P-orbitals are the second type of electrons to appear in the atomic structure. These appear as two lobes positioned either side of the nucleus. They always appear in sets of 3. In this article, we will explore the geometric reason for this configuation.
What is a P-orbital?
The electron cloud surrounds the atomic nucleus, and is composed of different shells, found at a specific distance from the centre. These are sometimes expressed as sets of concentric rings, each of which contains a certain number of electrons. Once a shell is filled, subsequent electrons begin to fill the next shell. This diagram of the atom is called the Bohr model, and is a popular expression, still used by the majority of Chemists today. However, a more accurate description of the atom is produced by the S, P, D, and F model, based on the Schrödinger equations. This reveals that each shell is composed of 4 different orbitals types, each of which exhibits its own geometry.
These orbitals are each composed of 2 electrons that are assigned either an UP or DOWN quantum spin number. As each shell forms, another set of orbitals is added to the set. The first shell only contains an S-orbital. The second shell contains both an S-orbital and a set of P-orbitals, and so on. This pattern continues up to the 4th shell, which contains all 4 orbital types. In the 5th shell, only a set of S, P and D orbitals form. In the 6th shell, the S orbitals combine with only 3 P-orbital atoms, before the radioactive elements begin. This means there are 4½ P-orbitals sets that account for 27 elements on the periodic table.
P-orbital account for 6 elements in a single shell, and are always found on the right-hand side of the periodic table from groups 13 to 18. The final element is called a noble gas, as it will not form bonds with other atoms. Except for Helium (2), which is formed of an S-orbital, all nobles gases arise once the full set of P-orbital elements complete. This is important as it structures the shells in the electron cloud. Each noble gas signifies the end of one shell, after which electrons begin to fill the higher shells. Note that D (blue), and F (Green) orbitals do form in the same shell as a complete set of P-orbital elements, but only once the shell above is filled with other S or P-orbital electrons. For more information, see our post on Harmonic Chemistry.
Orbitals are comprised of two opposite nodes, each of which can contain an electron with an UP or DOWN quantum spin number. The nucleus sits directly in the middle of the orbital. P-orbitals are sometimes described as being ‘dumb-bell’ shaped. The electrons in each lobe continuously ‘swap’ places in quantised steps, without any in-between stages. Electrons are always UP or DOWN, and magically jump from one side of the orbital to another. This quantisation of electron spin is covered in more detail in our article on the 4D electron cloud.
The P-orbitals are the second type of orbital to appear after the S-orbitals, which is the only one to resemble the sphere. When we compare the two types, we can see that the S-orbital splits into 2 to create the P-orbital type. This is similar to the way cells divide at the biological level.
In Atomic Geometry, we view the electron cloud as a 4th Dimensional entity. Just like the P-orbita, an S-orbital is also comprised of two electrons. From the perspective of the 4th dimension, this can be viewed as a 4D sphere, which appears in 3D as two spheres occupying the same space. When rotated on its time axis, the spheres swap places. The UP and DOWN spin values are considered to be the opposite field orientations of these 2 spherical bodies that comprise the 4D form. P-orbitals are also considered to be 4D in nature. However, unlike the S-orbitals, each sphere is off-set to one side of the nucleus in 3D space. This is called a 4D Torus, which you can read more about in our article on Orbital Geometry.
Many people still hold the outdated view that the electron is a some kind of particle that is rotating around the nucleus, like a planet orbits the sun. This idea was discredited back in the 1920s, when the electron cloud model of the electron was being refined. At the end of this period, the concept of probability was introduced to explain the shortcomings of the Bohr model, and to reconcile the wave like nature of matter.
This describes the electron cloud using a graph that has the distance away from the Nucleus defined on the horizontal axis, and the probability of finding the electron at that location in the vertical axis. The probability distribution of the first P-orbital shows a simple curve, that starts with zero at the atomic nucleus, and peaks in a single location, denoting the most probable location for the electron. When we consider this from a 4D geometric perspective, we find it is represented by a horned torus. The cross-section of the torus shows the two P-orbital lobes sit adjacent to each other. Note that the distribution graph only defines the radius of the electron cloud, whereas the Torus shows its diameter.
As subsequent P-orbitals appear in each shell, we find additional peaks begin to form in the probability distribution curvature. The second P-orbital that appears in the 3rd shell exhibits two peaks. Therefore, the torus widens to accommodate an inner and outer ring. The outer peak can be perceived as a ring torus, similar to a doughnut with a hole in its centre. Whilst not often portrayed on the probability distribution graph, the inner peak is in the inverse spin state to the outer. However, as negative probabilities do not exist, the peaks generally only appear in the positive.
As each of the P-orbital sets begin to fill subsequent shells, so the number of peaks increases for each one. The 3rd P-orbitals exhibits three peaks, and the 4th four. The final P-orbital only forms halfway before the radioactive elements begin. Therefore, these elements only contain a single electron in each P-orbital.
Note that the 1st P-orbital appears in the 2nd shell of the atom. D-orbitals appear in the 3rd, 4th and 5th shell and F-orbitals appear in the 4th shell only. For most of this article we will refer to the P-orbitals by their order of appearance, rather than the shell in which they appear for simplicitie’s sake.
P-orbitals appear in sets of 3, each with an x, y or z orientation, which defines the axis of 3D space. This forms a cross that exhibits a 90° angle at its centre. If we connect the adjacent end points to each other, it forms an Octahedron.
The Octahedron is one of the 5 Platonic Solids, that are the only regular 3D forms that can be constructed. Comprised of 6 corners and 8 triangular faces, it is the only solid that incorporates both the square and the triangle in its construction. This can be clearly seen when viewed from either its corner or its face. The square and triangle are the only regular 2D shapes able to tile the plane with just two colours. Geometrically, this gives the Octahedron the unique quality of combining theses two shapes in 3D space.
