Atomic Geometry - In2infinity

Atomic Geometry

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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 if you could hold a 3D model of an atom in your hands. Well, now you can! Atomic Geometry can be created using simple card, scissors, and glue, that form 3D polyhedra. When created with specific side lengths, these polyhedra can be nested inside one another in such a way as to reflect the geometry of the electron cloud. This tactile exploration of the atom is a fantastic tool for younger children to comprehend the depths of atomic theory. We have found through running workshops in atomic geometry that 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.

The theory of Atomic Geometry introduces the field of Geoquantum Mechanics, which is able to produce the worlds most accurate prediction of the atomic radius of all stable elements on the periodic table. This article introduces the foundational concepts of Atomic Geometry and briefly touches on Geoquantum Mechanical theory.

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 an 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

Atomic Geometry offers a completely new perspective of the Atom, which was conceived in 2015, by Colin Power, co-founder of In2Infinity. The inspiration for this model came about through his exploration of compass construction. Throughout the history of humanity, this 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.

Now, for the first time, these geometric techniques have been applied to the structure of the electron cloud. It is no surprise that it brings truly revolutionary perceptions of the Atom, that solves many mysteries of the quantum world.

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 look 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, 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, 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 complete the world’s first geometric model of the Atom without the need of a computer. Instead, it can be build as a tangible object. 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 atomic shells, the stability of the atomic structure, and the mechanisms by which a wave can collapse into a particle.

Today, Atomic Geometry is still in its infancy. However, we believe we have finally gathered enough evidence to suggest that it is a viable theory for scientific investigation. We therefore encourage people to build the geometry and start educating themselves about the concepts of Atomic Geometry, 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 atoms 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. In fact there are only 6 main orbital shells that can form stable atoms, with the 7th being comprised of naturally radioactive non-stable elements. 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

However, this 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 appears to be, it is conceptually simple to understand and easy for chemists to work with. However, 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

However, there is an important fact that most people, even those who have studied physics, seem to miss. The shells of the Atom contain ‘sub-orbital‘ types. 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 imaginary concept based on the extension of Aufbau principle. In reality, the Atomic Structure can expand up to element 83 before it becomes unstable (radioactive).

A Geometric Persective

Thus far, all models of the Atom have been created by gathering data through energetic means. By using Alpha, Beta and Gamma rays, scientists have acquired a vast amount of data to produce an understanding of the Atoms components and structure. 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, the field of Quantum Geometry, that extends concepts of Atomic geometry, is able to determine dynamic pathways, whereby energy can be quantised via certain geometric processes. From this geometric perspective we can explain why the electrons are quantised to specific energy levels, and why the atomic structure is completely stable. 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 appear with the ‘highest probability’. Current theories of Quantum Wave Mechanics have noted the relationship of these orbital formations to Spherical Harmonics. Yet despite consistence experimental evidence that confirm this to be true, the exact reason for this phenomenon lacks 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 we have even examined the geometry of the orbital types themselves, it already becomes a clear that a numerical mechanism is at play that 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 limit 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 2 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 4 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

Examining the shapes of each sub-orbital reveals 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. Thus 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 3D 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 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 as it does not exhibit a boundary. Whilst the 90° angles of a square are found upon its outer container, the 90° angles of the cross are found surrounding the dot at the centre. Combined with a square the cross divides the square into 4 quadrants. The cross is a simpler form than the triangle, being construct from one dimensional lines. We call this 0², a concept that is extensively outlined in our new theory 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 plain)
  4. Triangle (3D , smallest shape)

F-orbitals pairs are formed of two equally overlapping triangles that generate a very familiar image called the Star of David. Similarly, the Cross (e.g. Celtic) is another prominent symbols in religious and spiritual cultures. Now, for the first time, we can identify the same symbolism embedded into 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 can map this expansion using a simple drawing compass construction. By setting the compass opening to a fixed distance, we can easily create a set of 7 interlocking circles. This image is termed the Seed of Life, a symbol found throughout the  ancient world, and it still in use today by a vast number of cultures and religions.

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) that can be used to 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 indicative of F-orbitals.

Viewed like this we perceive the an exact reasons for the geometry of D-orbitals to proceed the appearance of F-orbitals. This also suggests that the a nature by which space is formed is quite different from the existing models in use today.

