Most people think of the electron as a tiny particle that circulate the atomic nucleus. However, in truth, there are 4 types of orbital S, P, D and F, each of which create a different geometric pattern. In this article, we examine the 2D orbital geometries of the electron cloud, and provide a simple geometric reasoning for the different configuration.


When the structure of the atom first began to be revealed by scientific exploration, it started to become apparent that it exhibited a nucleus surround by a field of electrons. Shortly afterwards, it was discovered that the electron cloud itself was quantised into discrete bands. However, even this was not the complete picture. Upon closer examination, it was revealed that these quantised states came in four types, S, P, D, and F, each of which exhibited a specific geometric configuration. 

When we examine the limitation of these orbital types from a purely geometric perspective, we find that each can be represented by the dot, line square (cross) and hexagon, after which the pattern terminates. Furthermore, the principles of 2D geometry dictate can explain the limitation and order of appearance of these orbital types. This is derived from the fact that there are only two types of regular 2D tessellation that can emerge using just two colours. The square and triangular plane. By examining the relationship between 2D and 4D geometry, we can produce a new model of the electron cloud, that can explain the reason for the two types of quantised spin values for the electron and this limitation of the electron cloud, i.e. all matter in the universe.

KEy Points

  • The shape of the atomic orbitals are defined by 2D geometry
  • The 1/2 spin of the electron is explained in terms by the rotation of 4d object
  • The order of appearance of the orbitals is explained by the principles of goemetry

What is an orbital?

Traditionally, an electron orbital is considered to be a probabilistic area of space which determines the likelihood of finding an electron at that particular point around the nucleus. Each orbital represents a specific energy level of the electron.  When an electromagnetic wave strikes the atom, certain frequencies are absorbed, and the electron ‘jumps’ into a higher orbital shell. The atom now becomes energised. Subsequently, the electron ‘falls’ back down into the original shell, and a new electromagnetic wave is emitted.

This can be considered as the vibration of an atom, but with one key difference. The electron does not ‘move’ in space, it literally jumps. This means there are certain areas surrounding the atom where the electron will never be found. It is this fundamental nature of ‘quantisation’, which means the energy levels are divided into discrete steps. Similar to a piano keyboard. Each note produces a particular frequency, that is tuned to all the other notes. As we move from one note to the next, there are jumps in the frequencies, that are based on the original ‘root note’.

This means the electron will never be detected in the space between orbitals. These are called ‘nodes’. This fact has been experimentally observed as fine lines within the visible spectrum of light. When monochromatic light is projected onto a particular type of atom, so it will absorb certain frequencies. This is a called the spectral emission of an atom, which is unique to each element on the periodic table. By analysing the different spectral emission lines, scientists can ascertain the composition of various different materials.

Types of Orbital

Whilst many people suppose that the electron is a tiny particle that orbits the nucleus. In fact, this is not the case, a point discussed in more detail in a 4D geometric model of the electron cloud. The electrons appear like a field that covers a certain area surrounding the nucleus. In most cases, this is more accurately described as a sphere. However, orbitals also come in four different types, labelled S, P, D and F. Only the S orbitals are spherical in nature. P-orbitals are sometimes referred to as dumbbell shaped, as they are comprised of two spherical areas either side of the nucleus. Most D-orbitals are generally composed of four lobes, which form a cross shape. Finally, F-orbitals appear with six lobes in a similar configuration to a hexagon.


An orbital consists of two electrons, each assigned a different quantum spin value, either up or down. When comprised of a single electron, it moves from one portion of the orbital to the opposite configuration. This means that the orbitals appear in ‘quantised’ states. When the second electron fills the orbital, then each ‘swaps’ places with the other in a continuous rotation. We can consider this as the vibration of the atom.

There are a variety of ways that we can view this concept. One is imaging the electron cloud from an electromagnetic perspective. A magnet has a north and south pole. The magnetic field flows from one pole to the other, creating a torus field, just like an S-orbital. The magnetic pole of the orbitals flips direction in a continuous motion.

It is a curious, yet often unrecognised fact, that exactly the same phenomena occurs with our sun. The magnetic pole of the sun flips every 11 years, which radically affects the climate on earth. The field of the sun reaches across millions of miles, enveloping the entire solar system, and is typically termed the Heliosphere. This interesting correlation between the microscopic domain of the atom, and the macroscopic domain of the sun is not normally considered by mainstream scientific thinking.