The final P-orbital element that completes the octahedral set is always a non-reactive noble gas. Present atomic theories do not offer a specific reason for this, other than the P-orbital set is complete. From the geometric perspective, this is explained by the formation of the octahedron itself. Octahedra can be compiled in sets of 6 to form a larger version of itself, leaving tetrahedral spaces at the centre of each face. Electrons can appear within the octahedra, but not in the tetrahedral spaces. This fractal structure doubles in size with each iteration, each containing the previous set.
This structure is also found in the nature of Electromagnetic waves. These consist of two fields, orientated at 90° to each other. As each expands and contracts as it propagates through space, they can be mapped to an octahedron. We explore this notion in great detail in our wave solution to the photoelectric effect. We suggest that at scales far smaller than the atom, the Quantum foam is structured through an octahedral fractal. From this view, the octahedral nature of the P-orbitals is present, even before the electrons begin filling the different shells.
By way of explanation, we can image an empty glass that is invisible until it is filled with water. The glass can only contain a certain amount of liquid before is overflows. The glass sit inside another lager glass, that catches any liquid overflowing from the first.
In this analogy, the glass is a fixed form, however, this is not true of the electron cloud. Once a certain energy level is reached, the next orbital form of two peaks, which indicates that the ‘glass’ has morphed into two different sized containers. This second model explains both the order and formation of the S, P, D, and F Orbitals, and the nature of successive orbitals to form an extra peak in the probability distribution graph.
The concept of 4D geometry is not as complex to grasp as it might first appear. The simplest case of a 4D hypercube, can be envisioned as one cube nested inside another. A similar concept applies to other 4D polytopes, such as the Octaplex, which is described as a Rhombic Dodecahedron nested in a Cuboctahedron.
The glass used in the previous example can be perceived as a 4D entity, that changes shape in accordance with the principle of least energy. In chemistry, this concept is called hybridisation and appears in the field of molecular geometry. The same idea is applied here to a single atom. This introduces a new perspective of the atom that considers the geometry of both the nucleus and electron cloud as a single 4D phenomenon.
All atoms are emersed in vacuum energy, a quantum foam that fills space at the smallest scales of reality. The quarks in the atomic nucleus exhibit a 1/3 spin, which increases the energy density forming the boundary of the proton or neutron. The 3 quarks of each nucleon collectively exhibit a 1/2 quantum spin value, which is reflected in the nature of the electron cloud.
This view does not see electrons as small particles appearing and disappearing magically as they move from one shell to the next. Moreover, the electron cloud is a 4D energy field. This identifies quantum spin values as the rotation of 4D geometric space. More information regarding the formation of boundaries at the atomic scale can be found in our article on 4D Quantum Foam.
As noted previously, each P-orbital can be viewed from the perspective of a 4D torus. When the entire P-orbital set is considered, we find that each lobe is formed where the two of the torus fields intersect. The ring of each torus follows the edge of the octahedron.
This offers an intriguing explanation as to how the electron can magically move from one lobe of the orbital to another. A torus can be composed by rotating a 2D circle around a central point. When two circles combine in 3D space orientated at 90° to each other, it forms the geometry of a sphere, divided into 4 parts. A 2D object does not directly manifest in a 3D space, as it lacks a third dimension, therefore has no thickness to it. A piece of paper, which is often used as an analogy for 2D space, is not a 2D artefact, as it has a thickness. Only when two or more 2D objects combine at different orientations are collective able to exhibit the dimensions of 3D space.
In the case of the P-orbitals, the combination of torus fields produces a lobe at each corner of the octahedron. Due to the geometric nature of the polygon, each is composed of 2 circles, orientated at 90° to each other. As the circles rotate around the torus, they intersect, and the electron ‘appears’ in that particular lobe. However, when they do not intersect, they disappear out of 3D space, and so are unable to be measured, giving the appearance that the electron has magically jumped from one side of the orbital to the other. From the perspective of 4D, the electrons are moving outside the 3D dimension.
The torus (left) is made of a circle rotating through a second with a central axis. Both maintain an angular relationship of 90° to each other. When two toruses combine (centre), it forms two lobes at either side of the torus. The circles at the centre and the two lobes are defined by two circles both orientated at 90° (right).
Electrons fill the orbital sets in a particular order. First, each electron occupies a different orbital until half the set is filled with all UP electrons. Subsequently, the remaining lobes are filled by the DOWN electrons to complete the set. Present theory can offer no explanation as to why this should be. The torus model suggest this is because each lobe is formed from the fields of two interlocking toruses. When the second electron fills the lobe, a third torus is added to the octahedron, which completes the entire set, ready for the next electron to fill the 3rd lobe.
This follows a certain geometric pattern. The first P-orbital creates a single lobe, the next forms 2 lobes indicative of the line, and the third creates a triangle. As the electrons move in resonance between opposite lobes, so the triangle appears on one side of the octahedron and the other. The orbitals have transitioned from, a dot (0D), to a line (1D), to a triangle, (2D) before completing the 3D form.
Once the triangle is created, the next electron fills an opposite a lobe with a DOWN spin, appearing in the inverse, to produce the first 3D orbital structure, the half pyramid. The next electron completes the pyramid, with the final electron completing the octahedron. All regular polyhedra have an even number of faces and corners, which is due to the nature of 3D space. Similarly, the electron cloud produces orbital shapes that create one half of the structure, before producing a mirror image with the remaining electrons.
