by 2, or (η – 1) / 2. If we divide the longer side (a) by the shorter side (b), just as we did with the Deltoid Icositetrahedron, we find the result is close to √2. This shows an interesting correlation between these polyhedra, where the difference between the two the Tribonacci Constant.
Deltoid Icositetrahedron: (A ÷ B) ≈ (η ÷ √2)
Pentagonal Icositetrahedron: (A ÷ B) ≈ (√2)
Viewed like this, we can recognise that the relationship between the geometry of the most ferromagnetic element, Iron, and that of the highly electrically conductive element, Copper, differs by a factor of the Tribonacci constant. However, in Atomic Geometry, Zinc (30) is also ascribed the same geometry. So why isn’t this element conductive?
To answer this, we need to take a closer examination of the nucleon counts and distribution of the D-orbital elements in the 1st set. Copper (29) and Zinc (30) may exhibit the same geometric shape, but the number of neutrons in each atom not the same. Furthermore, different atoms are only stable with a certain number of neutrons. These are called isotopes.
Within a substance, a single type of atom will exhibit a ratio of these isotopes, which is defined as an overall percentage for each type. Some atoms, such as Scandium (21), Vanadium (23), and Manganese (25) are single isotope elements. Notice how normally this applies to atoms with an odd number of protons. This includes Cobalt (27). However, Copper (29) strangely breaks this trend, being the only D-orbital with an odd number of protons to exhibit 2 isotopes. In a copper wire, only around 1/3 of all atoms will consist of nucleases of 36 neutrons. The other 2/3 will all only have 34 neutrons. Silver is another example of a double isotope element with an odd number of protons.
This nature of a multiple isotope is quite common in the 1st D-orbital elements with an even number of protons, which tend to exhibit 4 stable types, except Zinc (30), that has 5. When we examine the different isotopes for the last 5 elements on the D-orbital set, we find only Cobalt (27) exhibits a single isotope.
Isotopes of the 1st D-orbital set. Magnetic and conductive elements
However, this does not really paint an accurate picture, as the percentage distribution of each isotope varies wildly. For example, around 90% of iron will exhibit 30 neutrons, with the other 10% divided amongst the other 3 types. The 5 isotopes of Zinc (30) are divided so that just under half will exhibit 34, neutrons, then just under a third exhibit 36, and then just under a 5th, 38, until only 0.6% will exhibit 40 neutrons. In order to gain a deeper appreciation of these subtle differences, we can map the isotope distribution of each element on a graph.
We can see that Iron (26) has a very narrow isotope profile, with 90% of all isotopes exhibiting 30 neutrons. Cobalt (27) is an odd numbered element and only exhibits a single isotope. Compare this to Nickel, whose ferromagnetic quality falls dramatically, whilst is conductive properties rise above that of Zinc (30). Whilst just over 2/3 make up isotopes with 30 neutrons, the remaining 1/3 are distributed, diminishing up to 36 neutrons. Just under 1% of all Nickel atoms exhibit 37 neutrons.
When we examine the Isotope distribution of Nickel, to those of Copper and Zinc (30), we realise the peaks correspond to each other. The smallest isotope of Nickel with 30 neutrons is most prevalent at around 68%, only 1% less than the most common isotope of Copper (28), which has 34 neutrons. The 2nd isotope of Nickel account for around 26% of a compound, which is again just over 1% of the second isotope, this time of Zinc (30), with 36 neutrons. This correlation becomes much clearer when we overlay the distribution curvature of the Nickel atom over the Copper and Zinc distribution pattern.
By placing the neutron distribution of the D-orbital elements of a single graph, we can see a trend in the overall wave pattern. The peaks from electrically conductive Copper (29) can be connected to the peaks of the 3 magnetic elements. This forms 2 parallel lines, one that run from Iron (26) to the first peak in the copper distribution, with a second running from Nickel (28) to the second peak of copper. The third line is projected from the peak of the single isotope of Cobalt (27), which appears with 32 neutrons, and produces a line through both the Copper (29) peaks.
This shows us there is a correlation between these elements and the percentages by which each isotope is express in a material. We can extract the isotope distribution wave for Iron (26) the most magnetic element, and Copper (28) the most conductive of the set. Around 91.8% of Iron exhibits 30 Neutrons, which forms a large single exhibits a single peak. This is unusual in the 1st D-orbital set, for an element comprised of an even number of protons.
When combined, the 2 peaks of the Isotope distribution of Copper and single peak of Iron, produce a waveform that is similar to the Bloch electron waves, that explain the nature of electricity flowing through a wire. Iron produces a large wave that is truncated by the Copper wave. We could swap the Iron isotope of with that of Cobalt and still combine it with copper to achieve a similar result. Finally, Nickel, whose isotope distribution is main concentrated at 2 peaks, can also be superimposed to create a similar wave form to the Bloch Electron waves.
Whilst this provides an interesting correlation, it does not provide enough substance for a scientific proof. The exact ratio of frequencies that forms a Bloch electron Wave, is still not fully understood. However, there is another clue that is found in the nature of the Tribonacci Constant, however, to explain this we first need to explain a principle that is often overlooked in the formation of these N-bonacci numbers. Note N-bonacci refers to all variants of the Fibonacci numbers that include successively higher terms, such as Tribonacci includes 3 terms so the Four-banacci includes 4 terms, and so on up to infinity. It does not matter is the first term is larger or smaller than the first, if it is a negative number, or even a decimal fraction. No matter what, you will always get phi. If 2 numbers are added, subtracted, multiplied or divided, it produces a 3rd number. We can even take the result and any of the number that created and use those as terms for a Fibonacci calculation, and still get phi. Therefore, any number pair resolves to phi, or 1.618. You can try this for yourself by downloading this simple Fibonacci calculator made from a simple spreadsheet.
