Atomic Geometry

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⏱ 15 min read
Atomic Geometry

Atomic geometry is the world's first comprehensive geometric model of the atom, that visualises the electron cloud through three dimensional polyhedra. Whereas other models of the atom describe the atom from the perspective of energy, this model observes the geometric patterns of space generated by the 4 types of Orbital S, P, D and F. Drawing inspiration from Schrödinger's wave equations, we find the electrons fall into distinct geometric arrangements, which can be translated directly into Platonic and Archimedean Solids. These are perfectly nested inside of each other to produce a 3D representation of the stable elements on the Periodic table.


✔ Atomic geometry is the world 1st completely geometric model of the atom.
✔ S, P , D and F orbitals are 2D shapes found in the completion of the Seed of Life Mandala
✔ Orbitals combine to form geometries that can be represented as a nested set of Platonic and Archimedean Solids


Image you could hold a 3D model of an atom in your hands. Well, now you can! We are presenting a 3D geometric model of the atom that you can create just with a drawing compass, ruler, card, scissors, and glue. It is based on 3D polyhedra of specific side-lengths, that can nest inside of each other, just like Russian dolls, to accurately reflect the spatial arrangement of the electron cloud. This tactile exploration of the atom is a fantastic educational tool for students to comprehend the depths of atomic theory. With a lot of experience running workshops in Atomic Geometry children as young as seven years of age are able to cognise complex ideas such as the electron configuration, orbital shells S, P, D and F, the atomic nucleus, and electron pairing.

From the concepts of Atomic Geometry we will also dive into what we call Geoquantum Mechanics, where we suggest that we can predict the atomic radii of all stable elements on the periodic table to a much more accurate degree than what is currently proposed.

In the first part, we lay out the basic structure of the atom and its relationship to simple squares and triangles. Subsequently, we explore the four types of orbital S, P, D, and F from the perspective of 2D and then 3D. We will show how the electron configuration of the atom falls into a cube of empty space to form the atomic blueprint. Finally, we will provide a brief introduction into the field of Geoquantum Mechanics, and explain how we are able to predict the atomic radii so accurately.

In2Infinity - Atomic Geometry - holding model in hand

History of Atomic geometry

With Atomic Geometry we offer a completely new perspective of the atom, which was developed in 2015 by Colin Power, co-founder of In2Infinity. The inspiration for this model came from his in-depth exploration of compass construction. Indeed, throughout the history of humanity, this drawing technique has revealed many scientific laws and discoveries. From the ancient world of Plato and Pythagoras, Euclid’s ‘The Elements’, to great thinkers such as Leonard Da Vinci, and astronomers such as Johannes Kepler, all have employed the drawing compass in their exploration of the universe.

For the first time, we have applied these geometric techniques to the structure of the electron cloud. This journey of has brought many surprises that have revolutionised our perception of the atom, and new answers to the mysteries of the quantum world. And now, we would like to share them with you!

It all began in 2013, when I built my first set of Platonic Solids and nested them inside of each other. Whilst doing this, I rearranged the metaphysical concept of the five elements, breaking traditional convention. This gave me a completely new perception of these forms and their relationship to each other.

Shortly afterwards, in February 2014, I began writing a book on geometric concepts, drawing inspiration from ‘God’s secret formula‘ by Peter Plichta. This book exposed some interesting revelations about the periodic table: there are only 81 stable elements with a strange anomaly: Element 43 and 61 were missing. Being intrigued about this mystery, I started to turn my attention to the atomic structure.

I remember the day when I first found out that P-electron orbitals resembled the configuration of an Octahedron, one of the five Platonic Solids. On examining the other types of orbital shell, it became exceptionally clear to me that the atom was based on geometry. The fact that this geometric structure had not been recognised by mainstream science was quite astonishing to me. The orbitals clearly present simple forms: S, looks like a circle, P, a line, D, a square and F, a hexagon.

Amazingly, this pattern was exactly the same as a famous geometry that I had been drawing for years: the Seed of Life. In spiritual teachings, the Seed of Life is known as the ‘Blueprint of Creation’, in scientific terms, we suggest that it maps the atomic structure. Once I discovered this connection, I delved deeply into quantum physics with more vigour than ever.

In spiritual teachings, the Seed of Life is known as the 'Blueprint of Creation', in scientific terms, we suggest it maps the atomic structure. Click To Tweet

It was in 2015 when founding In2Infinity, that I began to rewrite my first book. This included many new concepts around the geometry of the atom. As I ploughed through various quantum physical concepts, I was surprised at how neatly they fit this new geometric model.

In may 2015, whilst preparing a course on Metatron’s Cube, I build the nested atomic set for the first time. This provided me with another great revelation: the atomic blueprint can be mapped in 2D using the Fruit of Life as a template. Each shell of the atom appeared to me in perfect order. Over the course of two weeks I built the whole set of solids and nested them to visualise a geometric model of the atom without the need of a computer. Instead, you can simply build it as a tangible object just out of card paper. As far as I am aware, this is the only model of the atom that you can hold in your hand.