This geometric model of the electron cloud offers a compelling explanation for the mysterious qualities of the quantum world. Viewed as a 4D field, we can explain the reason for the P-orbitals sets to form an octahedral structure, and how electrons are able to move from one lobe to the other without ‘rotating’ around the nucleus. It suggests their appearance is governed by the intersection of two torus fields, rotating in unity. The ‘quantum field’ itself is formed from intersecting 2D planes, which defines 3D space.
This goes against the common assumption that 2D space can only exist inside a 3D space. Instead, we see that 2D space combines to form a 4D space, which defines 3D space. As the octahedron is the only solid that can embody both the equilateral triangle and square within its structure, it naturally arises as the first solid in the electron cloud, after the sphere. Our article on the 4D nature of the electron cloud covers this concept in greater detail.
The radius of each atom is not a clearly defined value. The electron field expresses regions of less density as it moves towards the orbital boundary. Furthermore, atoms cannot be individually isolated and measured like objects in our everyday experience. They can be placed into a crystal lattice or molecule with other atoms of the same type, and their radius determined by halving the distance between each nucleus.
Other methodologies are calculated from theoretical models. The two most common of these are the Bohr and Van dar Waal Radius. Each of these expresses wildly different results from each other. The Van dar Waal Radius is much larger than the Bohr model, as it tries to treat the atom as a solid sphere. The Bohr radius is based on the notion of electromagnetic forces, that attract the electrons towards the nucleus. As the atom increases in atomic number, so more electrons and protons are added to the nucleus.
The increased size of the atomic centre causes the electrons to contract more towards the protons, which reduces the radius of the atoms across each row of the periodic table. Once the P-orbital forms, the radius expands in size as a new pair of S-orbitals appear in the next higher shell. This leads to a generalised rule, that atoms get larger for each row (shell) of the atom, but smaller moving right to left along each row.
The problem with this idea, is that it is not true of all datasets. Notice that in the images above, that noble gasses at the end of each row are depicted as being much larger in table 2 compared to table 1. This difference arises due to the fact that they do not form bonds with other atoms, and should therefore exhibit a larger Van dar Waal radius. The Bohr model, on the other hand, suggests the radius should be smaller as there are more protons in the nucleus compared to the previous element. Ignoring the uncertainties of the radii for the noble gases, we can compare the experimentally measured values of the other P-orbital elements to those predicted by the Bohr model.
Values derived from Wikipedia atomic radius data page. Note: experimental data is accurate to ±0.05Å.
In some cases, a close match is found. However, the rate of decay for the P-orbital elements is much less than the Bohr prediction. Furthermore, there are quite a few examples where the radii of certain elements remains the same. The 3rd, 4th, and 5th element in the 2nd P-orbital set all exhibit a radius of 1Å. Similarly, the same elements in the 3rd P-orbital set all have a radius of 1.15Å. In the 4th set, the elements level out at 1.4Å. In each case, the Bohr Model continues to predict a smaller radius, which pushes its value outside the margin of experimental error.
Before we consider the geometric model of the atom in more detail, we should review some of the principles of 3D polyhedra. The regular solids each define three types of sphere. The In-sphere is formed from a sphere that touches the centre of each face. Slightly larger is the Mid-sphere, that touches the mid-point of each side. Finally, the largest is the Out-sphere, or Circumsphere, that encompasses the whole form, touching its corner points.
The in-sphere and out-sphere of the Octahedron and Cube are exactly the same size. However, the Midsphere is different due to the Octahedrons triangular sides, which makes it slightly smaller than the square face of the Cube. The ratio of expansion from the In, Mid, and Out-sphere is √1.5 then √2 for the Octahedron, which is reversed in the Cube. We can see that the ratio, 1, √1.5, √2, and √3 are key intervals in the geometric nature of both the Octahedron and Cube.
Geometric Ratio of the P-orbital Radii
We can re-examine these curiosities of the P-orbital radii through the lens of geometry. What we find is that most of the radii can be expressed in terms of ratios that are found within the In, Out, and Mid-spheres of an Octahedron or Cube.
As each orbital set begins to complete, its radius seems to settle at a specific value. This is most notable for the 2nd and 3rd P-orbital sets, where elements 3, 4, and 5 all exhibit the same radius. By the 5th element, we find that the radii across all sets unifies into a geometric ratio, starting a 0.5Å through to √2Å.
We find there are also some radii that appear slightly larger than the geometric ratios. These appear in the first half of the P-orbital set. For example, the radius of the 2nd P2 element does not correspond directly to a geometric ratio. However, the value 1.1 happens to be close to √1.25, which is the square root of the preceding value, and is key in the formation of the Golden ratio.
Similarly, the 1st P4 element’s radius of 1.55 can be found by dividing √3 by √1.25. These orbitals appear in the shell above the F-orbital set, whose radii happen to settle at 1.75Å, close to √3. Subsequent elements in the shell seem to average out with a radius of √2, with a slightly increased radius of 1.45Å for element 2 and 3. We can apply the value √2 to these elements, with a possible explanation for their enlarge size stemming from the previously formed D-orbital sets, whose radius generally varies from 1.45Å to 1.3Å, and which are found between the P2 and P4 orbital sets.
We can compare the geometric perspective of the P-orbital radii to the Bohr Model. The experimental data is accurate to within ±0.05Å. We can add this to the graph, (yellow strip), to show where any calculated results stray beyond an acceptable value.
We can see, the Bohr model consistently tails off towards the end of each set, predicting a radius that is outside the margin of error for the experimental results. The geometric results, on the other hand, remain within the tolerance of the experimental values. Presently, there is no theoretical model of the atom that can accurately predict the radius of the elements on the periodic table. Yet, once viewed from the perspective of geometric ratio, a new picture begins to emerge.