The construction of the Fibonacci number series is usually depicted as starting from 0, 1. As successive terms are added together so it produces the next number in the series. Therefore, we find the number expand 0, 1, 1, 2, 3, 8, 13, and so on. However, it turns out that this is only a ‘special case’. In fact, we can choose any 2 numbers and adding successive terms produce an infinite variety of Fibonacci type numbers, which when divided will begin to approximate the Golden Ratio. When we say any to numbers, we really do mean ‘ANY 2 NUMBERS’. Think about this from the standpoint of number theory. Regardless of which 2 numbers we choose, the results, when divided, will ALWAYS approximate phi as the previous term is added to the first to create the next number. You may think that this constant will be an impossible large number, but in fact it is not. In fact, the limit of the Bonacci constants is 2. From the perspective of number theory, this equates to mathematical insanity. But actually is reveals something quite profound, which you can find out more about by studying our new system of 4D mathematics, which resolves the Continuum and Reimann hypothesis, and even destroys fundamental assumptions such as prime numbers, the idea the pi and e are transcendental, and even the notion of algebra, that is the basis of all mathematical and scientific comprehension of our universe, (which is enough to give anyone who is ingrained in the field a mathematical heart attack.)
Now, exactly the same thing can be said of all other N-bonacci numbers. For example, the Tribonacci involves 3 terms, yet no matter which numbers we choose, the result always produces the same constant. This is an important realisation, as every calculation that you can possibly perform can be broken down into 3 terms. 2 numbers which are operated upon, with a 3rd which produces an answer. Naturally, we can continue adding more terms up to ∞-bonnacci, whereby we would be including every single number that can ever be created throughout the entire number line, including all permutations of all decimal fractions. Yet, the constant will never exceed the number 2. That’s quite a mind-blowing concept when you think about it, but is easy to demonstrate through a simple Excel spreadsheet, which you can download on view of Google Docs here.
If you include an infinite number of terms in the N-bonacci calculation, the result will never exceed the number 2.
Whilst these claim all seem pretty mind-blowing, yet we still have not arrived at the final point, which is even more outrageous. Whilst Fibonacci numbers are generally considered as an additive function, we can also find an alternative expression by substituting all the additive expression for subtraction. In the example of the Fibonacci expression, this just becomes (1-1=0), (1-0=1), (0-1=-1), (1- -1=2)… and so on into infinity. Starting with 1 and 1, this produces a series of numbers similar to the normal Fibonacci numbers, with terms alternating between positive and negative integers.
Subtractive Fibonacci numbers
0, 1, -1, 2, -3, 5, -8, 13, -21…
This means that when 2 terms are divided, the result is -Φ, or -1.618… You might think that if we extend the number of terms to 3, then the result will be the Tribonacci constant expressed in the negative. However, amazingly, this is not the case. Rather than settle on the Tribonacci constant, what actually happens is an erratic set of numbers which seems to defy prediction. When traced over time, the result of different numbers produce a wavelike phenomenon, which can change dramatically with even the subtlest of different numerical inputs. Even with all inputs set to 1, the results for each n-bonacci series produces a radically different result. Notice that as we increase the number of terms, that wave pattern changes dramatically.
But what has this got to do with quantum physics, or how electron work? To answer this, we should understand that the only mathematical model that exists today that can be used to describe quantum phenomena are the Schrödinger equations. However, these are so complicated that even the most simple atoms require vast amounts of computing power in order to produce any kind of accurate results. We cannot even accurately model the helium atom, which only consists of 2 electrons.
Yet, when we examine these equations in simple terms, we find that they are combining waveforms that evolve in 3D space. When added together, these create a diverse range of wave patterns, similar to those found in the Tribonacci Calculator. The key difference it that we only need to input 3 numbers and the calculator will automatically generate that wave function.
The Schrödinger Equations are the basis of Bloch’s theorem, which is used to try and predict the behaviour of various waves that propagate through a lattice. Notice that by the time we reach the 6-bonacci, what starts to emerge is a waveform that is similar to the Electron Waves that are predicted as the periodic function of this theory. Furthermore, if we select the inputs 29 : 26 : 3 = 0, which are the elemental numbers of copper and Iron, then a wave function emerges that is akin to the transport mechanism ascribed to an electrically conductive wave.
This approach to number is based on our new theory of 4D Math, which was only conceived at the beginning of 2022. As such, there is a vast amount of uncharted territory that springs from these ideas. Yet, initial correlations do seem promising, which can offer a simple alternative to the complexity of the Schrödinger approach.
Unfortunately, the moderators of certain mainstream mathematical forums have not taken kindly to this new mathematical system, preferring to ban, censor and delete any questions or presentation of the theory, rather than allow it to be exposed to the wider mathematical community. If you want to try a psychosomatic experiment, you can try and post a question relating to our new mathematical concepts, and see for yourself. But we warn you, expect full scale resistance, name-calling, censorship, anger, and aggression.