The atomic blueprint can be mapped in 2D using the Fruit of Life as a template. Click To Tweet

The further I examined this structure, the more I began to see a completely different perspective of reality. The atom was no longer a nucleus, surrounded by an apparently chaotic and probabilistic electron cloud, instead, the cloud was highly structured. Through investigation, I began to see reasons for the quantisation of the atomic shells, the stability of the atomic structure, and the mechanisms by which a wave can collapse into a particle.

While the concept of Atomic Geometry is still in its infancy, we have finally gathered enough evidence to begin outlining this geometric theory viable for scientific investigation. We encourage students, scientists and teachers to build this 3D geometry of the atom, which I believe will advance our understanding of the atom and the universe at large.

Colin Power

The Atom

Almost everyone today is aware that reality is made up of atoms. Each is formed of a nucleus made of protons and neutrons that accounts for over 99.99% of the atomic mass. This is surrounded by an electron cloud, which falls into distinct shells spaced at specific distances from the nucleus. This is often depicted as a dot surrounded by circular orbits of electrons, termed the Bohr Model. There are only 6 main orbital shells that make up all stable atoms, with the 7th being naturally radioactive, forming the non-stable elements of the periodic table. Each shell determines where an electron can and cannot exist, i.e. the space around the atoms is quantised. This is why we use the term ‘Quantum Physics’ as a methodology for investigating the nature of atoms.

In2Infinity - Atomic Geometry - the Atom Proton Neutron Electron and Bohr Model

The Electron Cloud

The Bohr Model was disproven in the 1920’s when Louis de Broglie discovered that electrons exhibit wave-like characteristics. So why is it, that the Bohr Model is still taught in schools?

The answer is, it is conceptually simple to understand and easy for chemists to work with. The downside is, most people have an outdated and inaccurate picture of the atom, and how it really works.

The wave-like qualities of the electron led Werner Heisenberg to the hypothesis that they exist as a wave of probability. This was supported by the fact that the exact location of the electron defied measurement. As both light and electrons seemed to exhibited wave-particle phenomena, it was concluded that the act of measurement (observation) itself collapses the wave into a single particle. This established the ‘electron cloud‘ as a function of probability, which is the prevailing model of today, called the Copenhagen Interpretation. Extensions of this theory conclude that the electron has the possibility of appearing anywhere in the universe. Yet, the exact mechanism that accounts for this still remains an unsolved mystery.

In2Infinity - Atomic Geometry - the Atom Probability Cloud

While the attention is often on the strange behaviour of electrons, we want to highlight their spatial limitation. While most learn about the atomic shells in school, we can dive into more detail and highlight the sub shells, called ‘sub-orbitals‘. These have been given the arbitrary labels of S, P, D and F. It is important to note that these are the only four that have ever been observed to exist. Often, sub-orbital G is placed along side this set, but this is a purely theoretical concept based on the extension of the aufbau principle. In reality, the atomic structure can expand up to element 83 before it becomes unstable (radioactive).

A geometric perspective

Thus far, all models of the atom have been created by gathering data through energetic means using alpha, beta and gamma rays. This has resulted in the discovery of sub-orbital shells with specific configurations. It is this spatial arrangement which creates the characteristics of compounds and molecules within the discipline of molecular geometry. However, so far, the underlying principles have never been explored from the perspective of geometry in any detail. Yet when we do, we find the results are quite surprising.

Atomic Geometry represents the sub-orbitals using simple 2D and 3D geometry. By doing so, it reveals that the electron cloud is highly geometric. Additionally, extending into the field of Geoquantum Mechanics we were able to model dynamic pathways, whereby energy can be quantised via certain geometric processes. From this geometric perspective we can explain why electron are quantised to specific energy levels, and why the atomic structure is completely stable. So far, these questions are still unsolved by traditional quantum theory.

Atomic Geometry Video

To gain a clearer overview of Atomic Geometry, its principles and explanations, we have created a short animation video that shows how these four different orbitals combine to form an interlocking structure, based on Platonic and Archimedean Solids.

Electron Orbitals

The S, P, D and F-orbitals of the electron cloud define precise spaces around the nucleus where electrons can appear with the ‘highest probability’. Current theories of Quantum Wave Mechanics have noted the relationship of these orbital formations to spherical harmonics. Despite consistent experimental evidence, this phenomenon lacks so far a clear conclusion.

In2Infinity Atomic Geometry Electronorbitals

Square number code

When we study the electron cloud, it reveals that sub-orbitals appear in a specific order and number. This is termed the ‘Electron Configuration‘.

The first is the S-orbital, which appears once in each shell. Subsequently, P-orbitals appear in sets of 3, D-orbitals in sets of 5, and F-orbitals in sets of 7. It is interesting to note that the number 7 also appears to be the limitation for the maximum number of shells up to Uranium (92), the last naturally radioactive element.