Octahedra structure of the P-orbitals
Now that we have established a connection between the atomic radii and geometric ratio, we can perform a few simple experiments and to see if we can find a clear explanation for each element. To begin, we can recognise that each most of the orbital sets tend to settle at a particular radius. The exceptions are the first orbital set, that continue to reduce in size to 0.5Å. Hydrogen (1), the first element on the periodic table, has a radius of 0.25Å. Whereas the 2nd P-orbital set levels out with a radius of 1Å. This creates a doubling pattern, just as should be expected from sets of octahedra nesting together to form ever larger structures. The 3rd P-orbitals appear after the formation of the D-orbital set. This seems to interrupt the doubling sequence. Instead of a radius of 2, the 3rd P-orbital is scaled by a factor of √3, to produce a final radius of 1.154, (2÷√3). The 4th P-orbital set appear after a second set of D-orbitals. Both exhibit a similar radius of √2.
As the P-orbitals exhibit an octahedral structure, we can use each of these radii to produce a circumsphere that encompasses an appropriately sized Octahedron. This creates two more possible radii for the mid and In-sphere of each octahedra. We can also produce another dataset by aligning the P-orbital radii with the In-sphere of the octahedron. This creates a set of larger values for the Mid and Out-sphere.
By analysing the data created from this set, we see the foundations for the structure of the electron cloud starts to be defined. An Octahedron with an in-sphere of 0.5Å defines the radius for the 1st P-orbital element, Boron (5), 0.866Å. Additionally, the radius of Oxygen(8), 0.612Å, is produced in its mid-sphere. The radius of the remaining two elements, Carbon (6) 0.707Å, and Nitrogen (7) 0.66Å are found contained within the mid-sphere of the Octahedron enclosed in a sphere of 1Å, and the in-sphere of the Octahedron enclosed in a sphere of 1.154Å, (highlighted in red). Therfore , all the radii of the 1st P-orbital elements, excluding Neon (10), are defined by the ratios found in the first 3 octahedra.
When the In-sphere of the Octahedron is set to 1Å, its mid-sphere is 1.22Å, (highlighted in green). This defines the radius of the first and last elements in the 2nd P-orbital set. Furthermore, the out-sphere of 1.732 (√3), roughly defines the F-orbital set. Therefore, all the 2nd P-orbital radii, except the 2nd element, Silicon (14), are found in the Octahedron with an in-sphere of 1.
The 3rd P-orbital Octahedron does not double in size. Instead of measuring 2Å, the radius is reduced by a factor of √3, 1.732, found in the out and mid-sphere 3rd and 4th Octahedra. The radius of the 2nd element in the 3rd P-orbital set, Germanium (32), 1.22Å, is also defined by the mid-sphere of the orbital below. As the final elements all exhibit a radius of 1.54Å, the size of all the 3rd P-orbital elements, (except gallium (31), 1.3Å), are found within the dataset. Interestingly, the out-sphere of 2Å is the radius of Strontium (38), an S-orbital Element that appears directly after these P-orbitals complete.
The D-orbitals generally appear around the 1.4Å mark, which also the radius of the 4th P-orbital Octahedron. This value also found in the mid-sphere of the 3rd P-orbital set.
Finally, the 4th pair of P-orbital octahedra defines the ratios 1, √2, and √3, within their structure. The number 1 is the radius of the 2nd P-orbital set, √2 defines the 4th P-orbital set and the average D-orbital radius, and √3 defines the radius of the last 7 F-orbital elements.
From the 20 P-orbital radii measured, 17 are found within this simple octahedral model. The three radii not accounted for include, Silicon (14), radius 1.1Å, Gallium (31), radius 1.3Å, and Indium (49) radius 1.55Å. However, it is worth noting that these elements are all found at the beginning of the P-orbital sets, whereby only one or two orbital lobes have appeared. Only by the 3rd set do enough orbital lobes form to generate an octahedron. This can also be noted in other orbital formations, such as the first F-orbital elements, Lanthanum (57), and Cerium (58), which form cross shaped D-orbitals, rather than hexagonal shaped F-orbitals.
This suggests there is a geometric relationship between the sizes of the different P-orbitals. When considered from the perspective of the ’empty glass’ model of the atom, the structure of the electron cloud already exists, even prior to the formation of a particular P-orbital set. The radius of the 1st orbitals are therefore defined by the others in the shells above, even before the electrons begin to fill each shell. This is a completely new concept, which unifies the structure of the electron cloud as a single geometric construct.
Nested cubes and octahedra
The Cube and Octahedron are called ‘Platonic Duels’, as each exhibits the same number of corners to the others faces. An octahedron can be nested inside a cube so that each of its 6 corners defines the centre of the cubes faces, and vice versa. This means that the in-sphere of the octahedron touches the centre of each face in 8 locations, which defines the corners of a cube. The same can be said of its out-sphere.
From this view, we should view the In and Out sphere of an Octahedron, as a Cube, rather than another octahedron. We can perform a similar geometric experiment as the one previously conducted with only an Octahedron, but this time incorporate the nested with a Cube. As before, we can begin with a sphere of radius 0.5. This provides two types of data sets, that begins with either the Cube or Octahedron. For each set, both the in and out-spheres are the same. Only the mid-sphere changes for each set. We can begin by compiling the structure with an Octahedron or Cube, each with an in-sphere of 1. This produces two sets of radii.
When we examine the results, we find that the radii of 1st P-orbital set are clearly defined (bold). The exception is Nitrogen (7), with a radius of 0.65Å. Furthermore, all the results above 0.866, where the 1st P-orbitals begin, define the radius of 7 out of 12 of the stable S-orbital elements. The exceptions include, Hydrogen (1) 0.25Å, Lithium (3) 1.45Å, that form before the P-orbital set.