An atom with a single electron in its outer shell acts in exactly the same way at the pole reveal of the sun, only at a much quicker rate. However, when a second electron completes the orbital, the atom acts in a slightly different manner. Now each electron swaps places, which produces an equilibrium in the electromagnetic field. 

This is clearly demonstrated by the Stern–Gerlach experiment, which shows that only atoms with a single electron in the outmost shell will be deflected when passed through a magnetic field. Notice that the atoms only ever exhibit either an up or down orientation, which forms two distinct lines on the photographic plate. This experiment was the first to demonstrate the concept of electron spin, and is discussed in greater detail in our article that explores the electron cloud from the perspective of the 4th dimension.

Spherical Harmonics

There are many ways in which the electron cloud can be visualised. One of these is through the idea of harmonic resonance. The simplest of these is to represent space as a flat, circular drum. The centre of the drum vibrates up and down, representing the electron moving from one polarity to the opposite.

S-orbital vibration

The example above shows the vibration of the first S-orbital of the atom. On the left, we see the different polarities represented on a sphere as red and blue. On the right, the drum version accentuates this notion, making it easier to comprehend. The drum model of the orbitals vibrations can also emulate the P, D, and even F-orbital shapes.

Notice how the P-Orbital (left) exhibits two points that oscillate either side of the centre of the drum. The D-Orbitals (right) have 4 oscillating points that come in pairs. By comparing these images to the orbital shapes, we can see the relationship. The up electron appears on the top surface of the drum, whereas the down electron on the underside.

Whilst this helps us to comprehend the vibrational nature of the atomic orbitals, the model can also be expressed in 3D spherical space. This tends to create a much more accurate interpretation of the different orbital types. Spherical harmonics are often used by physicists to model the atomic orbitals of different atoms. These are commonly depicted in a table that shows each orbital shape in the various possible configurations and orientations.

Here we can see the S, P, D, F orbitals arranged in ascending order. Notice that as the orbitals move through the different types, so the variations of each one also increases. This reflects the degrees of freedom associated with each shape. The sphere can be rotated in 3D space without appearing any differently. Whichever way is it rotated, it maintains is spherical form. The same cannot be said of the P-orbital. This shape can be rotated on its x, y or z axis, and the form will look different. However, if viewed end on, it will look like a sphere. If it rotates on this central axis, the shape will not change. However, a D-orbital has extra degrees of rotation. When viewed from the same angle, the other two lobes will protrude. Therefore, unlike the P-orbitals, it changes its appearance when rotated on the axis. This is the foundational principles of spherical harmonics, which is derived from the notion of the symmetry of each orbital type.

In quantum physics, four quantum numbers define orbital types and their position within the electron cloud. The number denoted by ‘M’ represent the shell, also term the atomic number. The second quantum number, ‘L’, defines the orbital types, (S, P, D or F). The third quantum number ‘n’, the Azimuthal quantum number, defines the orientation of the orbital. It can have a + or – value, which moves to left or right across the row, respectively. The final number is the value of electron spin, which can be either up or down. In this way, the state of any electron in the electron cloud can be perfectly defined. As these are always whole numbers, the conclusion is that the atomic shells are quantised by whole numbers. 

electron spin quantisation

This exploration of the atomic orbitals provides a relatively accurate description of the electron cloud. These descriptions arose when the wave-like nature of matter was discovered in the 1920s. Experiments, conducted by Davisson and Germer, confirmed the wavelike behaviour of the electron. Afterward, the concept of the atom had to be reformulated to include this property. 