The expanding numerical pattern of the orbitals of 1, 3, 5 and 7 is an odd number series, which conveniently falls into a triangular formation – our first geometric insight into the atomic structure. More geometry becomes apparent when we detail the orbitals within each sub-shell. When adding up all orbital pairs per shell, it exposes a numerical pattern from 1, 4, 9 and finally 16. 16 is the maximum number of electron pairs that can be found in a single shell.

In2Infinity - Atomic Geometry - Squaring Pattern Orbital Shells

This construct of 1, 4, 9, and 16 can be created from a square number series, i.e. 1², 2², 3² and 4². Before examining the geometry of the orbital types themselves, it already becomes clear that there is a numerical mechanism at play which limits the nature of the electron cloud. The combination of odd numbers 1-7, are compounded with each new shell, to generate square numbers which relate to the number of electrons.

The electron cloud is structured through a combination of odd numbers 1-7, that are compounded with each new shell, to generate square numbers series. Click To Tweet
In2Infinity - Atomic Geometry - Quantum Numbers Geometry
NOTE: Each orbital consists of two electrons which makes an ‘electron pair‘. In Quantum Theory, these are differentiated from each other by their spin (up or down). Furthermore, the quantum state of an electron is qualified by four quantum numbers.
  1. N = 1, 2, 3, 4, 5, 6, 7 = shell (energy level)
  2. L = 0, 1, 2, 3 = orbital type (sub-shell, 0 = S, 1 = P, 2 = D, 3 = F,)
  3. M (L) = – L…L (type of orbital L, i.e. 1S, 3P, 5S, 7F / electron pair)
  4. M (S) = +1/2 or -1/2 (spin of electron within electron pair)

2D orbital geometry

When we start examining the shapes of each sub-orbital, we find even more startling geometry. S-orbitals are the only type that are completely spherical. P-orbitals divide the sphere into two equal lopes, one up and one down. The pattern continues as the lobe divides again to form the cross-shaped D-orbitals. So far, this process appears to follow a simple pattern, driven by a process of division. However, when it comes to the F-orbitals the pattern stops, and instead the electrons fall into a hexagonal arrangement.

In2Infinity - Atomic Geometry - SPDF Orbital Space

Orbitals and Dimensions

Whilst the reason for this geometric pattern can be extrapolated from complicated Schrödinger equations, geometry does offers a far simpler solution. We can map the configuration of different sub-orbitals in accordance with conventional geometric laws that govern the expansion of dimensional space, from zero to 2D:

  • The dot exhibits zero dimension (0D).
  • The line is one dimensional (1D).
  • The triangle, square, hexagon and circle are two dimensional (2D).

In geometric terms, the S-orbital can be related to a dot (or circle), from which dimensional space begins to expand. As P-orbitals exhibit two distinct lobes, the dot has divided, to generate a line (1D). The dot divides again to create the square, or more accurately, the cross shaped D-orbital. A single electron from this pair now occupies two lobes, or a line, that cross the second electron at 90°. Finally, the F-orbitals break the pattern of division, forming a hexagon. In this final stage a single electron occupies three lobes, to form a triangle, the smallest regular 2D shape.

In2Infinity - Atomic Geometry - SPDF Orbitals defined in Geometry as cot, line, cross and hexagon

The Cross and the Square

The dot, line, triangle and square are the most fundamental forms in geometry which appear to structure the electron cloud. But why do hexagonal F-orbitals appear after the cross-shaped D-orbitals?

To clarify this, we need to introduce aspects of our new concept called ‘Inverse Geometry’. This proposes the existence of a geometric form, generated by two lines that cross at 90° to each other. The ‘cross’ is distinctly different from the square. While the 90° angles of a square act as a container, the 90° angles of the cross surround a dot at the centre stretching into infinity without a boundary. When we combine the square and the cross, it divides the square into 4 quadrants. The cross is a simpler form than the triangle, being construct from two one dimensional lines at 90°. We call this 0², a concept that is extensively outlined in Universal Math. We propose that this is the reason why the cross shaped D-orbitals proceed the hexagonal shaped F-orbitals, that conform to a 2D triangle. The progression of dimensional space is therefore:

  1. Dot (0D)
  2. Line (1D)
  3. Cross (1D on a 2D plane)
  4. Triangle (smallest 2D shape)

F-orbital pairs can be created from two equally overlapping triangles, a hexagram. In symbolism, this is known as the Star of David. Similarly, the cross (e.g. celtic) is another prominent symbols in religious and spiritual cultures.Are these symbols showing us the secrets of the very fabric of reality?

In2Infinity - Atomic Geometry -Symbolism - D orbitals depict Celtic Cross and F orbitals Star of David

Orbital Structure and the Seed of Life

From the perspective of geometry, we draw this expansion from overlapping circles. Once we reach 7 interlocking circles, the Seed of Life, a symbol found throughout the ancient world, appears.