Golden and silver Ratios and the magnetic and conductive nature of the elements
Next, let us consider for a moment the ferromagnetic elements. Out of the 3, Iron (26) produces the strongest magnetic field, followed by Cobalt, and then Nickel (28). This can be gauged in emu’s per cm³. The results of each bear a striking similarity to geometric ratios. This is particularly apparent for Iron (26), with a field strength of around √3, or even more accurately, √0.5+1. This can also be express as the Silver ratio but substituting the +1 for +2, and then dividing by 2. Cobalt (28), roughly √2. Nickel exhibits a much smaller magnetic force that is roughly √2÷3. This means that the difference between the magnetic strength of Iron (26) and Nickel (28) is ((√2+1) x3)÷3. The Silver Ratio is expressed through the magnetic properties of these elements.
However, this geometric similarity does not end there. Each is also electrically conductive to a certain degree. When measured in cm Ω, and scaled to the appropriate magnitude, we find the Iron has a conductivity close to one, followed by Nickel (28) with a value around (√2), and finally Cobalt (27) around √3. These are the ratios found between the In, Mid, and Out-sphere of a Cube.
Similar to the ferromagnetic atoms, 3 element stand out as exhibiting conductive properties. After Silver (47) is the most conductive, followed by Copper (29) by just the smallest degree. Measured in cm Ω, both values approximate 1/Φ, or 0.618, raise by a power of 10, (which becomes the ratio 5√5-5). Notice the similarity between the golden ratio, formed from √5, whereas the silver ratio is based on √2. Scaled to this magnitude, the conductivity of Zinc (30) that comes directly after Copper, becomes close to Φ, or 1.618. Gold (79), the 3rd most conductive element, has a conductivity of 4.52, which is close to Φ²×√3.
Comparing these results, we can begin to see that the Golden Ratio seems to be expressed through these electrically conductive elements, with Zinc (30) scaled to a value of Φ, whereas the Silver Ratio is found in the Magnetic nature of Iron (26), Cobalt (27) and Nickel (28). This in centred around √2 in Cobalt (27), whose conductivity is √3. In the theory of Atomic Geometry, these 27 protons represent a Cube (3³) whereas it’s 32 Neutrons form the corners of the 5D Cube, or more accurately the Rhombic Triacontahedron. The ratio between its magnetic and conductive properties are in a ratio of √2:√3, which is the difference between the mid and Out-sphere of Cube.
It is an interesting fact that Cobalt has a greater conductive capacity than Zinc (30). On the table of element ordered by their nature of conductivity, Cobalt (27), Zinc (30), and Nickel (28) appear together in sequential order. Applying our Geometric ratios, the sequence of conductivity descends from √3, to Φ, and then √2. Notice that the ratio gets reduced from the out-sphere of the cube (√3) , into the mid-sphere (√2), stopping at phi (Φ) as it passes through zinc (30).
When we place the two datasets together, we can begin to see that beginning with a ratio of 1:√3, for Iron (26), the ferromagnetic properties drop from √3 to √2, as the conductive capacity of Cobalt increases to 1 to √3. After this, the magnetic properties of Nickel (28) drops off considerably, as the conductivity falls to √2.
This is where there is a considerable change in the relationship between electron and magnetic forces. Copper (29) has no ferromagnetic qualities, but has an unusually large conductive capacity. This occurs as the D-orbital shell suddenly complete prematurely, removing an S-orbital electron from the outer shell. After this, Zinc (30) has a much lower conductivity of around Φ. Clearly, this mysterious jump must arise from some particular process in the atomic structure, that has its foundations in the Φ ratio.
If we imagine, for a moment, that the Electro conductivity of Copper were to fall to 1÷Φ (0.618), then we can see that the mysterious peak is amplified by a factor of 10. What this suggests is that the transition from magnetic to conductive elements can be attributed to the Phi ratio, which can somehow amplify the conductive field, providing copper with its strangely large conductive capabilities. It is interesting to note that Carbon (6), which is often considered to be an insulator, can also be a good conductor of electricity when organised into certain geometric crystal lattices. Yet, it only has a conducive value of 0.0061 cm Ω, which is 1÷100Φ, at the magnitude presented in the model.
When we place these Elements along with Silver and Gold, in order of conductivity, we notice a curious curvature. The Elements in the D-orbital row descend from √3, ending at Φ with Zinc (30). We notice that both Silver (47) and Copper (29), which are the most conductive, exhibit a value of around 1/Φ cmΩ, raised to a power of 10. Sliver (47) appears directly in the same position in the D-orbital set above copper, and also exhibits an unusually large radius, which happen to have a radius of Φ.
In the same place in the 3rd D-orbital set, we find Gold (79). This has a slightly lower conductivity, which is close to Φ²×√3. This also has a value that is close to the Silver Ratio (√2-1) multiplied by 20, and divided by the Tribonacci constant (η). As these 3 element exhibit a much larger conductivity than other on the periodic table. We can begin to see how Gold represents a particular unison, where the Golden, and Silver ratios, blend with the Tribonacci constant, and √3.
Brillouin Zones and the Geometry of Electron Waves
Whilst the examination of the waves that propagate through a lattice can be modelled through as a type of waveform, in truth they are distributed in all directions throughout the structure in all directions. To understand the effect that each has within a 3D lattice, we can map the atomic structure onto a 2D plane.