However, there are a few problems with this model. The most prominent of these is the quantisation of up and down spin. Spherical harmonics demonstrates a smooth curvature between the up and down states of an electrons, motion. The peaks gradually transform from one state to the next. This means there is a point of equilibrium between the two states. This can be more easily perceived in the drum model, where the surface reaches a mid-state, becoming flat between each oscillation.

only the up and down spin states exist for the electron

In the image above, we can see that in between the up and down state of the electron, there is a zero point. This implies that the electron at this point will be neither in an up nor down state, which is not what is observed. The smooth transition between the up and down spin is an inherent quality of classical electromagnetism. If the electron did behave in this manner, then the Stern–Gerlach experiment would not have produced the result of two discrete lines on the photographic plate. Instead, we should observe an even distribution of the electrons. The concept of electron spin acts more like a square wave than a sine wave.

sine wave vs square wave

Aside from the Stern-Gerlach experiment, there are several other reasons that suggest this it true. The energy possessed by every electron is the same. This is a fixed scientific constant, that is expressed as the unit of elementary charge (e). A sine wave would express as a varying charge as the up or down spin approaches the zero line. In a smooth wave, the charge value should diminish. 

In fact, all electrons are exactly the same throughout the universe. This is an important property in the quantisation of the quantum field. This problem of the quantised nature of electron spin was originally resolved by the suggestion that the electron particle is rotating, which would create a north-south Pole, similar to that of the sun or the earth. However, if that were the case, then the rotation would exceed the speed of light, and so the theory becomes untenable.

In the standard theory of quantum mechanics, the solution is to abstract the electron cloud as a probabilistic field. This concept was first proposed by German physicist Max Born in 1926. The Born Rule is a foundational principle of quantum mechanics. However, this does not really solve the paradox of electron spin, moreover, it only produces the theoretical framework that enables quantum mathematics to determine the likelihood of the outcome of particle interactions.

Probabilistic theory is a key part of the quantum mechanical consensus. It suggests it can offer the most accurate predictions of the atom. However, the claim is not without is criticisms. To start with, quantum probability is often only accurate for hydrogen-like atoms. These are atoms that only have one electron on their outermost shell. Secondly, quantum mechanics does not perform well at predicting the experimentally measured radius for most atoms. Even the hydrogen atom, with an experimentally measured radius of 25 picometres, comes out at 53 picometres, when calculated using the Bohr radius. The margin of error increases massively for the next atom, Helium, which has an experimental radius of 120 picometres, compared to the calculated Bohr radius, which expects the radius to reduce in size to 35 picometres. This fact only becomes apparent when we examine the data table of the radius of various atoms.

Furthermore, the complexity of quantum calculations compared to those of other sciences means that mistakes are frequently found in the formulations. These are often only identified once improved experimental measurements are performed. Additionally, quantum calculations have to be ‘normalised‘, in order that the probability function fits the desired result. Closer examination of the development of quantum mechanical theory show a disturbing trend in altering the calculations in order to fit experimental results, rather than predicting the outcome, as is the norm for scientific enquiry.


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Orbital geometry and the 2d plane

The unsettling truth of the matter is that there is no solid scientific theory in mainstream science that can accurately describe the quantised nature of electron spin. Resonance models cannot explain the quantisation effects of the atomic orbitals, whereas QED and other theories need to rely on probabilistic mathematics, which need to be normalised in order to produce results close to experimental values. So is there an alternative that can explain the nature of electron orbitals and the quantisation of spin?

In fact, there is. Based on the principles of geometry, the orbital shapes can be described to perfection. This can be achieved without the need for complicated of equations. The first orbital type appears as a simple sphere. We can represent this as a single dot. The next orbital to emerge is the P-orbitals, where the sphere now doubles to become two spheres. Similarly, when the dot ‘divides’ into 2 dots, the line is created. In 2D Euclidean Geometry, a line is always the shortest distance between two points. The D-orbitals form next, whereby the number of dots doubles to create a new line that is then rotated at 90° to the first. Connecting the ends of the lines now creates a square. The final F-orbital type adds another line. These now become evenly spaced, establishing a 60° angle. Joining the ends of each line creates a hexagon.

InInfinity AtomicGeometry SPDFOrbitalsdefinedinGeometryascot,line,crossandhexagon

In terms of the atom, no other orbital geometries have ever been established after the F-orbital type. Theoretical G-orbitals are sometimes considered, but have never been observed. But why should there only be four types of orbital, and why should they emerge through these particular patterns?