The process of drawing the Seed of Life reveals exactly how the sub-orbitals of the electron cloud are able to form. Click To Tweet

The process of drawing the Seed of Life reveals exactly how the sub-orbitals of the electron cloud are able to form. After the first circle, which represents the S-orbital, a second is added to generate a form called the Vesica Piscis. This form defines two points (nodes) from which we can create a line, the shape of a P-orbital. Adding two more circles to each node produces a form called the Trion Re. The four overlapping circles generate nodes that divide space into 4 quadrants. This creates a cross that matches the geometry of the D-orbitals. Notice, there is no outer square boundary. Finally, the triangle and hexagon come into existence from the completion of the Seed of Life Mandala, which is indicative of F-orbitals.

Viewed like this, it makes sense that the geometry of D-orbitals proceeds the appearance of F-orbitals, which gives a very different perception of the formation of space.

In2Infinity -Atomic Geometry - Seed of Life maps SPDF Orbitals

Positive and Negative Space

These observations also allow us to see the second dimension in a new light. D-orbitals occupy a 2D plane, but are constructed from a pair of one-dimensional lines. On a square tessellation, the cross will divide each square into four. F-orbitals consist of three one-dimensional lines arranged in a hexagon, which divide a hexagonal tessellation into triangles, the final boundary of the atomic structure. So, why are there not more sub-orbital shapes?

Again, we can find answers to this in the rules of geometry. The only two regular shapes that can tessellate a 2D plane with just two colours is the square and the triangle.

An electron can only fall into one of the two states, up or down. In order to differentiate these states requires a space that can be uniformly divided into just two ‘colours’. None of the surrounding space can exhibit the same state (colour), in the same way. This is similar to the mechanism of a computer where each byte is stored in an on/off state.

Therefore, it seems reasonable to assume that electron pairs can only exist within a lattice that is isotropic in nature: a completely flat space that is uniform in all directions from a point of origin, divided into a positive (up) and negative (down). This enables electrons to appear in quantised states, restrained by the two types of regular 2D space, the triangle and square.

Based on this we suggest that the electron cloud is dividing the space around the nucleus in accordance with the laws of 2D geometry, and it is this that accounts for the two opposing quantum states of the electron, up and down, and the geometric orientation of S, P, D, and F-orbitals.

The electron cloud is dividing the space around the nucleus in accordance with the laws of 2D geometry, and it is this that accounts for the two opposing quantum states of the electron, up and down. Click To Tweet
In2Infinity - ATomic Geometry - 2types of 2D square and triangle up and down electron spin

Such a realisation is a quantum shift in thinking, because it unifies the concept of 2D and the atomic fabric of space in a completely new way. Through precise geometric principles, we have established a deep connection between the rules of 2D space and the structure of our reality.

3D orbital geometry

Having unified the sub-orbital types with geometric laws of the second dimension, it is not surprising that this extends into 3D. Each sub-orbital is comprised of a specific number to form a complete three-dimensional set. Once the shape is filled, electrons will jump to the next shell of the atom. In the next part, we will consider these formations from the perspective of 3D.

The 5 Platonic and 13 Archimedean Solids

Before we proceed with a 3D geometric explanation of the electron cloud, we need to be clear about the limitations of 3D space. Just as 2D space is limited to only two regular polygons, 3D space is limited to only five regular polyhedra, called ‘Platonic Solids’. They are very unique as they all are made from the same regular polygons, all edges have the same length and all corners have the same distance to the center.

Three of these, the Tetrahedron, Octahedron, and Icosahedron have triangular faces, whilst the Cube has square faces, and the Dodecahedron is pentagonal.

In2Infinity - Theory - Atomic Geometry - 5 Platonic Solids 2D shapes

The five Platonic Solids we transform into a set of 13 semi-regular polyhedra, called the Archimedean Solids through the process of truncation, explosion and twisting. Aside from the Truncated Tetrahedron, 12 fall into two distinct categories. One is based on the Octahedron and Cube with octahedral symmetry, and another six are derived from the Dodecahedron and Icosahedron with icosahedral symmetry.

If you are not familiar with these forms, you can explore them in great detail in our Guide to Sacred Geometry.

In2Infinity - Theory - Atomic Geometry - 5 Platonic Solids Archimedean Solids


The spherical S-orbitals are the first that appear in the atom. Each S-orbital is comprised of two electrons orientated in opposite directions, up and down. These orbital types are important in the formation of simple atomic bonds, as they are the first to appear on the outermost (Valence) shell.

In2Infinity Atomic Geometry S-Orbitals

S-Orbitals and the Torus

We can perceive S-orbitals from the perspective of a circle (2D), sphere (3D), or torus (4D). The circle is a shadow projection of a 3D sphere onto a 2D space, the sphere a 3D representation of a 4D torus. Let’s go through each dimension and how it relates to the electron pair.