The radii of the two S-orbital elements, Helium (2) 1.22Å, and Beryllium (4) 1.061Å that form before the P-orbitals, appears at the centre of each table. Sodium (11), and Magnesium (12) have a radius of around 1.8Å and 1.5Å respectively. These appear between the formation of the 1st and 2nd P-orbital sets. The Magnesium (12) radius appears in two locations within both the Cube and Octahedral based structures. This element is key to the process of photosynthesis, and features in our wave solution to the photoelectric effect.
The two largest radii provide a close match to the final set of stable S-orbitals elements, Caesium (55) 2.6Å, and Barium (56) 2.15Å, which are the largest on the periodic table, defining the limitation of the electron cloud. This subject is covered in great detail in our article of S-orbital geometry. When we compare these results to the experimentally determined values and the theoretical Bohr model, we find a much greater correlation using this simple geometric interpretation.
Comparison of the experimentally determined radii compared to the Geometric and Bohr model
We can apply the same technique to the 2nd and 3rd P-orbitals, basing the structure on a radius of 1 and 1.154, to produce another set of radii. This defines the sizes of another 2 S-orbital elements, Lithium (3) 1.4Å, and Strontium (38) 2Å. The Lithium (3) radius is the same size as the 4th P-Orbital set, and also defines the average size of the D-orbitals. Finally, the radius of the F-orbital elements, which start at 1.85Å for the first half, dropping to 1.75Å for the second. Both sit just above 1.83Å, and 1.732Å found within the data sets, as the mid-sphere of the largest Octahedron from the previous 1st P-orbital set, and the in-spheres of the 2nd P-orbital sets.
These 3 dataset span the radius from the 1st to the 4th P-orbital sets, and define the Average radii of around half of the stable elements found on the periodic table. Aside from a few S-orbital radii, and certain D-orbitals that vary slightly from 1.4Å. The results from an impressive match to the experimental data set. This is particularly notable for the D and F- orbitals, that the Bohr mode predicts should be much larger. We can graph the data and compare the results.
From the spacing generated from the Cube and Octahedrons, we are able to define the atomic radii of 51 out of the 83 stable elements periodic table. This makes a compelling argument that the electron cloud is most accurately described through geometry, rather than energy.
Cuboctahedron and the P-orbitals
The faces of a Cube and Octahedron are combined into a single polyhedron called the Cuboctahedron. This Archimedean Solid is unique, as its adjacent corners are the same distance away from each other as they are from its centre. For this reason, it is sometimes referred to as the Vector Equilibrium. It is the ideal shape to nest spheres in 3D space, termed hexagonal close packing. Together with the Cube and Octahedron, these three shapes are found to produce almost all crystalline atomic structures.
The Cuboctahedron also exhibits an out and mid-sphere, however, it exhibits two types of in-sphere due to having both square and triangular faces. With an out-sphere of 1, the mid-sphere will have a radius of 0.866, which defines the difference between the beginning of the first P-orbital set and the end of the second. Furthermore, if the mid-sphere is set to 1, the out-sphere will measure 1.154, which defines the difference between the end of the second and third P-orbital sets.
The first and second P-orbital radii reduce in size to 0.5Å, and 1Å, falling into a 1:2 radio. This doubling pattern should be expected from sets of nested octahedra. After this, the first set of D-orbitals form (20-30), prior to the appearance of the 3rd P-orbital set (31-36). This seems to break the doubling pattern, to produce a radius of 1.154, instead of 2. The D-orbitals generally fall into a radius of around √2, (1.4Å). This limits the radius of the 3rd P-orbital set, with the first, Gallium (31) 1.3Å, and then Germanium (32),1.25Å. After this, the next 3 elements all exhibit exactly the same radius of 1.154Å. The out and mid-sphere of the Cuboctahedron perfect define spacing.
When we extend the pattern produced by nesting the out sphere to the mid-sphere of the Octahedron, and add the two types of In-sphere to the dataset, we find that a mid-sphere of 1.154, produces an out-sphere of 1.333. This is extremely close to the radius of Gallium (31), and within the margin of experimental error. Furthermore, the radius of the next element, Germanium (32), 1.257Å, appears in the mid and in-spheres of the Cuboctahedrons above.
The highest numbers in the dataset roughly define the radius of the pair of S-orbital elements that form between the 3rd and 4th P-orbital sets, Rubidium (37) 2.3Å, and Strontium (38) 2Å, (highlighted in yellow). Additionally, all D-orbital elements fall between 1.55Å and 1.3Å (highlighted in green), which is defined by the Out and Mid-sphere of the 3rd iteration up from 1. The F-orbital sets, with two main radii of 1.85Å and 175Å, are also approximately defined with a value of 1.777 (highlighted in blue).
We can continue to read the chart through its diagonal to find all the radii for all P-orbital sets. The 4th set starts with Indium (49) 1.55Å, which falls to 1.45Å for the two subsequent elements. The mysterious increase in radius, not explained by the Octahedral model, can be accounted for in the Cuboctahedral set.
The 3rd Orbitals similarly fall from 1.3Å, 1.25Å, settling at 1.154Å. Again, these anomalous radii are found in the Cuboctahedral model. The same can be said of the 2nd P-orbital set that begins with Aluminium (13) 1.25Å, followed by Silicon (14) 1.1Å, after which the radius settles at 1Å. The Cuboctahedral dataset produces values of 1.257, 1.089, and 1, which is a very close match. Finally, we can identify all the radii for the 1st P-orbitals, starting at 0.866, 0.707, 0.65, 0.612, and finally 0.53. All of these fall well within the margin of experimental error.
Note that the Last P-orbitals finish with a radius of √2, which does not appear in the Cuboctahedral set, but is found in the Cubic and Octahedral models. We can compare this dataset formed from the Cuboctahedron to the experimentally determined radii, and Bohr model. The result is that the geometric model is an almost exact match for the experimental data.