Many of the lattice structures, particularly those of the D-orbital element, are organised into either a body centred (BCC), or face centred (FCC) Cubic configuration. When viewed face on, these will form square arrangements that can be represented on a 2D Grid.
This notion is commonly referred to as reciprocal space, or in quantum mechanics, k-space. This reciprocal lattices are used extensively in the modelling of electromagnetic waves, called Brillouin zones. The nature of electrical conductivity relies not on the particles that from the atom, rather the hidden wave that resonate through the material. By modelling the reciprocal space in a lattice, scientist are able to predict many of the behaviours of different elements in a compound. This animation show how the first 6 Brillouin zones are created on a square grid.
When we examine the construction of this reciprocal space, we find that it expands through a series of geometric ratios. Beginning with a red square, side length 1, the 1st Brillouin Zone forms a larger yellow square rotated as 45°, with a side of √2. After this, the next blue area forms a cross, with its corners removed. This is very similar to the geometric construct for the Silver Ratio. Notice that the 4 missing corners when combined have the same surface area as the inner square of 1.
As the next zone completes (orange), we find 2 squares form, rotated to a similar degree at the square faces of the Snub Cube, which expresses the Tribonacci ratio. These squares are slightly larger than the previous one, with side lengths of √5. These rotated square are formed of right-angled triangles that are in a ratio of 1, 2 and √5. This is one of the ways the Golden Ratio can be constructed. In the final image, the squares form an octagonal shape, that measure a distance of 3 on the x, and y-axis, with sides measuring 1 and √2. This form therefore encodes both the Sliver and Golden Ratio within its stricture.
Using this template generated from these first 6 Brillouin Zones, we can tile a 2D surface. When 4 are placed together, small space form between each unit. At the centre we find the shape of an octagon. As we expand the tapestry, we find these octagonal spaces from a hidden geometric pattern that is the reciprocal to the Brillouin Zone. As the tapestry grows in size we notice that the ratio between zones and spaces between zones is a combination of 2 sequential square numbers. The same mathematical mechanism is also found in the structure of the shells of the atom that also expand through sequential square numbers.
Beautiful as this geometric structure is, it is important to remember that it represents the lattice structure of a compound of elements. Through this framework, energy, i.e, electron waves, are transmitted through seemingly incomprehensible distance, at least from the perspective of a tiny atom.
Besides the Square matrix, there is only one other regular 2D tapestry that can be formed. The Hexagonal Plain is constructed from sets of smaller triangles. It also exhibits a specific geometry related to its Brillouin Zones. However, these can be arranged in different orientations, to produce a variety of tapestries. When the two main types are superimposed, the gaps in the circle disappear, forming the shape of 7 nested concentric circles. This motif is famous in the world of sacred Geometry, and is termed the Flower of Life. Within this discipline, 6-pointed Stars that appear in the centre of the Brillouin Zones is popularly known as the Star of David.
As we expand this hexagon tapestry, it begins to take on an irregular shape. This is due to the Rotation of the Hexagon, We can see the same rotation of the rectangle inside the seed template. As the Hexagon Doubles in size from the original, we find that the centre of each new set of 7 becomes rotated. The distance of the side length of between the centre of each set now becomes √7. Its appearance within the Brillouin Zones of the hexagon structure is an interesting as it has a value very close to ³√(3√33) which appears in the formulation of the Tribonacci Constant.
Each Brillouin Zone is generated around each point on the matrix. By overlaying the template on each adjacent point on the grid, a geometric fractal emerges. This structure can be categorised into 5 main types. The first is a single cell that forms over the centre (yellow). After this, the next adjacent 4 points produce an overlapping motif on the x and y-axis (blue). Notice that the 2 square appear face on in the vertical and horizontal direction, which forms the Sliver Ratio. Where the zones overlap at the centre, they form an octagonal shape that appears over the central point. The 3rd set of Zones are place on the diagonals to the central point. Now the 4 non-rotated squares appear to form a larger square with a relative side of √8+2, double that of the Silver Ratio.
The 4th set of Zones (purple) begins to produce a larger space at its centre. Notice how the non-rotated squares overlap, again forming a motif similar to the Silver Ratio. Finally, the last arrangement, (green), see the non-rotated square align in a 3×3 grid, which is large enough for the square of the original to fit. Only the rotated squares that have a side of √5 overlap each other.
When we superimpose all of these 5 arrangements over each other, the combination of zones produces a motif at its centre formed of 4 small octagons. We explored this diagram previously with through our geometric approximation of the Tribonacci Constant, derived from a fractal of the Silver Ratio. Notice that each Octagon is rotated 22.5° inside the square frame, just like the midsection that rotates as the Rhombic Cuboctahedron Transforms into its Dual. The Deltoid Icositetrahedron is ascribed to Iron (26), which exhibits powerful ferromagnetic properties.
The square lattice only has a single configuration, which generates an Octahedron in its empty spaces. The fractal forms a square tapestry, that is divided into triangles through each midsection. Amazingly, these form the template for the golden Ratio. Thus, the square tapestry seems to exhibit both the Golden and Silver Ratio, which hints towards a deeper relationship to the Tribonacci Constant, through its fractal nature.