In 2D, the two electrons appear opposite each other on the endpoints of the circle’s diameter, which refers to the up and down configuration. On a 3D sphere, these electrons will be exactly opposite on its surface, which represents the traditional view of particle physics. In a 4D torus, the electrons are still opposite but follow the flow of the field, explaining the up and down orientation.

The 4D torus is a dynamic field, which is in continuous motion and typically represents electromagnetic fields, who exhibit a north and south pole. Within these torus fields there is a unidirectional flow of energy, going up and down. We can represent this by simply drawing an arrow that stretches from the ‘down’ orientated electron, to pass through the centre (nucleus) and reach the ‘up’ orientated electron. The arrow represents the flow of energy through the centre of a toroidal structure.

In light of these different dimensions, we propose that the electron spin and the existence of electron pairs are a consequence of a 4D toroidal dynamic. As our perception of reality is limited to 3D, 4D phenomena do not appear to us as physical objects, but rather as electromagnetic fields found around planets, and solar systems. We also suggest that they exist around each galaxy. In fact, they seem to exist on every scale, a realisation far deeper than it may first appear.

The holy grail of quantum physics is to unify quantum gravity, the quantised phenomena of the micro scale, with the theory of general relativity, the smooth curvature of time-space at the macro scale. Could 4D geometry help us resolve this conundrum? We believe that the answers lies within geometry, which we have started to outline in Atomic Geometry and will reveal in more detail in our other ideas and theories.

The Vesica Piscis and the Bohr Radius

The Bohr radius (a0) is a scientific constant, which defines the radius of a hydrogen atom, the distance between an electron to the nucleus. It is derived from mathematical calculation rather than actual measurement. According to this standard, the radius of Hydrogen (1) is 53pm, and Helium (2) 31pm. When we divide 53 by 31 we get a very close approximation to √3.


53312=2.923 32=3


The ratio of √3:1 is very crucial when it comes to geometry. We find it in a prominent spiritual symbol known as the Vesica Piscis as the distance between the nodes (√3) to the circles radius (1).

Helium is the second element and the first noble gas on the periodic table. This non-reactive element produces an impenetrable boundary, upon which subsequent atoms can form. It is this nature that allows two hydrogen atoms to form a water molecule. If helium (2) was not a noble gas then oxygen (8) should bond to each other, yet such a molecule never occurs in nature.

√3 is not just found in the Vesica Piscis but also in the Seed of Life. The 7 equally overlapping circles can be encompassed in a larger circle, twice the size. The template gives rise to a hexagon within the central circle. Inside of the hexagon, we can draw a smaller circle, which will have a diameter of √3.

Based on the combination of the Vesica Piscis and Seed of Life, we can compose the image of a 4D torus. The Vesica Piscis represents the torus seen from the side, and the Seed of life represents the torus seen from above.

Note that in the image above, the central curves inside the Vesica Piscis are a ‘double’ line. Due to perspective, The one behind is obscured by the one on front. The resulting image produces a torus that is divided into six equal sections. The north and south pole are separated by a distance of √3. From a 4th dimensional perspective, the Vesica Piscis and Seed of Life define a torus that exactly maps the difference in atomic radii of hydrogen and helium.

The Vesica Piscis and Seed of Life define a 4D torus that exactly maps the difference in atomic radii of hydrogen and helium. Click To Tweet


S-orbitals account for the first 4 elements on the periodic table. After this, the first set of P-orbitals appear. Starting with boron (5) this set includes the essential components for biological life, namely, carbon (6), nitrogen (7) and oxygen (8). These elements appear on the far left of the periodic table. It is important to note that, with the exception of helium (2), all noble gases occur when a full set of six P-orbital electrons complete the shell.

P-Orbitals and the Octahedron

It is an intriguing fact that P-orbitals always appear in sets of three. Each is orientated at 90° to each other, forming a three-dimensional cross, that spans an x, y, and z axis. It is within this spatial arrangement, where we find the probability cloud of the electron. When we superimpose an Octahedron over the centres of each spherical P-orbital we find that it matches it perfectly. As one of the five Platonic Solids, the Octahedron seems to play a crucial role in the atomic structure, as it defines all noble gases after helium, making sense of the octet rule, whereby which atoms undergo bonds.

In2Infinity Atomic Geometry P-Orbitals make an Octahedron

4th Dimensional P-Orbitals

P-orbitals are often described as exhibiting two distinct lobes, which has been widely adopted in disciplines such as Molecular Geometry, as it explains the shapes of molecules and compounds. But when we examine it closer, we find that this is not the case as this description is based on a 3D perception of the atom. In 4D, each lope becomes connected through a torus ring. This means that the set of p-orbitals looks more like three interlocking torus fields, mapped onto an Octahedron.

The set of p-orbitals is 3 interlocking torus fields, mapped onto an Octahedron. Click To Tweet

To conclude, we have suggested that S-orbitals are 4D in nature, which explains why electrons fall into pairs. Extending this to the P-orbitals, we find that each lobe is constructed by the intersection of two torus fields positioned at 90°to each other.