1st - P-orbital Radii
2nd - P-orbital Radii
3rd - P-orbital Radii
4th - P-orbital Radii
What we can see it that the Cuboctahedral space provides an explanation as to how the P-orbitals transition from the S and D-orbital types as each lobe becomes filled with electrons. This explains the anomalies found in the Octahedral and Cubic models.
The Cuboctahedral model expands at the rate of 2÷√3. This value is also found as the ratio between a cubes side length and the distance of its vertex. When combined with the other ratios of the Cube and Octahedrons, (1, √1.5, √2, and √3), we are able to define all radii for every stable element on the periodic table. As no other atomic theory is able to provide such an accurate prediction.
Geonuclear physics of the 1st P-orbitls
Thus far, the geometric model has provided a compelling argument for the observed radii of all the elements on the periodic table. Through the Octahedral, Cubic, and Cuboctahedral nesting, we are able to reproduce all the P-orbital elements, and define the range of radii of the D and F orbital elements. However, we can also go one step further by examining the atom as one whole structure.
Our new theory of Geonuclear physics considers the number of Protons and Neutrons found in each atom in terms of geometric configuration. The shape of each nucleus affects the radius of the electron cloud. The S, P D and F model is based solely on the hydrogen atom. The Geonuclear model expands on this foundational structure by taking into account the nucleonic count of each atom.
It is an interesting fact that some elements are only stable more neutrons than protons in their nucleus. Boron (5), for example, is stable with 5 Protons and 6 Neutron. The number 10 is specifically avoided. The reason for the proton/neutron relationship has no clearly defined theoretical basis. Geonuclear Physics tries to offer a coherent solution founded on geometric principles.
The 11 nucleons of Boron (5) can be arranged so that the 5 protons form a tetrahedron, 4 at each corner, with one resting at the centre. The 6 neutrons create the corners of an octahedron, located at the tetrahedrons centre. A Tetrahedron with a side of √2 fits inside a cube with a side of 1, producing an out-sphere of 0.866, and a mid-sphere 0.5. This defines the radius of Boron (5), and Fluorine (9).
Carbon (6) has 12 nucleons, and so will fill the 12 points of a Cuboctahedron. Nitrogen (7) has 14 nucleons, which is the number of faces on a Cuboctahedron. These are defined by corners of the Rhombic Dodecahedron, which is the dual of the Cuboctahedron. Together they form the Octaplex, a 4D polytope described previously.
The Rhombic Dodecahedron itself is formed of a compound of a Cube and Octahedron, and produces the template for the 4D hypercube. This is represented as 2 cubes nested inside each other, and has 16 corner points. These can be ascribed to the next element, Oxygen (8), The 8 Protons and 8 Neutrons are each ascribed to one of the two cubes. Finally, Fluorine (9) has 19 nucleons with is the number of points need to form an octahedron made of 6 smaller octahedra. The extra neutron sits at the centre of the structure, leaving other protons and neutrons to fill the external frame.
Except for Oxygen (8), each of these forms contains an Octahedron within its structure. The Cube that encloses Boron (5) will also contain the Cuboctahedron of Carbon (6). Its dual, the Rhombic Dodecahedron, contains a second Cube, and an Octahedron. The 2 Cubes go on to form the Hypercube of Oxygen (8), leaving the single Octahedron to form Fluorine (9).
This P-orbital set contain the three elements that are at the foundations of Biological life. Carbon (6) often forms hexagonal rings with other carbon atoms, to build cellular structures. The hexagon is found within the Cuboctahedron. Nitrogen (7) and Oxygen (8) are unique in the set, as they form the diatomic molecules in the air we breathe. These elements achieve this through forming double and triple bonds, which binds two atoms together as a non-reactive molecule. This phenomenon is strange, as each molecule exhibits 14 and 16 electrons, whereas normally non-reactive molecules should have an electron count of 18, such as is found in Argon (18). This fact is key to our ecological and biological systems, and yet there is presently no concrete scientific answer that might suggest why this might be so. From a Geonuclear perspective, these particular atoms are the only one in the set to be comprised of a hyper-cubic nature. The Rhombic Dodecahedron, like the cube, is also able to fill 3D space, which creates the template for the 4D hypercube to manifest. This offers an alternative explanation for the double and triple bonds formed by these two elements.
Each of these Geonuclear structures contains a relationship to a Cube, an Octahedron, or both. By setting the side length to a consistent value of √2÷2, (0.707Å), for each shape, we find that the out-sphere almost exactly defines the radii for the P-orbital set. Note that Boron (5) is assigned a side length double that of the others (√2), so that the octahedron within the structure has a side length of √2÷2.
The results provide an almost exact match to experimental observations. In this case, only the out-sphere of each shape is used in each case to identify the radii for each atom. This Geonulcear model provides an accurate description of the first P-orbital set, in terms of radius, and at the same time, offers clear explanations for the nature of Carbon (6), Nitrogen (7) and Oxygen (8) to form molecules of a specific orientation and atomic number.
The geometry of the Nobel Gases
The transformation of an Octahedron into a Cuboctahedron can be achieved through a geometric process called the ‘Jitterbug‘. The faces of the Octahedron rotate as the form expands, forming gaps that go on to create the square faces of the Cuboctahedron. As the process unfolds, the spaces first become two equilateral triangles, forming the Icosahedron. Once the cube octahedron is formed, it provides the foundation for another Octahedron twice the size. This process shows how the Octahedral structure can double in size, as found in the first two P-orbitals.