The principles of geometry can provide us with a very simple and logical explanation. There are only two types of regular 2D tessellation, which can be created using just two colours. The one which is the most familiar to us is the square tessellation, which is the chequered plain, similar to that of a chess board. The second, which is not quite so familiar, is the triangular tessellation. Six of these triangles can be grouped together to form a hexagon. However, the hexagonal tapestry needs three colours in order to tile the 2D plane. If we recall, electrons can only appear with either an up or down spin. The limitation of these two types of states is therefore perfectly expressed through the only two types of regular 2D space.

two types of regular D tesselation

This concept is so simple that you might be wondering why no one has considered it before? Firstly, it is worth noting that in standard mathematical thinking, it is often believed that the second dimension can only ‘exist’ when it is embodied in the 3rd dimension. This view begins to shift that belief. Secondly, in the domain of physics, results are typically determined through the experimental measurement of a single point in space. The measurement problem in Quantum Mechanics is derived from this very nature. The notion that the electron is a particle has always been assumed. When the wavelike nature of matter was established, the notion of wave-particle duality, already formulated due to Albert Einstein’s suggestion of the photon, was adopted.

The idea that such a simple concept could describe the complexities of the electron cloud seems not to have been thoroughly considered, as quantum mathematics generally enjoys a more complicated explanation. However, these simple principles do begin to provide an insight as to why the electron can only occupy two distinct states. It also explains the reason for the pattern of emergence for each orbital shape.

At first, it appears as if each orbital follows a doubling, pattern. From a dot that divides into two, to form a line, then a line that divides again to form a cross. We might expect the next division to create a double cross (octagon) on the 2D plane. However, in the transformation from the D to F-orbital, we find that this doubling sequence does not follow suit. Instead, we find the F-orbital falls onto the triangular tapestry. After this, the pattern terminates, and we reach the end of the orbital types. Therefore, we find that the principles of 2D geometry define both the limitation of the orbitals shapes, and the evolution of each one.

Electron obtials and compass construction

The art of compass construction has been found throughout history in the exploration of geometry. This powerful drawing technique produces geometric designs without the use of measurement. Similarly, we can say the universe itself does not use mathematics to calculate emergent phenomena. Moreover, everything is relative to the smallest scale. In science, this has been quantified as the Planck length, which is derived from the Planck constant. This unit defines the minimum distance that a measurement of physical reality can be ascertained.

Planck Length = √ ħG/c3
Where ħ is the reduced Planck constant, G is the gravitational constant and c is the speed of light

The art of compass construction always produces the same pattern. This emerges from the combination of circles, the scale of which is based on the original opening of the compass to a distance of ‘ONE’. The size of the opening is irrelevant. What is important is that one set the compass dimension is not changed. After the first circle is created, the point of the compass is placed somewhere on the circumference and a second circle is drawn. This forms an image called the Vesica Piscis. Where the two circumferences cross, two nodes form. These become the central points for a second pair of circles. This creates a new set of nodes. We can continue adding circles to more nodes that appear on the circumference of the first initial circle, until we arrive back at the start point. This completes an image that is called the Seed of Life. By examining this process, we find it accurately emulates the four types of atomic orbitals.

InInfinity AtomicGeometry SeedofLifemapsSPDFOrbitals

The Seed of Life image appears throughout the world, from ancient architecture, etched into temple walls, to modern art and design. What seems to be largely unrecognised it this curious relationship between the nature of 2D compass construction and the four types of atomic orbitals. It is interesting to note that much new age thinking and spiritual knowledge have referred to this mandala as the ‘Blueprint of creation’. Not only is it formed of 7 circles, which is akin to the number of days in the Biblical Story of Creation, it also produces the octave in space. A larger circle placed around its outer edge will be exactly twice the original size of the initial circle that formed the image.

The 4D electron

Whilst the correlation between the principles of 2D geometry can perfectly describe the evolution of the various orbitals types, it does not by itself solve the curious nature of electron spin. In order to explain this, we need to examine the nature of 4D geometry. At first, this might seem like quite a daunting prospect. However, the foundational mathematical premise is not as complicated as you might think.

4D geometry is simply an extension of one dimension above 3D geometry. Let us begin with a 1D line with a distance measurement of 2. When we square the number, the result is 4. The line now becomes the side length of a square comprised of 4 smaller squares. We have increased dimension from the 1st to the 2nd to define the plane. If we raise the number 2 to the power of 3 then the result is 8. This is the number of cubes in 3D space that form a larger cube. We can see a simple pattern emerging. Each time we raise to a higher power, the result increases the dimensional number. Therefore, when 2 is raised to the power of 4 we get the result 16. This is the number of smaller cubes that comprise the 4D hypercube.