In2Infinity -Atomic Geometry - Seed of Life maps SPDF Orbitals

Positive and Negative Space

These observations allow us to construct a slightly different view of the 2nd dimension. Whilst D-Orbitals occupy a 2D plane, they are constructed from a pair of one dimensional lines. When placed upon a square tapestry, the cross will divide each square into four. F-orbitals consist of hexagonal lines, which divide a hexagonal tapestry into small triangles. This produces the final boundary of the atomic structure. So, why are there not more sub-orbital shapes?

The reason can be explained by examining the rules of geometry. The only two regular shapes that can tessellate a 2D plane with just 2 colours is the square and the triangle.

An electron can only fall into one of two state, 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 a each byte of a computer needs to be stored in an on/off state within a specific transistor.

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, that is divided into a positive (up) and negative (down). This enables the electrons to appear in quantised states, restrained by the two types of regular 2D space, the triangle and square.

Viewed like 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 2nd dimension, it is not surprising that this extends into 3D. Each sub-orbital is comprised of a specific number to form a complete set. Once complete, the next shell of the atom begins to fill. In the next part, we will consider these formations from the perspective of 3D.

The 5 Platonic & 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. It is an interesting fact that, just as 2D space is limited to only 2 types of regular 2D tessellation can fill a plain with just two colours, so there are only five regular solids. These ‘Platonic Solids’ have the characteristic of being formed from the same equal-sided shapes. Three of these, the Tetrahedron, Octahedron, and Icosahedron have triangular faces, whilst the Cube has square faces, and the Dodecahedron pentagonal.

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

From 5 Platonic Solids another set of semi-regular polyhedra, called the 13 Archimedean Solids can be derived. Aside from the Truncated Tetrahedron, the other 12 fall into two distinct categories. one based on the Octahedron and Cube, that exhibit octahedral symmetry, and another six derived from the Dodecahedron and Icosahedron, that exhibit icosahedral symmetry.

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

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


S-orbitals are the first to be formed in the atom, and are the only ones to form the geometry of a sphere. Each S-orbital is comprised of two electrons, that are 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

S-orbitals can be considered from the perspective of a circle (2D), a sphere (3D), or a torus (4D). When we draw a circle using a simple drawing compass we can also consider it as a shadow projection of a 3D sphere onto 2D space. A line that passes through the circles center will divide it into two equal halves. Where the line crosses the circles circumference two nodes are defined. Just like electron pairs that appear in up or down configuration.

As this model is shadow form of a 3D sphere, we can see that the two nodes will mark opposite points somewhere upon its surface. This is representative of the traditional view held by particle physicists.

Geometrically, a torus is a 4th dimensional sphere. These geometries form electromagnetic fields, that are in continuous motion. For this reason all magnets exhibit a north and south pole. Within these torus fields there is a unidirectional flow of energy. We can represent this by simply drawing an arrow that stretches from the ‘down’ orientated electron, to pass through the centre (nucleus), to reach the ‘up’ orientated electron. the arrow now represent the flow of energy through the centre of a toroidal structure.

We believe that this 4D toroidal perspective of the S-orbitals provides a far more accurate description, that clearly defines why electrons fall into pairs. The electron orbitals are actually types of electromagnetic fields, that are 4D in nature. This is the reason why they evade precise measurement. We exist in a 3D space and so 4D phenomena is beyond our direct perception. We can use instruments to detect these fields so we know they exist. However, that do not appear to us as physical objects.

We observe electromagnetic torus fields around planets, solar systems, and we believe they exist around each galaxy. Therefore, to suggest that an S-orbital shell should take the same form is just a natural extension of logic to the smaller scales of reality. This realisation is far deeper than it may first appear. For it describes a geometric pattern that is inherent at different scales within the universe. The present goal of quantum physics is to solve the problem of quantum gravity, that also seeks to unify the theory of general relativity at the microcosmic scale with the nature of quantised phenomena at the microcosmic scale. Could it be that 4D geometry reveals a pattern of phenomena that could help resolve this conundrum? Based on the theory of Atomic Geometry and our other geometric theories, we believe the answer is yes.

The Vesica Piscis and the Bohr Radius

The Bohr radius (a0) is a scientific constant, use to define the radius of a hydrogen atom. It is derived from mathematical calculation rather than actual measurement. It is a quantity that defines the most probable distance of the electron from a theoretically infinitely dense single proton nucleus. According to this standard, the radius of Hydrogen (1) is 53pm, and Helium (2) 31pm. When we divide 53 by 31 the answer is a very close approximation to √3.


53312=2.923 32=3


This ratio √3:1 is highly prominent in the discipline of compass construction. After a circle is created, (radius 1), the next step sees the addition of a second same sized circle centred somewhere upon the circumference of the first. This form, known as the Vesica Piscis, is the foundational starting point of all compass only construction. It is a very prominent symbol that has been embraced by religions and spiritual culture.