The Hexagonal Tapestry, on the other hand, generates an overlapping fractal of the hexagonal array, forming a triangular lattice, that form hexagons, which are finally divided into 12 parts. This can be formed by 4 triangles rotated at 90°, or by 3 squares rotated 120° to each other.
Brillouin Zones and Atomic Geometry
The modelling of Reciprocal space of a 2D plain provides a useful tool for comprehending the electromagnetic wave structures when modelling the Brillouin Zones in a 3D space. Iron will from a body centred cubic (BCC) lattice structure. The three highly conductive element, Copper (27), Silver (47), and Gold (79), all conform to a face centred cubic (FCC) configuration. Each of these lattice produces 2 completely different Brillouin Zones. Irons BCC structure produce a reciprocal geometry of a Rhombic Dodecahedron, which forms the template of the 4D hypercube, whereas the conductive element form a Truncated Octahedron. When we consider this relationship, we find that it is the Octahedron that defines the corners of both Solids. In Atomic Geometry, this shape is applied to the P-orbitals, whereas the Cube is related to the D-orbitals, that defines the geometry of the lattice itself.
In the construct of a 4D Cube, the Rhombic Dodecahedron orientates the centres of 6 other Cubes, which together with the larger Cube that it nest inside, creates the ‘cubic faces’ that form the 4D solid. Similar to the Cuboctahedron that nest 12 spheres in space, 12 Rhombic Dodecahedra can be nested together inside a Truncated Cuboctahedron.
Returning to the square grid of the Brillouin Zones, we can see that a single unit will fit perfectly inside the projection of a Truncated Octahedron. Furthermore, the projection of the Rhombic Dodecahedron will fit over the square units, orientated at 45° to the central square (red and yellow). We can therefore begin to recognise the 3D nature of the Brillouin Zones within its 2D representation.
Therefore, we find that the pattern created by the Brillouin zone on the square lattice is formed of a Truncated Octahedral honeycomb in 3D. Each unit contains a matrix of Rhombic Dodecahedra, that are space 1 unit apart. At the centre of each unit, (the red square), a Cube is found, This can be truncated to form a cuboctahedron, which appears when viewed from its face as a square rotated a 45° within the square of the cube. Over this, the Rhombic Dodecahedron is found, which appears as 4 square arranged into a larger square, again rotated at 45° to the Cube.
Each corner of the Rhombic Dodecahedron falls at the halfway point of the adjacent 4 squares. This is the space where the Cubic faces of the 4D cube are located. Each of these is ‘shared’ between the adjacent cubes in the grid. Thus, we can begin to see how the separation between cubes is also exactly 1 unit, which forms the fabric of the 4D Hypercubic Space. As the Rhombic Dodecahedron is ascribed to the Brillouin Zone of Iron, we can ascribe 4D cubic space to the magnetic field. Each 4D unit of space is encapsulated by the Truncated Octahedron, which is the Brillouin Zone of Copper, and so is ascribed to the electric (conductive) field.
In the theory of Atomic Geometry, Iron is ascribed the form of the Deltoid Icositetrahedron. What we notice is that this form and the Rhombic Cuboctahedron, that produces the antiferromagnetic qualities of Chromium (24), can also be found within the lattice. The shadow projection of each form can be aligned with the rotated squares (purple) that form at the final boundary of the Brillouin Cell. Therefore, the geometric form we have ascribed to Iron and Chromium are both found to encapsulate the 4th Brillouin Zone (orange) of the lattice.
With the geometry of Iron identified within the Brillouin Lattice, we can now examine the reasons for the conductive properties of Copper. This nature is ascribed to ‘Fermi Surfaces’ which appear as spots at the centre of the hexagonal faces of the Truncated Octahedron. Within the Brillouin Lattice, these hexagonal surfaces are found in the space between blocks, (yellow on the truncated octahedral honeycomb diagram below).
When we overlay the matrix created by the Brillouin zones, we find that the evolution divides the area into squares, whose corners are connected be a vector to the midpoint of the opposite side. This is termed the Sandreckoner Diagram, which is an important tool in the sphere of geometry. This simple design is able to divide a square into 4, 9, 16, 25, 49, and 121 smaller squares, and also incorporates the golden Ratio (Φ). The length of each vector is that stretches from the corner of the square to the midpoint of the opposite side is √1.25 compared to a side of 1. Whereas the formula that generates Φ is √1.25±½.
Notice that at the centre of the image are 4 octagonal shapes, (green bottom right below). These form the rotational points for a rotated Truncated Octahedron, which is √1.25 larger than the original square. When rotated, the Truncated Octahedron aligns itself with the Rhombic Cuboctahedron and its dual, as shown above. What we are beginning to see is a relationship forming between the orientation of the Iron Lattice (magnetic field), and the Copper Lattice (electric field) which are unified by increasing the unit size by a factor of √1.25. The ±½, is represented as the relationship at the midpoint of the 4D hyper cubic matrix, i.e. half the distance between each hyper cubic cell on the original grid. Thus, we can see that at the heart of the relationship between the magnetic and electric field is none other than the Golden Ratio.
The 8 sides that are found at the centre of this rotation are the same shape as the ‘holes’ created when a collection of Brillouin Zones are placed in a tapestry. Examination of this shape reveals that is it the orthogonal projection of a Tetrahexahedron, which is the Dual of the Truncated Octahedron. Thus, we find that the Brillouin tapestry is formed of this pair of ‘reciprocal’ geometries.