As you can see, the mathematics of 4D geometry are actually quite straight forwards. Viewed like this, we can see that the power merely represent the dimension. The 4D hypercube is really just two 3D cubes. In 3D space, these actually appear in exactly the same location. As we are unable to reproduce a 4D object in 3D space, we often use the technique of shadow projection to explore these higher dimensional space. Each 4D shape has an x, y, and z spacial dimension, and a W time dimension. When the 4D hypercube is rotated in the W axis, the result is that the two cubes swap places.

What is interesting about the 4th dimensional hypercube is that its volume can also be calculated by the multiplication of two 2D plains. For example, the number 2² = 2. When we multiply two planes together, the value, 2² × 2² = 16 or 24. The same can be said of all square numbers. This novel depiction of 4D space presently has no recognition in the current mathematical literature. Yet, it does provide a model of 4D that emerges out of the multiplication of two 2D plains. This provides us with a geometric explanation for the quantised nature of electron spin.

We can demonstrate this by imagining two squares planes of opposite colour orientation. One square sits in the manifest portion of 3D space, whereas the second sits beyond 3D in the 4th dimensional space. The two squares swap places, similar to a 4D cube, when it is rotated on it’s w-axis. The square plane in 3D disappears and is replaced by the second square plane. In this way, the colours swap in quantised steps, just like the electron that  ‘magically’ changed location without moving.

This model is the only one of its kind that can properly explain the quantised nature of electron spin. Unlike the classical wave model, at no point will the electron ever be found in the zero position. Instead, the quantisation of reality is explained by the rotation of a 4D object in 3D space.

This is based on the Euclidean notion of 4D. This differs from the conventional 4D spaces developed by quantum scientists such as Minkowski space to try to model spacetime. The key difference it that this model shows how time itself can be quantised into discrete units. Moments of time are ‘rendered’ in 3D space sequentially. If we examine the model of the 4D hypercube, we can see that each of the cubes appears centred at specific moments of time. The two cubes are inverse of each other. The first orientation shows an electron with an up spin, the second in the exhibits a down spin. If we focus on the central section, we can begin to see this quantised effect in action.

tesseract rotation

This idea begins to radically redefine the notion of time. It explains why time only seems to have a forwards motion, a fundamental truth of reality outside bizarre quantum theory that desperately tries to reconcile the particle nature of reality. It explains the quantised states of the up and down electron spin. Furthermore, it is also completely coherent with mathematical theories that explain the nature of quantum spin through the notion of spinors, which require a 720° rotation to complete a single rotation in 3D space.

Notice that in the video above, the calculations of the wave function suggest that ψ must be squared in order for the system to work. This removes the minus sign from the asymmetric wave function. Just as we have squared the two types of 2D plane. However, the mathematical notion that negative square numbers produce a positive result is challenged in our theory of Geometric Maths. This new mathematical system re-examines the foundational principles of number theory and the categorisation of numbers, and is able to resolve the Continuum Hypothesis, through the perspective of the 4th dimension.

In quantum theory, the spinor has to ‘rotate’ 720° in 3D space to complete a single rotation. Yet at the same time the electron cannot be rotating, as that would mean that it would be moving faster than the speed of light. However, the 4D perspective can easily resolve this issue, without comprising our logical notion of spacetime. This 4D model of the electron cloud resolves all of these issues. However, there is one key difference. Rather than seeing the electron as a particle, the model suggests that the electron cloud is a 4D field.

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electron Particle Vs Field

Probably one of the great mistakes of modern quantum theory is the dismissal of the Aetheric field. This notion was believed to be true throughout the majority of human scientific thinking, from the times of Ancient Greece, right up to the end of the 19th century. When Albert Einstein resolved the photoelectric effect by introducing the particle notion of light, the consequences led to irrational proportions to reconcile the already established observations of the wave qualities of light, with those of the particle.