Each circle defines the others centre in perfect unity. The two nodes are formed where the circles overlap produce a distance or √3 compared to the initial compass opening of one.

Helium atoms are the first noble gas found on the periodic table. These non-reactive element produce 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) would should bond to each other, yet such a molecule never occurs in nature.

√3 is not just found in the Vesica Piscis. Even more famous in the world of compass construction is an images called the Seed of Life. This image flows out of the Vesica Piscis by adding circles to new nodes that are create upon the circumference of the first circle. After the completion of the 7th circle, a perfect unity is formed, whereby a larger circle, twice the size, can be placed to encompass the emblem. At the centre we can draw a hexagon. By placing the compass at the centre, and contracting its opening we can draw a second circle, that sits inside the hexagon, with 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 represent 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 resultant 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 the Hydrogen and Helium atom.

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 begin to appear. Starting with boron (5) the first 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 6 p-orbital electrons complete the shell.

P-Orbitals and the Octahedron

It is an intriguing and undeniable fact that P-Orbitals always appear in sets of three. Each is specially orientated at 90° to each other, forming a three dimensional cross, that spans an x, y, and z axis. This orientation produces a definite structure inside of which there is a probability that the electron will be found. Connecting the centres of each spherical P-orbital lobe defines an  Octahedron. This is one of the 5 Platonic Solids, which are regular forms. Its appearance in the atomic structure, and the fact that they define noble gases points to a deep realisation about the nature of space itself.

In2Infinity Atomic Geometry P-Orbitals make an Octahedron

4th Dimensional P-Orbitals

Whilst P-orbitals are often describes as exhibiting two distinct lobes,  closer examination of quantum theory suggest this is not the case. This traditional view is from a 3D perspective. It has been widely adopted in disciplines such as Molecular Geometry, as it explains the shapes of  molecules and compounds.

However, when it comes to examining the nature of the electron cloud, this 3D model is insufficient. Instead we must adopt a 4D image of each orbital. Viewed like this, each lope becomes connected through a torus ring. Therefore, the correct expression for a set of p-orbitals is 3 interlocking torus fields, mapped onto an octahedron.

The correct expression for a set of p-orbitals is 3 interlocking torus fields, mapped onto an octahedron. Click To Tweet

We have suggested that the S-orbitals are 4D in nature and that this is the reason why electrons fall into pairs. This view is supported by the fact that P-orbital lopes are actually 4D torus fields. When considering a completed set, we find 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 construction. There are 8 triangular faces that are mapped onto 3 interlocking squares.

As we have shown, the square and triangle are the only shapes to tessellate a 2D plain with just two colours. This structure allows for electrons to exhibit a dual state of either up or down. With the emergence of P-orbitals as the first atomic form, after the sphere, we find their Octahedral formation is the only suitable regular polyhedra to combine both types of 2D.

From this view the 2nd dimension of space lays the foundations for 3D space. This is contrary to the presiding mainstream view that a 2D space can only exist within a 3D space. However, this assumption lacks a clear geometric process that would validate such a claim. Instead, the idea that the two regular 2D planes, triangle and square, combine with an octahedron, which is the first type of 3D space. Therefore, the reason that P-orbitals produce an octahedral form is because they are adhering to the qualities regular 2D space, from which electrons appropriate their dualistic nature.

Octahedral Space

The combination of 2D plains to produce 3D space is a novel concept within current geometric thinking. However, within the theory of Atomic Geometry we introduce the idea of Octahedral Space, that acts as the matrix of reality upon which the electron cloud is structured. Furthermore, we suggest the Octahedral space is a specific type of 3D space that has different qualities to 3D cubic space. We suggest that Octahedral Space is constantly in effect around the nucleus of an atom even before is becomes filled with energy from an electron. By adding energy to a hydrogen atom, scientist are able to capture images of the various electrons configurations, s, p, d and f. If we are able to generate this images from a simple hydrogen atom, then it stands to reason that the space that this energy is filling is always present. When the space is devoid of electron energy, then the space remain undetectable. When energy fill the shell, so the shape becomes visible. Just like a crystal clear glass, whole shape becomes apparent when filled with a coloured liquid.