We can examine the nature of this reciprocal cell, by placing the shadow projection of the Tetrahexahedron in the central space. When we compare this to the Shadow projection of the Deltoid Icositetrahedron, (the shape of Iron), we find a very close match. The Deltoid exhibits a more cubic geometric form. This can be nested around the Tetrahexahedron at 2 different scales, whereby the 1st encloses the sold, and the 2nd touches the inside of the Truncated Octahedron. Together they produce a 4D template, in the same way that the 4D cube is represented by 2 cubes nested inside one another.
When compared to the Brillouin zones, the 1st, (red below), falls into the purple, orange, and green zone, with each corner touching the corner of the yellow zone, which represent the Rhombic Dodecahedron in the Grid. When the Truncated Octahedron is superimposed over the cell, we find the other 4 points appear at the centre of each hexagonal face, exactly where the Fermi Surfaces appear.
The 2nd, larger version, has 4 corners that touch the outside of the square cell, which is where the centre of the Rhombic Dodecahedra are located. The remaining 4 points are now located at the midpoint of the ’empty’ kite shaped spaces that extend away from the centre in each 45° angle. This produces a fractal overlay that unifies the geometry of the Iron atom, (magnetic Field), to the Rhombic Dodecahedral, (4D Cubic) space, and the Tetrahexahedron, which is the reciprocal of the electric field.
In Atomic Geometry, the Antiferromagnetic element, Chromium (24), is ascribed the form of a Rhombic Cuboctahedron. When aligned on the Brillouin Lattice, with either the central square centred on the square face of the Truncated Octahedron, or the outer frames aligned to the square of the cell itself, we notice that does not produce a very good match. Rotating the form does improve the similarity, but still the corners of the Rhombic Cuboctahedron are slightly out of alignment with the grid.
After Iron, the D-orbitals see a slight reduction in
the radii, as the Deltoid Icositetrahedron collapses into the Snub
Cube. As a consequence, the magnetic properties of these atoms is
reduced. The Snub Cube already exhibits the two rotated squares in the centre of its shadow projection, which can be superimposed perfectly over the 6th (purple) Brillouin Zone. However, this creates an outer shell that exceeds the boundary of the Truncated Octahedron. We can reduce the size of the Snub Cube, so the inner square matches the Tetrahexahedron, yet the outer shell provides a slightly awkward fit for the Truncated Octahedron. Only by rotating this smaller form is a direct fit made between the Tetrahexahedron, and Truncated Octahedron. This offers a geometric explanation as to why Cobalt, and Nickel exhibit ferromagnetic properties to a lesser degree than Iron, but have greater conductive properties.
This becomes more apparent when we overlay a Pentagonal Icositetrahedron, the shape of the Copper atom, onto the grid. Suddenly we find that both sizes match the frame of the Tetrahexahedron, and Truncated Cube, and can even be rotated within the square cell without compromising the boundary of the Truncated Cuboctahedron. This unique quality is what we believe gives Copper is unique conductive capabilities. The Pentagonal Icositetrahedron can exist at various scales and rotations that match the Truncated Octahedron, and its dual, which lie at the heart of the electromagnetic field.
Having identified the geometries of Iron and Copper within the single cell units of the Brillouin Zones, we can examine the structure formed over a wider area. A grid composed of 9 units produces a template onto which the Rhombic Cuboctahedron, Snub Cube and their duals can be overlaid. At the centre of the Grid is a Cube around which the of shadow projection of each solid is placed.
The square at the centre of the Rhombic Cuboctahedron encapsulates the 3rd and 4th zone, (blue and orange). Its vertical and horizontal sides touch the boundary of the adjacent cells. As before, the form can be enlarged and rotated, however, this off-sets the corners of the outer frame, bringing them out of alignment with the grid structure.
Its dual on the other hand, the Deltoid Icositetrahedron, can be nested over the Cube to perfect fill the centre square. Even when rotated and projected at a slightly larger size, the sides of come into alignment with the Rhombic Dodecahedron, (Yellow). Notice how in the first position, the corners touch the midpoint of each side of the Rhombic dodecahedron, which, as it grows in size and rotates, starts to align its faces (top row right below).
The Snub Cube forms a similar outer shell to the Rhombic Cuboctahedron, with the shortest of the outer sides running parallel to the 3rd (blue) Brillouin Zone. This form does not rotate like the Rhombic Cuboctahedron, as the square faces at the centre are already rotated due to its snub geometry. This means it will naturally scale in size so that the inner square maps to the Tetrahexahedron. Now, the shorter sides will cut through the middle of the adjacent Cubes.
Finally, the Duel of the Snub cube, the Pentagonal Icositetrahedron, can be placed over the centre cube, filling the same area as the Deltoid. This too can be rotated without interfering with the boundary set by the adjacent Rhombic Dodecahedra.
When we compare these geometries, we can see that the dual tend to produce a better fit at a much smaller scale. They can also be rotated and aligned with the Brillouin Zones, without interfering with the adjacent Rhombic Dodecahedra. As these geometries represent Iron and Copper, we propose that the electromagnetic field is geometrically orientated to accommodate these rotating solids.