Thus far, we have avoided the radii for the noble gases as these elements do not form bonds, which means there is no hard experimental data for these elements. The debate whether their radius should be small or larger from the previous element has yet to be resolved. The Bohr model, based on the attractive force of the electron to the proton, predicts a smaller radius. The Van der Waal radius is based on the outermost shell of the atom, and suggests the radius should increase in size due to the neutrality of the atom.
From the perspective of Geonuclear Physics, we can examine the nucleonic count for the 4 stable noble gases and see if there is any kind of geometric correlation. There are four noble gasses are Neon (10), argon (18), Krypton (36) and Xenon (54). We can list these along with the number of protons and neutron found in each element.
The number of protons is always equal to the number of electrons, which gives each element its atomic number. However, as the elements progress, we find that more neutrons need to be added to the nucleus in order to maintain a stable structure. This fact is particularly interesting when it comes to Argon (18), which has 40 nucleons, Whereas potassium (19), that direct follows it, only has 39. This anomaly occurs just before the 1st D-orbitals begin to appear.
The nucleon count of Neon (10) is exactly half that of Argon (18). The same ratio is found between the scale of the first P-orbitals sets in which these elements are found. Out of the 5 Platonic Solids, only the Dodecahedron exhibits 20 corner points. Its dual is the Icosahedron, which has 20 faces and 12 corners. The Dodecahedron is the Duel of the Icosahedron, whose in-sphere touches the centre of each of its 20 faces, which defines the 10 corners of a nested Dodecahedron. This is the correct number of points to match the nucleonic count of Neon (10).
Based on the radius for each Octahedron found in the P-orbital sets, we can produce an Icosahedron of the same side length through the Jitterbug transformation. We can find the in-sphere to define the Dodecahedron, and try to make a prediction for the sizes of the noble gases according to this geometric principle.
As a comparison, we can use the Van dar Waal Radius, as it predicts a larger size for the noble gases. However, as these radii are generally much larger than the experimental values, we need to scale the results. By taking the radius of the previous element and multiplying it by the radius of the noble gas, we can determine the ratio between the two elements. We can use this to scale the Van dar Waal radius to fit the experimental dataset.
For example, Fluorine (9) and Neon (10) have Van dar Waal radii of 1.47Å and 1.54. If we divide 1.54 by 1.47, the result is 22/21. Now we can multiply the experimental radius of Fluorine (9), 0.5Å by 22/21 to find the equivalent radius of Neon, which in this case will be 0.523Å. We can repeat the process for the other noble gases and compare the results to the Dodecahedral model.
The results produce a very close match, within the margin of experimental error. Whilst there is no hard experimental data that can prove the radius of the noble gases, the geometric theory does offer a prediction based on the nature of the Dodecahedron. The model is supported by the number of the nucleons found in each noble gas.
Neon (10) has a total of 20 nucleons, one for each corner point of the Dodecahedron. Argon (18) exhibit exactly double this number, and is represented by two nested dodecahedra, similar to the 4D Hypercube. Krypton (36) has 84 nucleons, which is the next in the series of Dodecahedral numbers after 20, in the same way that 19 follows 6 in the Octahedral numbers. 84 can also be composed of two 20 point Dodecahedra similar to Argon (18), with an extra 44 nucleons remaining to forming a 3rd order Octahedron, which is the correct size to contain the octahedral from the two previous P-orbitals. Note octahedral numbers are 6, 19, 44.
The final noble gas, Xenon (54), with 131 can be complied into a first and second order dodecahedron (20 +84), leaving 27 more points to used to form a cube (3³). This orbital set appears after two sets of D-orbitals, whose cubic nature can account for the appearance of the Cube within the set
The implications of the noble gases to form a dodecahedral structure offer new insight as to why they should be non-reactive. The Octahedron/Tetrahedron, the Cube and Rhombic Dodecahedron are all able to fill space, creating larger versions of themselves at each step. The Cuboctahedron unifies all three of these shapes, and offers the ideal arrangement for packing spheres in space. However, the same cannot be said of the Icosahedron/Dodecahedron. These shapes contain a pentagon in their structure, which does not tile the 2D plane perfectly, like the square or triangle. Instead, two side lengths must be employed, that are in proportion to each other through the golden ratio. This tapestry tends to expand in scale from a central point. This geometric perspective provides an interesting alternative explanation for the inert properties of the noble gases. They simply do not fit into the fractal structure of space.
Should a methodology to determine the radii of these noble gases one day be discovered, and if the predictions of this Geonuclear model were shown to be true, the implications would be quite profound. For it provides explanations for the behaviour of the various elements from chemical and biological phenomena, the structure of the electron cloud, radius of atoms, right down to the geometry of the atomic nucleus, all combined into a single theoretical framework.
Geometric compounds and space
Platonic duels can be compounded so that each side length divides the other at its midpoint. This unifies the mid-spheres of the two polyhedra. In terms of the Icosahedron and Dodecahedron, this produces a compound with triangular face divided into two. Notice the same face is also found on the 19 point octahedra, that forms in the Geonuclear model of Fluorine (9).
Using the Octahedron as a basis, we can map the different radii of the jitterbug transformations, adding a Dodecahedron into a compound with the icosahedron. For each of the 4 P-orbitals, this creates the following datasets.
Yellow = S-Orbital, Red = P-Orbital, Green = D-orbital Blue = F-Orbital, Highlighted = 5th P-orbital.
Examining these radii, we notice that the Out-sphere of the Dodecahedron matches the mid-sphere of the Cuboctahedron. In the first P-orbital set, this reproduces the radius of Oxygen, (0.612Å). In the 2nd P-orbital set, a value of 1.225, which is the radius of Helium (2). The 3rd set sees the value 1.414, the rough radii for the D-orbital elements and the 4th P-orbital Octahedron. In the final 4th dataset, the two polyhedra unify at 1.732, which is roughly the radii of the F-orbital set.