Unifying 2D and 3D

The Octahedron is the only one of the five Platonic Solids that has the unique characteristic of combining both the square and triangle within its form. There are eight triangular faces on its surface, which encompass three interlocking squares. As mentioned previously, the square and triangle are the only shapes that can tessellate 2D, filling it with just two colours, the postulated reason why the two electron spins exist.

What this shows us is the idea that the 2nd dimension of space lays the foundations for 3D space. The two regular 2D planes, triangle and square, combine to form an Octahedron, the first type of 3D space in the atom after the sphere. Is the reason why P-orbitals are octahedral? Are they adhering to the qualities of regular 2D space from which electrons appropriate their dualistic nature? The concept that a combination of 2D planes can produce 3D space is a novel concept. Within current geometric thinking 2D space is defined by a 3D, which is a definition exactly inverse to what we observe in the atom.

Octahedral Space

When it comes to 3D, we naturally imagine cubic space. Yet, other spaces exist. Atomic Geometry has been framed around the concept of octahedral space, which we propose to be a kind of matrix that structures the electron cloud. This matrix seems to exist even before it is filled with electrons. The reason why believe this is so, is because the S, P, D and F-orbitals can be created only from a hydrogen atom. By charging it up beyond its spherical electron cloud, the expansion of energy follows this geometric matrix that we see as the four types of orbital. When it is devoid of electron energy, it remains undetectable. Just like a crystal clear glass, it only becomes apparent when we fill it with a coloured liquid.

All noble gases except helium, follow the nature of octahedral space, which act as boundaries, where molecular bonding does not take place. This is structured by six electrons that can appear somewhere within the lobe of a P-orbital. This geometric perception of the atom reveals the nature of electrons, and why they should be confined to specific quantised areas around the atomic nucleus.

Octahedral space can be expanded to an infinitely large array, producing ever larger octahedrons at each stage. Yet, the atom is limited to just five complete P-orbital sets. Only half of the sixth P-orbital is stable after which the structure become unstable, collapsing through radioactive decay.


The first D-orbitals appear in the 3rd shell of the atom, between the S and P-orbitals. This is not obvious when we look at the traditional periodic table, where the D-block begins in the 4th row (shell). This is because the elements are laid out in terms of their increase in energy levels.

If we analyse it in terms of space, the arrangement would look quite different. After the noble gas argon (18), the next two electrons produce an S-orbital in the 4th shell. Subsequently, the first set of D-orbitals appear in the 3rd shell. Therefore, all D-orbital elements have two S-orbital electrons that can form bonds independently (exceptions to this are the elements that defines the Aufbau Principle).

This is why D-orbital element can create such a wide variety of metal alloys. They can form molecular configurations independently of their S-orbitals that appear in the shell above. Most of these metals can be oxidised, when a free oxygen atom forms a bond with the outer S-orbitals, producing the phenomena we call rust.

Another important fact is that there are only three sets of D-orbitals that are comprised of stable atoms. The fourth set (elements 103-112) are highly radioactive, and do not appear in nature. They can only be manufactured within the lab, and tend to exist for just a fraction of a second. No element beyond 100 has ever be synthesised in any kind of macroscopic quantity observable by the human eye.

Therefore, only 3 sets of D-orbital electrons form stable atoms, which appear in the 3rd, 4th, and 5th shell. This suggests that the atom does not expand uniformly, rather, with each successive shell, the next type of orbital appears. This pattern continues up until the F-orbitals in the forth shell. After this, successive shells have one less stable orbital type.

D-Orbitals and the Cube

Out of the five D-orbitals, three fall upon the same x, y, z axis as the previous set of P-Orbitals. Each lobe of these ‘cross’ shaped orbitals are located above and below the existing P-orbital. Viewed like this, the D-orbitals are derived from the division of a P-orbital (line) into a cross (two intersecting lines). A simple process of division.

When these D-orbitals are combined they divide a Cube of empty space into eight parts. We can model this geometrically as set of eight small Cubes, complied to form a larger Cube.

In2Infinity Atomic Geometry D-Orbitals make a Cube

Atomic Duals

We came up with the term ‘cubic space’ , which has distinctly different qualities to ‘octahedral space’. The Cube is unique amongst the set of the Platonic Solids, as it is the only form that can fill space uniformly by itself. This space-filling property is descriptive of the space that we experience in daily life.

Objects are orientated in space, and can move through space without changing shape or dimension. Cubic space, as a uniform structure, is the only regular solid that can fulfil this function. Through such a uniform matrix, relative distances in space can be metered and measured.

Based on the foundations of the octahedron, which embodies both triangle and square planes, its platonic dual, the cube can form. The geometric pattern of the ‘matrix of space’ is perfectly described through the order and appearance of the electron orbital types.