The Pentagonal Icositetrahedron is also found to scale to a much larger degree when centred over the ’empty’ spaces in the grid created by the Tetrahexahedron. This comes about by the realisation that the other small kite shaped spaced that are formed between Brillouin Cells are the same shape as those found at the centre of the projection of the Pentagonal Icositetrahedron. This allows us to map small versions of the solid over these points. In the diagonal space between each cross, 4 larger versions of the solid can be placed with corners that touch the kite shapes found in the smaller versions.
Furthermore, we find that the Tetrahexahedral spaces can also be scaled so that the outer corners of the projection now centre in the middle of each cube. Together with the Pentagonal Icositetrahedron this forms an interlocking fractal design. The 4 empty spaces that are not occupied by the Pentagonal Icositetrahedra are filled with small Tetrahexahedra which produce 9 lager versions of the solid centred over the other remaining holes.
These 2 tapestries, create an interlocking foundation that we suggest appropriates the electron wave. A fractal unit that can be scaled perfectly to much larger sizes. Interestingly, the overall structure now appears to resemble the image of a set of P and D-orbitals, where the small Pentagonal Icositetrahedra form the lobes of the P-orbital Octahedron, whilst the larger fill in the spaces in between, such as is the orientation of the D-orbitals.
We can centre this fractal matrix on either the holes, or the cubes on the grid. When centred on the Cube, the larger Pentagonal Icositetrahedra become more spaced out, which removes the overlap between solids. When we consider these differences, we can see that the previous example is more indicative of the BCC structure, with 4 large touching Pentagonal Icositetrahedra surrounding a smaller one at the centre. This second structure creates space at in the middle of the 4 solids, that can house a smaller Pentagonal Icositetrahedron at its centre, which is akin to the FCC structure of Copper.
Furthermore, when we place the Tetrahexahedra at the centre, we find we can create 6 sets in a nested structure, whereby 4 of the corners of the largest solid touches the centres of the small Pentagonal Icositetrahedra, positioned in the vertical and horizontal planes. The remaining 4 corners touch the corners of the Rhombic Dodecahedra. This forms a series of squares whose spacing are defined by the Tetrahexahedra.
We have examined the nature of the proposed geometry of the D-orbital sets and found that the Duels of the Rhombic Cuboctahedron, and Snub cube produce a beautiful yet informative fractal structure. In the theory of Atomic geometry, the other important solid is the Icosidodecahedron, whose 30 corners are ascribed to the neutrons of a number of D-orbital elements, including Iron, and Nickel. When examining its shadow projection, we notice that is falls into 2 types of orientation. When the 2 are superimposed at 90°, the result is an image similar to the Pentagonal Icositetrahedron.
This performs quite a close match to both the Cubes on the Brillouin Zone, and the reciprocal spaces, produced by the Tetrahexahedron. Notice how the triangular faces overlay each other, forming a 6 pointed star, (hexagon), on the diagonal of the plane of the cell. Amazingly, we find that 6 of these can be placed in a ring around a central one, which is indicative of the hexagonal plane. This comes in two types of orientation, forming a ring of 12. In Atomic Geometry, the F-orbitals are formed from exactly this kind of geometric arrangement. Therefore, within this fractal structure, we are able to identify the blueprint for all the different orbital types.
Having examined the fractal structure of the Brillouin Zones, and the Atomic Geometries found within, we can return to examine the structure of the D-orbitals in greater detail. Of particular interest is the transformation from the Rhombic Cuboctahedron, and Snub Cube into their dual and the effect that have on the relationship between the Cube and Octahedron, which compound to form the Rhombic Dodecahedron.
In the example of the Snub Cube, the centre of each square face becomes the corner of the dual. These are the corners of an Octahedron, which now produce the out-sphere of the dual. In effect, the Cube becomes reduced in size, as the octahedron take prominance within the solid.
In our post on the D-orbital Geometries, we explain in great detail how the Truncated Octahedron and Tetrahexahedron can form at the centre of the Rhombic Cuboctahedral model of the D-orbital set. This begins with a Cube, side 1, which is exploded to for the Rhombic Cuboctahedron. This creates a 2nd cube, side √2, that nests inside with its corners touching the triangular faces. When the Rhombic Cuboctahedron transforms into the Deltoid Icositetrahedron, the mid and in spheres exhibit a radius of 1.3Å, and 1.22Å respectively. As the out-sphere of the Cube (side √2) is also 1.22Å, the ratios of the D-orbital model does not change. However, the Octahedron now forms that has a side of roughly 2, which compounds perfectly with the √2 Cube, forming the Rhombic Dodecahedron. Thus, we find the structure of Iron now produces the same geometry as the reciprocal space found in its BCC structure.
With the next elements, we find the radius drops to 1.35Å, as the Snub Cube forms, this causes the Octahedron to diminish slightly in size, hence these elements begin to lose the ferromagnetic qualities. Consequentially, the mid and in-spheres are reduced in size to, 1.24Å and 1.157Å. The mid-sphere of the Rhombic Dodecahedron is around close to this at 1.154Å, which produces the size difference between the 2nd P-orbitals with an average radius of 1Å and the 3rd with an average radius of 1.54.
When the Snub Cube Transforms into its dual, the Octahedron again grows in size as the corners push through the centre of each square face. This Octahedron has a mid-sphere of 0.94Å, which when truncated produces an out-sphere of 1, aligning with the sphere created by the Cuboctahedron in the original D-orbital model. The Truncated Octahedron forms the FCC Brillouin Zone, which is responsible for the extremely conductive properties of the copper atom.