In our new theory of the 4D Aether, we explain in great detail the wave only solution to the photoelectric effect and ultraviolet catastrophe, which lie at a heart of this issue. It is ironic that in order to make quantum mechanics work, the Aether field needed to be reintroduced. However, instead of explaining the vacuum energy in simple terms, the notion of probability was introduced, purely as a theoretical mechanism to preserve the particle nature of light and the electron.

However, the quantum field is in truth just a more abstract concept of the traditional view of the Aether field. In its original description, the Aether was supposed to exhibit a stationary position in the background of space. Other Aether theories explained it as being fluid. When viewed from the 4D geometrical perspective, we can satisfy both the static and the fluid aspects of the Aether. The notion is actually very simple. All atoms are emersed in a quantum foam, which is the foundational source of all energy in the universe. This notion has inherited various different names, including, the energy of the vacuum, cosmic microwave background, and the quantum field, depending on which discipline is describing it. However, at the core, the premise does not change.

Once we reintegrate the notion of the Aether into the scientific model, some very simple explanations for the orbital types begins to emerge. A single dot that disturbs the surface of water will create a series of concentric rings, that expand as a circular wave, into infinity. In 3D space, this becomes a spherical wave, which accurately describes the nature of light. This is the reason the intensity of light diminishes by the square of its distance. As the wave expands, so the surface of the sphere expands, which is defined geometrically as the inverse square law.

the intensity of light explpained

In terms of the electron field, we find that the first orbital geometry is also represented by a sphere. However, this field does not expand through the vacuum at the speed of light. Instead, it is contained within the radius of the atom. In standard theory, this is due to the quantised up and down states of the electron, which requires a 720° rotation to complete a single rotation in 3D space. This property is inherent for all fermion particles. Light, on the other hand, acts like a normal vector. The 360° rotation is in a 1:1 ratio with 3D space, called a boson. This key difference is what defines the electron shells, and prevents the electron field from collapsing into itself. Without spin, the notion of matter would not exist.

single rotation boson VS double rotation fermion

In simple terms, this describes the fundamental difference between light and the electron cloud. The double rotation of the electron acts as a container for the energy of the vacuum. This increases the energy density inside the field compared to the energy density outside. This explains why only certain frequencies of light are absorbed by the electron cloud. 

A particular frequency is completely absorbed, which causes the electron to jump into a higher energy shell. In terms of 4D, this motion is triggered by that particular frequency, which produces a single rotation of the 4D field. As the 4D field rotates, it moves outside 3D reality, and reappears at a higher energy level. On the second rotation, the electron ‘falls’ back down, emitting a new wave of light. The process is governed by the nature of 4D rotation, and the ratio of spin between the light wave and the electron cloud. This concept is concurrent with the fact that the speed of light (c) is limited due to the electromagnetic resistance found in the vacuum of space.

This explanation of the quantum field describes how the electron cloud is able to maintain its ridged structure. It defines how the electron can magically jump from one shell to the next, and unifies the nature of absorption and emission of the electromagnetic waves through the notion of 4D rotation.

s and P orbital geometry

The S, P, D, and F orbitals appear in a specific order in the electron cloud. The first shell of the electron cloud is completed by two types of atom, Hydrogen (1) and Helium (2). These comprise a single S-orbital. In terms of 4D geometry, this forms a 4D sphere, where each electron exhibits an up and down spin. Helium (2) is a noble gas, which means it does not form bonds with any other atom. In other words, the electrons become ‘fixed’, establishing an impenetrable boundary. They cannot be removed from the electron cloud to create molecules and compounds. The energy is now contained within this hypersphere. This completes the first shell of the atom, upon which, all subsequent elements are founded.

hydrogen and helium
The 4D sphere. In 3D each appears in the same location, and ‘swaps’ places when rotated on its w-axis.

When we consider the nature of the electron as a 4D field, we can begin to see that this boundary acts as a container for the vacuum energy. When the up and down electron spins are complete, the 4D sphere falls into magnetic equilibrium. The field processes both an up and down state at all times. This explains why atoms that exhibit two electrons in their outer S-orbital shell are not deflected by the magnetic field in the Stern-Gerlach experiment.