D-orbitals and the Cuboctahedron

Let us next consider the spacial arrangement of a combined set of P and D-orbitals. To help us we can imagine a Cube of empty space. An Octahedron can be placed inside of a Cube in such a way that its 6 corners touch the centre of each face of the Cube. This is because the Cube and Octahedron are ‘Platonic Duals’. The Cube has 6 faces and the 8 corners, the Octahedron has 6 corners and 8 faces.

If we consider the position of the three D-orbitals, we find that they fall on the centre of each square. By connecting the set a new form appears, the Cuboctahedron. This Archimedean Solid is comprised of the faces from both the Cube and Octahedron. We will discuss this form in more detail in the section on F-Orbitals.

By considering these orbitals as a collective occupying a cube of space, we can relate it to the geometric forms that underpin their appearance. The P and D-orbital configuration of the atom is exactly modelled by an Octahedron and Cuboctahedron nested inside a Cube of space.

The P and D-orbital configuration of the atom is exactly modelled by an Octahedron and Cuboctahedron nested inside a Cube of space. Click To Tweet

However, there are still two more D-orbitals to be accounted for, which we shall look at next.

D-Orbital Torus and the Rhombi-Cuboctahedron

With three of the cross shaped D-orbitals dealt with, let us look at the orientation of the 4th. This orbital is rotated at 45° to the existing D-orbital along the x, and y axis. When the two are viewed together, it produces an Octagon.

The final orbital is of a completely different nature to the rest, as we suggest it expresses the nature of a torus field, with a lope extruded in a north and south orientation. The 45° octagonal D-orbital appears on the same plane as the torus ring. In consideration of theses geometric qualities, we postulate that it is the Rhombic-Cuboctahedron, which serves as a container. The midsection is an octagonal prism that can rotate freely, whilst the two ‘caps’ are held in place. Out the whole set of 13 Archimedean Solids, the Rhombic-Cuboctahedron is the only one that exhibits this quality. On top of that, it is the perfect form to map the final two D-orbital electron pairs.

In2Infinity Atomic Geometry D-Orbitals make a Torus Rhombicuboctahedron


The final orbital type are F-orbitals. These appear extrapolated from the order of elements in rows at the bottom of the periodic table. Just as with the D-orbitals, the periodic table suggests that the F-Orbital appear in the 5th and 6th shell of the atom. However, there is only one stable set of F-orbitals, which appears spatially in the 4th shell. The second set (elements 89-103) are radioactive.

NOTE: It is quite strange that within this radioactive block, two elements, Thorium (90) and Uranium (92) still exist on planet earth. Technically, these element should have decayed into non-existence, if they were created at the point of the Big Bang, just like all the other radioactive elements of this group. By accelerating the decay of (or depleting) Uranium or Thorium all other ‘naturally’ occurring radioactive elements (91 and 89-84) are generated. The heat that is being emitted in this reaction is commonly used in the generation energy in nuclear power plants. Any elements above 92 include Plutonium (93), however, this is only found as a trace element embedded in Uranium ores. Beyond this, we have all ‘artificial’ elements up to 100 that have only been synthesised in the lab and never in macroscopic quantities. Atoms beyond that point exists for only fractions of a second, collapsing within the blink of an eye.

F-Orbitals and the Cuboctahedron

Whereas D-orbitals from a ‘cross’, the most common orbital configuration found in the F-orbitals are hexagonal. There are four sets in total that have been defined to fall along an x, y and z axis. At this juncture, our Atomic Geometry model takes a different view of these orientations. These four hexagonal rings are the perfect fit for a Cuboctahedron. We have shown that an Octahedron combines triangular faces with a the internal geometry of three squares. The Cuboctahedron has both, square and triangle faces, with an internal geometry made of four hexagons.

In2Infinity Atomic Geometry F-Orbitals make a Cuboctahedron

Viewed like this, the complete set of orbitals follow a simple expansion, from the triangle and square, to fulfil the blueprint with the hexagon. In 3D, this transformation follows the sequence of an Octahedron, transforming through the Cube into a Cuboctahedron.

The Seed of Life expands though two mores stages, to create the Egg and Flower of Life. The first expansion adds 6 circles to the first, creating the Seed of Life, 7 circles, which represents the first set of P-orbitals, the first Octahedron. The pattern unfolds with another 6 circles, creating the Egg of Life, 13 circles, which encompasses the second Octahedron and the blueprint for the Cube, the D-Orbitals. Finally, another set six circle forms the Flower of Life with a total of 19 circle to manifest the third Octahedron and the Cuboctahedron. Within this teamplate we can also find the blueprint for other Platonic and Archimedean Solids, including the Rhombic-Cuboctahedron, that we have ascribed to the torus D-orbital.