This hypothesis comes through a much wider investigation into the geometric nature of the atomic structure, and its influence on the fields that are created within the crystal lattice structures. Each element exhibits a unique Brillouin Zone, which is found within its lattice structure. The 3 most conductive elements, Copper, Silver, and Gold, all exhibit a similar truncated octahedral zones, with roughly the same sized Fermi surfaces. When we examine the Brillouin Zones of the last 5 D-orbital Elements, we can see the geometric nature of these fields emerging. Iron creates the exterior frame of an octahedron, which is quite unique amongst the various elements. As zinc falls into a hexagon close packed (HCP) arrangement, is fails to produce the same level of conductivity as Copper. Its 30 protons are the perfect match for Icosidodecahedron, which shifts the lattice structure towards the hexagonal plane. Although Zinc (30) is also ascribed the same form as Copper, the Pentagonal Icositetrahedron, you might think that it should also exhibit the same conducive properties. However, as it has 30 Protons, it instead aligns with the 30 corners of the Icosidodecahedron, which is why it too exhibits a HCP crystal structure, which limits it conductive capabilities.
This geometric solution to the nature of magnetism and conductivity is the first of its kind, and is based on the theory of Atomic Geometry, which is the only atomic model in existence that can give precise reasons for the experimentally determined radii of the stable elements. Through this investigation, we can also explain the reasons for the magnetic and conductive capabilities of every atom, which unifies notions of the waveform proposed by Bloch’s Theorem, with the geometric nature of the Brillouin zone, whilst offering a set of simple tools that we believe will lay the foundations for a new era of solid state physics, and the production of new smart materials that will play an important role in overcoming the major problems, such as Boltzmann Tyranny, facing the technological development of our modern civilisation.
What does this tell us Electromagnetism?
Whilst it is a common misconception, that electrons are particles that travel down a wire, which produce the electrical effects, when we examine this theory in greater detail we begin to see that the theory is flawed. Instead, what is produce is a wave that travels through the lattice structure. Furthermore, we can more accurately describe this nature in terms of geometry, which, unlike the Schrödinger equations, is relatively easy to calculate. What emerges is that the Magnetic filed is governed by the Silver Ratio, whereas the electric field is formed by the golden ratio. Where these combine, we find the perfect interface between the two, which is expressed through the Tribonacci constant. When calculated using negative terms it produces wavelike qualities that find a close similarity to Block Waves, that lie at the foundation of the cutting edge of the theory of solid state physic.
A Geometric approach to the Electromagnetism
The Atomic Geometry model is the only one in existence that can resolve many of the inaccuracies that are common place in the present theory of electromagnetism. This includes providing a precise reason for the nature of paramagnetic, diamagnetic, antiferromagnetic and ferromagnetic qualities of every stable atom on the periodic table. It offers a clear unification between the accepted theories of Brillouin zones and Fermi surfaces, that when express in geometric terms provide a refreshing new set of mathematical tools, which can be modelled using a computer, or even produces as physical 3D solids. This makes Atomic Geometry both simple to implement, yet powerful in its application.
Carry on Learning
The wave model of matter was established in the 1920s, yet has always been accompanied by the belief it can also act as a particle. In our new 4D view clearly defines the notion of mass and charge that is presently missing from the mainstream scientific perspective, offering an intriguing insight into the underlying foundations of reality,
YOUR QUESTIONS ANSWERED
Got a Question? Then leave a comment below.
Wow, but I thought that electrons are moving particles. In school, everyone is taught this, surely this must be the case. After all, we have equations that prove it is true?
It is important to remember that a model can be formulated that works quite well for many situations. The reason the electron particle model is still taught today is because it is believed to be easier to understand than the wave model. Howver, is was disprooven, and today we understand that it is more accuratly defined by the wave model of the electron. What is new in our theory, is that as we have resolved the problem of the photoelectric effect, and ultraviolet catastrophe, without the need for any kind of particle, we have re-established the notion of the 4D Aether. Once introduced, the wave only model of electricity becomes possible, as now the very nature of quantum spin is attributed to the 4D rotation of the atom. Whilst particle physics claims to offer extremely accurate results, this is actually a superficial statement. For example, the hydrogen atom is supposed to exhibit a radius of 0.53Å, according to the Bohr model, whereas and the experimentally determined radius is only 0.25Å. In fact accurate prediction of the other elements also tend to fail.
Great article, but why do you believe your Tribonacci Calculator can replace the Schrödinger equations?
In science, there are a variety of different models that are used to explore universal phenomena. We suggest the Tribonacci calculator can offer new insight into the nature of the conductive and magnetic waves. For the purposes of this article, we have had to remain brief. However, you can download the Calculator and try out some experiments for yourself. For example, if you punch in the side lengths of the Deltoid and Pentagonal Icositetrahedron, you will see resonant peaks at 141 iterations, which is roughly equivalent to √2, i.e 1.41 x100. Other inputs often produce peaks at 38 which are the number of corners for the geometry of copper. As we mention, the field of 4D maths is still in its infancy, yet initial results to show an interesting correlation to the conductive and magnetic nature of the ratios we have covered in this article. As the filed develops, so we expect that the exact reasons for the different types of waveform generated by the calculator should become clearer. This is our conjecture, not a proof.