In the 2nd shell of the atom, another S-orbital forms, creating the next two elements, Lithium (3) and Beryllium (4). Geometrically speaking, this forms a double hypersphere. We can represent this as two circles one inside the other with a second set of circles off-set to the side, representing the 4D space.

helium and beryllium

However, Beryllium (4), unlike Helium (1), does not form a noble gas. Instead, we find that the outer two electrons can be used to form bonds with other atoms. Yet the inner shell still retains its noble like qualities, i.e. only electrons in the second shell can be utilised to form bonds.

After this, the next six types of atom form three P-orbital. The first three electrons fill one side of the set first, and then the remaining ones complete the opposite side of the P-orbital lobe. This shows us that each orbital type will complete first with a single electron, after which they begin to fill the other orientation of that set of orbitals types. This is called the Pauli exclusion principle.

P-orbitals appear outside the S-orbital sphere, which creates an extended ‘lobe’ on the electron cloud. These are more likely to form bonds with other atoms, as they are further away from the nucleus. This is the reason simple molecules and compounds create geometric shapes, which is the basis of molecular geometry. As P-orbitals come in sets of three, they can be mapped on the 2D plane using a mandala called the Flower of Life, which is a simple extension of the Seed of Life Mandala described previously.

Once the P-orbital set completes, the second noble gas, Neon (10), forms. Again, like Helium (2), none of the electrons in Neon can be used to make atomic bonds. This creates a second boundary, where the energy is confined by the orbital geometry.

Both S and P-orbitals share a geometric commonality. Neither are derived from the 2D plane, instead they are represented by a dot that divides to create a line. The D and F-orbitals were described as emerging from the 2D plane. If we consider the S and P-orbital types from the perspective of 4D, we find that both are derived types of 4D sphere. The S-orbitals are a 4D sphere, which is superimposed over the same space. However, the P-orbitals are formed of a 4D torus. Geometrically, both of these forms are akin to the sphere. In fact, P-orbitals are sometime visualised as a torus field, rather than two spherical regions, that appear at opposite sides of the nucleus.

Orbital p px animation

A P-orbital electron will ‘jump’ from one side of the orbital to the other in quantised steps. Again, the 4D perspective resolves this nature, through a simple logical explanation.  The electron disappears and reappears as it is moving in 4D space. Asides from Helium (2), all non-reactive noble gases on the periodic table are formed of a completed set of P-orbitals. These inert elements are of paramount importance, as they define the qualities of each atom, and their ability to form bonds. Once formed, the electrons become ‘fixed’. All subsequent atoms will retain the same P-orbital structure, and only the electrons of a higher energy level will form bonds with other atoms.

This 4D geometric perspective provides a much more comprehensive explanation for the structure of the elements on the periodic table than traditional quantum mechanics, which is often only applicable to ‘hydrogen-like’ atoms. The conclusion is that the 4D sphere, and its derived counterpart, the torus, are the geometric forms that are able to concentrate the vacuum energy, by the nature of their 4D rotational ‘spin’ properties.

hyper sphere
s-orbital Hypersphere
horn torus
P-orbital Horned torus

This explains why the electron radius has never been established. The electron is not a particle. It is a 4D field. This view also provides a reason as to why there are always exactly the same number of protons and electrons in the atom. Rather than perceiving the electron as a separate entity to the proton, the two now become unified by the 4D field. Protons also exhibit a half spin property, However, unlike the electron they are composed of 1/3 and 2/3 spin quarks. These increased spin values provide a mechanism for a great concentration of vacuum energy within the atomic nucleus. In our articles on the Quantum Foam, and our theory of 4D Matter, we explain how this nature gives rise to the notion of mass and charge in the universe. The 4D geometric picture resolves all of these quantum enigmas and much more.

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Orbital configuation

The first 20 elements on the periodic table are formed with just the S and P orbital types. The first shell forms from a single S-orbital. The second shell comprises another S-orbital, followed by a set of 3 P-orbitals, to create the second noble gas Neon (10). The third shell repeats this pattern to form the third noble gas, Argon (18). Then, in the 4th shell, an S-orbital electron pair produce a torus field around the whole structure. This orbital configuration can be mapped onto another mandala called the Metatron’s Cube, which is an extension of the Flower of Life.

metatrons cube and elements   on the periodic table

At this point, the pattern of orbital formation changes. Instead of forming another set of P-orbitals in the 4th shell, a set of D-orbitals from in the 3rd shell. Unlike the P-orbital set, the D-orbitals require an S-orbital in the shell above before they can appear. This fact is not clearly expressed, as the periodic table places the 1st D-orbital shell in the 4th row, instead of the 3rd. A similar concept applies to the F-orbitals, which only appear in the 4th shell once a set of S and P-orbitals form in the 5th shell above. But why should this be?