Octahedral space generates the noble gasses (except helium), that are boundaries, inside of which, electrons are not able to contribute to form simple bonds with other atoms. This is structured by six electrons that can appear somewhere within the lobe of a P-orbital. This point is defined as a place where two torus fields intersect at 90°. Each of these ‘nodes’ appear as one of the Octahedrons corners. Each corner of the Octahedron is formed from by a unification of both square and triangular 2D planes. Each point in Octahedral Space is a place where 4D phenomena interact. This gives the electron its up or down quality. This logical progression of geometry reveals a completely different perspective of what electrons are, 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 as each stage. However, 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.


D-orbitals first appear in the 3rd shell of the atom. On the periodic table these element are found situated between the S and P orbitals. It is surprising the number of chemist who are not aware of this simple fact. This is primarily to do with the way that the periodic table has been structured. As electron are add to each atom, so the amount of energy contained within also grows. The classical periodic table is laid out in terms of this increase in energy.

However, the shells into which each orbital forms is 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 forms in the 3rd shell. Therefore, all D-orbital elements should have two S-orbital electrons that can form bonds independently of D-orbitals bonds. (exceptions to this are the elements that defies 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 synthesized in any kind of macroscopic quantity that can be observed by the human eye.

Therefore, there are only 3 sets of D-orbitals electrons that can form stable atoms, which appear in the 3rd, 4th, and 5th shell of the atomic structure. This knowledge reveals that the atom does not expand uniformly. Rather, with each successive shell, the next type of orbital can be added. 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 in a set, 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 8 small cubes, complied to form a larger cube.

In2Infinity Atomic Geometry D-Orbitals make a Cube

Atomic Duels

We call this ‘Cubic space’ that has certain qualities distinctly different to ‘Octahedral Space’. The Cube is unique amongst the set of Platonic Solids, as it is the only form that can fill space uniformly by itself. The 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 Octahedral Space that combines the two type of regular 2D, Cubic Space, which is the platonic duel of the Octahedron, can be generated. 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 Duels’. Whereas the Cube has 6 faces and the 8 corners, the Octahedron has 6 corners and 8 faces.

If we consider the position of three of the D-Orbtials we can find that they fall into the centre or each side of the square. By connecting the set we can create a form called the Cuboctahedron. This important polyhedra 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 are able to clearly identify 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. It is simple geometric law.

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 3 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,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 it is depicted as a torus, with a lope extruded in a north and south orientation. The octagonal D-orbital sets are found to sit on the same plane as the torus ring. In consideration of theses geometric qualities we can identify an Archimedean Solid that is a perfect container. The midsection is formed of an octagonal prism that is able to rotate freely, whilst the two ‘caps’ are held in place. This is the only one out the whole set of 13 Archimedean Solid that exhibits this quality. and what is more 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 found in the atom 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-orbtials, that upon examination of the electron configuration appears in the 4th shell. Again we find the second set (Elements 89-103) are Radioactive.

NOTE: It is a curious fact that within this block two elements, Thorium (90) and Uranium (92) are found to still exist on planet earth. Technically, these element should have decayed into non-existence, if they were created at the point of Big Bang, just like other radioactive elements of this group. It is by accelerating the decay of (or depleting)  Uranium or Thorium that the other ‘naturally’ occurring radioactive elements (91 and 89-84) can be produce. The heat generated is commonly used in the generation energy in nuclear power plants. Any elements above 92 include Plutonium (93), however, this is only found in trace element embedded in Uranium ores. Beyond this the ‘artificial’ elements are found, which need to be synthesised in the lab. Only elements up to 100 have ever been synthesised in macroscopic quantities that can observed with the naked eye. 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 4 set in total that are suggested to fall along an x, y and z axis. At this juncture, Atomic Geometry takes a different view of these orientations. These four hexagonal rings are the perfect fit for a Cuboctahedron. We have shown the an Octahedron combines triangular faces with a the internal geometry of three square. The Cuboctahedron is formed of 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. However, in 3D this transformation appears as 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 Seed of life begins by adding 6 circles to the first. just like the first set of P-orbitals. The pattern now unfolds by adding 6 circle to the nodes around the outside. A second set of P-orbitals. This produces the Egg of Life, and the Blueprint for the Cube, or D-Orbitals. Finally another set six circle (P-orbitals) and the Cuboctahedron Manifests. Within these images we can also find the blueprint for other Platonic and Archimedean, including the Rhombic-Cuboctahedron, that we have ascribed to the D-orbital torus orbitals.

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 P-orbital, that generate 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 D-orbtals torus fields. There is a distinct difference between In the case of the D and F-orbitals Rhombic-Cuboctahedra. 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 type to exhibit a three dimension space. Closer examination reveals that each electron is contained with a tetrahedron. When the two interlock in 180° opposition to define the corners of a Cube.