F-Orbitals and the Rhombic-Cuboctahedron

There is a simple geometric process that determines the different orbital types. The division of a side length the into two equal parts. The process begins with a P-orbital, that generates an x,y,z axis in 3D space. The ‘cross’ shaped D-orbitals combine to divide a cube of space into eight equal parts. In doing so, they define the corner points of a Cuboctahedron. The hexagonal F-orbitals can also be orientated along the halfway point of the Cuboctahedron’s edge. Amazingly, the form that is generated is a Rhombic-Cuboctahedron. The exact polyhedra we use to identify the toroidal D-orbitals. There is a distinct difference between the D and F-orbitals. With the D-orbitals the form is generated by a torus field, whereas F orbitals derive the same from their hexagonal orbitals.

F-Orbitals and the Star-Tetrahedron

Amidst the F-orbitals we find a rather unique looking cubic shaped pair. These are the only orbital types to exhibit a three dimension space. Closer examination reveals that each electron is contained with a tetrahedron. When the two interlock at 180° opposition, they define the corners of a Cube.

In geometry, this shape is called the Star-tetrahedron. What is interesting about this form is that it contains an Octahedron at its centre. By adding 8 tetrahedra to each face of an Octahedron, the Star-Tetrahedron is created. Just as the P-orbitals begin the atomic structure with an Octahedron, the F-orbitals terminate at the Star-Tetrahedron.

Just as the P-orbitals begin the atomic structure with an Octahedron, the F-orbitals terminate at the Star-Tetrahedron. Click To Tweet
In2Infinity Atomic Geometry F-Orbitals make Star-Tetrahedron

Returning to our Cube of empty space, upon which we have mapped the previous orbitals, it is apparent that the star-tetrahedral orbitals can easily be located on each corner. In this way, each electron orbital can occupy a unique region in space. This supports the Pauli Exclusion Principle, which states that no two electrons can occupy the same quantum state. When perceived from the centre face of the Cube the electron distribution is limited to the corner points of a set of nested squares. This is a fractal image that can be repeated on into infinity. Each successive square either expands or diminishes in size by a factor of 1: √2. We call this the √2 fractal, and we have found it places an important role in the geometry of electromagnetic waves. This also accounts for the reason as to why one set of D-orbitals is off-set by 45°.

F-orbital torus and the Icosahedron

There is one last orbital left that we have yet to discuss. Whilst D-orbitals exhibit a torus orbital, within the F-orbital configurations we find a double torus. This can be viewed as a 5D hypersphere, which we have assigned the Icosahedron to.

Just as the Rhombic-Cuboctahedron is only one Archimedean Solid that exhibits a ‘rotational’ property, the same can be said of the Icosahedron from the set of 5 Platonic Solids. Deconstructed into three sections reveals a pentagonal middle prism.

In2Infinity Atomic Geometry F-Orbitals make a double torus Icosahedron

The pentagon contains the Golden Ratio (1:1.618), a particular proportion found throughout nature which relates to √5. Whilst many people have heard about the Golden Ratio, the Silver Ratio, is not so well recognised. The Silver Ratio is related to the Octagon, based on √2.

Our investigation of the atom has revealed there is an interplay at work, which is able to move electrons between the different shells and energy levels of the atom.

The Flower of Life image can be expanded through two more layers up to create the ‘Moon Flower’ (37) and ‘Flower of Heaven’ (61). These 61 circles create the blueprint for the Fruit of Life, 13 full circles, which in turn is able to produce a 2D projection of the 5 Platonic Solids. But that is not all. By connecting the nodes we can define a Cube (side-length 2), with an Octahedron (side-length √2), and Rhombic-Cuboctahedron (side-length 1) nested inside. The Star-Tetrahedron can also be mapped to each corner of the Cube, with the Icosahedron encompassing the complete image.

Similar Models

As a quick comparison between atomic models, we would like to present a table, which explains the various advantages and disadvantages of atomic geometry, compared to the Copenhagen interpretation, the Pilot Wave model and the MCAS model. The MCAS model is probably the least well known.

Therefore, as we can see, there are unique properties of Atomic Geometry. And the way it is being taught as a tactile experience becomes applicable for younger people to comprehend, and is completely in alignment with the logic of space and geometry.

Geometric Theory of the Universe

Atomic Geometry presents a complimentary model of the Atom that is compatible with existing models such as the Copenhagen interpretation and the De Broglie Pilot Wave Model. However, it also brings new concepts to the arena of quantum physics.

From the perspective of Atomic Geometry, we suggest that the nucleus of every Atom is surrounded by a particular type of geometric space. This gives rise to the quantised energy states, a fundamental characteristic of all quantum investigation. Yet, as to how this may occur has never been fully explained.

We propose that this space is not just three dimensional. In fact, orbitals have been noticed to exist in the fourth dimension and we postulate possibly the fifth. By this, we do not mean abstract concepts of dimension based on string theory rather than 1D, 2D, 3D, 4D and 5D axioms based on Euclidean geometry, such as the Platonic and Archimedean Solids.

In2Infinity - Atomic Geometry -. Euclidian Space 0D to 3D

About the Author
In2Infinity established in 2015 by Colin Power and Dr. Heike Bielek.