Here we also find the radii for the last 5th P-orbital elements, (highlighted in yellow). These 3 elements, Thallium (81), Lead (82), and Bismuth (83), are the last stable atoms on the periodic table. Interestingly, Bismuth (83) has a radius of 1.6Å, which appears in the mid-sphere of the Icosahedron and dodecahedron compound, and has the value for the golden ratio. This in turn defines the two lengths needed to form the pentagonal plane.
Through our investigation of the P-orbital elements, we have compared the in, mid, and out-spheres of the Octahedron, Cube, Cuboctahedron, Rhombic Dodecahedron, Icosahedron, and Dodecahedron. Through the jitterbug transformation, the Icosahedron is formed, which when compounded with the Dodecahedron defines the mid-sphere of the Cuboctahedron.
The Rhombic Dodecahedron is formed from the compound of a Cube and Octahedron when the corners are connected. This forms the blueprint for the formation of the 4D hypercube. Similarly, connecting the corners of the Icosahedron and Dodecahedron compound creates the Triacontahedron, which forms the template for the 6D hypercube. This form exhibits 32 corners, which also happen to be the maximum number of electrons that can fill a single shell of the electron cloud. When the Rhombic Dodecahedron is nested inside the Cuboctahedron, it forms the template of the 4D Octaplex. The outer Cuboctahedron then provides the foundation for the next cube and octahedral compound. These are the geometric principles of space, through which the initial Cuboctahedron ends up doubling in size.
We can conduct a similar geometric experiment as previously conducted and map the In, Out and Mid-spheres of these interconnected solids, scaled to the P-orbital octahedra. Each P-orbital is mapped to the out-sphere of the octahedron, which is compounded with the Cube in its mid-sphere. The result shows an amazing correlation to the experimentally determined values for every element of the periodic table.
We can see how the radii from the 1st P-orbitals also contains the radii for the preceding S-orbital elements. There is an overlap in the dataset with the 2nd P-orbitals, which contains the radii of the D-orbitals (green), and the lower bound for the F-orbital set. At the end of the row, the radii for the another 4 S-orbitals are found. The 3rd P-orbital set defines the other D-orbital radii and the upper bound of the F-orbitals, followed by the another 2 new S-orbital radii. The final set defines the radii for the final 2 S-orbital elements, and the 3 stable elements of the 5th and final P-orbitals, after which the radioactive elements begin. We can create a graph of the entire periodic table based on this simple geometric model, and make a comparison between the Experimental and Bohr models.
Zero values are noble gases which lack experimental data.
The result is a much closer match to the experimental values than the theoretical Bohr model. In each case, we can see that the P-orbital sets tend to level out whilst the Bohr model continues to fall in size. To compensate for this, the Bohr model predicts a larger S-orbital radius. This helps to bring the P-orbitals into alignment, but cannot account for the smaller radius of the D and F-orbital sets. The official explanation for this phenomenon is attributed to shielding by the electrons in the lower shells. Yet for the 1st P-orbitals the theory is untenable. From this simple diagram, we can clearly see that the Atomic Geometry model is the best fit to the experimental data. This offers a fresh new approach to atomic theory, based on the nature of geometric ratio.
What does this tell us about the atom?
Present scientific theory has compartmentalised the study of different aspects of the atom. The geometric model allows us to consider the atom as a single entity, and can provide a much more accurate prediction of the radius of each element. Unlike present quantum solutions, the geometric model is applicable to all elements across the periodic table, and is able to answer some of the most perplexing qualities and behaviour of different atoms.
A Geometric approach to the atom
The fact that this geometric perspective is able to resolve the atomic radii is an important step forwards for atomic theory. Presently, the development of new materials for quantum computing relies on being able to determine the size of a particular atom. By comprehending the nature of geometric principles, we can uncover the inner workings of the atomic fabric. The implications are far ranging, from the field of lasers, renewable energy generation, and battery storage, to finally being able to understand the reasons for the interactions of biological processes, and even solve the mysteries of life. This article introduces some of the foundational principles of Atomic Geometry, Geoquantum Mechanics, and Geonuclear Physics, which weave together to form the most coherent model of the elements that constitute the whole of existence.
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YOUR QUESTIONS ANSWERED
Got a Question? Then leave a comment below.
Wow, a fascinating article, but I still don’t understand how the electron can be a 4D torus?
The electron radius has never been determined. Presently, it is best described as a point of charge. In the 4D model, the two circular surfaces of the torus cross paths, which defines an x, y, and z axis. Between each, there remains a consistent angle of 90°, i.e. their intrinsic angular momentum is conserved. When the two overlap, the internal geometry of a sphere is formed. Any point within the sphere can be identified by defining a point on each of the circles surfaces. The point charges of the electron can therefore be measured. When the circles move apart, the spherical surface disappears, as does the electron. They recombine on the opposite side of the orbital, without changing angular momentum. As they overlap, they define a lobe on the opposite side of the torus. Again, each surface can reference a charge in 3D space.
This seems like quite a simple model. Why has no one thought of this before?
During the renaissance period, geometry featured prominently in scientific exploration, right up until Newton discovered his law of gravitation. At this point, the particle nature of light was first adopted. Similarly, the particle notion of the electron has been maintained, even though it displays wavelike properties. When Schrödinger first conceived the S, P, D and F orbital model, it was based on a geometric consideration of the electron cloud. However, the insistence of the electron particle introduced quantum probability into atomic theory, which obscures the fact that the orbital shapes are real phenomena. Since that time, particle physics has claimed to be the most accurate model of the atom. Yet that can only be considered true for hydrogen like atoms. However, as the radii of the theoretical models are normally used instead of the experimental values, this notion has never been challenged.