Present atomic theories have no explanation, however, the perspective of geometry does offer a simple answer. As we saw previously, S and P-orbitals are derived from the 4D sphere. However, D and F-orbitals are mapped onto the 2D plane. When we examine the nature of these orbitals, we find the electron appears in more than one location. In the case of the D-orbital set, two out of four lobes are filled with the electron simultaneously at any one time. With the F-orbital types, three out of six are filled.

D and F orbitals fields

The S-orbital hypersphere acts as a ‘container’ of energy, that increases the energy density within it. Inside the 4th S-orbital, the P-orbital set is already present in the 3rd shell. This leaves, areas within the sphere that can be filled by the D-orbitals electrons. This fact becomes clearer when the orbitals are expressed in 3D space. The three P-orbitals align on the x, y, and z axis, forming the shape of an Octahedron. This is a Platonic Solid that is the duel of the Cube. The P-orbitals fall into the centre of each face of the cube. This leaves the corner spaces, which can now be filled with the D-orbital electrons.

A similar process can be said of the F-orbital types, that form the hexagon. When perceived in 3D, they comprise the geometry of the Cuboctahedron. This form combines the sides of the octahedron and cube. It is sometimes referred to as the vector equilibrium, as each of its corners is exactly the same distance from its centre to its adjacent corners. This makes it the most efficient shape to nest spheres in 3D space. When the cube combines with the octahedron, the mid-points of each side meet. By connecting these nodes, the cuboctahedron is formed. This offers a simple geometric reason as to why the F-orbital types can only appear once the octahedral P-orbitals are formed in the shell above.

spdandf orbital configuration

In summary, S and P orbitals appear in the atom in sequential order. The P-orbital appear outside the S-orbitals sphere. D-orbitals appear after the P-orbitals, but must be enclosed inside an S-orbital in the shell above. F-orbitals appear after the D-orbitals, and are enclosed in a set of S and P-orbitals in the shell above. We can represent this nature using a simple 2D map.

spdandf orbital map



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What does this tell orbital types?

This model resolve numerous quantum paradoxes. It explains the quantised nature of electron spin, and why there are only four types of orbital. Additionally, we find that the configuration of the orbitals is explained by the evolution of geometric forms, which follows the principles of 2D and 3D geometry.

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A new approach to the atom

This model offers a new and innovative approach to the atom that does not require vast amounts of complex mathematics to produce an accurate depiction of the electron cloud, and its qualities. By comprehending the nature of geometry, we are able to explain the most intricate details of the electron cloud, that is in alignment with tradition quantum computations of the probability waves of the electron.

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Got a Question? Then leave a comment below.

john hammond
JOHN - Engineer

Does the nature of 4D explained by the multiplication of two 2D plains also apply to the 4D sphere?


The sphere is calculated using its radius. The volume of a sphere is r³ × 4/3π. This means the volume difference between a cube and a sphere differs by 6/π. The radius is defined as half the length of the side of the square. When we raise the radius to the power of 4, the ratio between the hyper sphere and hyper cube is 1:12π. For example, (24× 4/3π) ÷44 = 1/12π. 

The radius can be expressed as 2² × 2² = 24. However, if we calculate the 4D sphere by multiplying 2 intersecting circles, the calculation is (r²×2π)². The ratio to the hypercube becomes π²/4. The difference between 1/12π and π²÷4 is 3π². When we examine these ratios in terms of scientific constants, we find, 1/12π is an expression of Z0, the impedance of the vacuum. Whereas 3π is the speed of light × π².


You mention the 4D sphere and torus. Do the D and F-orbitals have a 4D form also?


Absolutely. The D-orbitals set can be viewed as a 4D hypercube, termed the tesseract. This is often represented as two cubes that nest inside each other. The F-orbitals can be represented as the 24-cell octaplex. This form is represented by the Cuboctahedron, combined with its dual, the rhombic-dodecahedron.

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