In Geometry this shape is called a Star-tetrahedron. What is interesting about this form is that it contains as its centre an Octahedron. 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, so the F-orbitals terminate it with a Star-tetrahedron.

Just as the P-orbitals begin the atomic structure with an octahedron, so the F-orbitals terminate it with a 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 we is is apparent that the Star-tetrahedral orbitals can easily be located on each corner. Is this way each electron orbital can occupy a unique region in space. This supports the Pauli Exclusion Principle, which states, no two electrons can occupy the same quantum state. When perceived from the centre face of the cube the electron distribution is found to occupy 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 expandes 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-Orbitals 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. However, in Atomic Geometry we assign the Icosahedron to this particular orbital.

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 is found throughout nature that is derived from √5. Whilst many people have heard about the Golden, the silver ratio, is not so well recognised. This is found within the Octagon and is based of created from √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 2 more layers up to create the ‘Flower of Heaven’. These 61 circle for the blueprint for the Fruit of Life, 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 compete image.

Geoquantum Mechanics

Thus far we have ascribed various polyhedra to each orbital type. In fact, we have identified the appearance of other solids such as the Truncated Cube and Octahedron, and Snub Cube.

Whilst a comparison of orbital types to fundamental solids provides a compelling model of the atom, in and of itself, it does not provide any kind of experimental evidence. In order to produce more support for the theory of Atomic Geometry, we should be able to tie in this geometric view with experimental data. As we are suggesting that the atom is a geometric construct is make sense that the sizes of each atom should follow a predictable pattern based on the polyhedra.

The discipline of ‘Geoquantum Mechanics‘, or ‘Geo Mechanics‘, is a field of study that examines the ratios created by the transformation of polyhedra through specific geometric processes. In the final part of this article, we will provide a very basic introduction to this fascinating area of study.

In, Mid, and Out Spheres

All solids exhibit an In-sphere, Mid-sphere, and Out-sphere, (or circumsphere). We prefer to use the words IN, OUT and MID to maintain clarity. Let us take the example of a Cube with a side length of 1. The diagonal of its side length will measure √2, and the distance between opposite corners will be √3. therefore, a sphere placed inside of the cube will have a diameter of 1, the IN-sphere. By increasing the size of the sphere, it will touch the centre of each side, producing a MID-sphere of √2. Finally the OUT sphere encompasses the Cube.

Sketching out the Atom

Armed with the knowledge of the In, Mid and Out sphere of a Cube, side length 1, we can begin to map the Geometry of the Atom.

The atomic radius of each type of atom have been determined experimentally through measurement to within a tollerance of 5 pico-meter (pm). Within the theoretical framework of quantum theory, two datasets have been produced that try to explain the variations in atomic sizes. We can make a comparison between these data sets and see how Atomic Geometry compares.

If we take a very rough overview is can be seen that almost every block exhibits points structure where a sequential series of elements share exactly the same radii. Putting aside the the P1 and P3 orbitals for a moment and turning attention to the other orbitals, we notice there seems to be a specific radius where each set levels off. P2 elements level off at around 100 pm, whereas P4, all the D-block elements level off at around 140pm. The first half of the F-block elements have radii of about 185 pm, falling to 175pm at around the halfway point.

If we imagine a cube with a side length of 200 then we find that each type of orbital radii roughly falls over the point where the IN, MID and OUT sphere are located. Whilst this is just a rough overview it a good starting point for scaling our atomic model the the values of the radii. Not only do the orbitals fall at around the correct radius, that also fall into the cube in the exact configuration we have allocated for each orbital type.

These values are only a rough guide, and in reality there is a tremendous variation in size for each type of atom. However, by using the principles of that transform Platonic into Archimedean Solids, we are able to generate a far more accurate match. So accurate in fact, that it supersedes all other existing predictions.

The Extended Jitterbug

We are not the first people to suggest that the atom follow a geometric structure. A similar idea of the atom was conceived of by Buckminster Fuller. He discovered that by collapsing the square faces of a Cuboctahedron into two triangles will create an Icosahedron. Continuing this process produces an Octahedron at the final stage. This transformation was termed the ‘Jitterbug’ and was suggested by Fuller to be in operation within the atomic structure.

Both the Octahedron and Cuboctahedron are 2 forms we have identified from the orbital geometries.

Within the theory of Geoquantum Mechanics we extend the jitterbug concept to include two new forms, the Snub Cube and Rhombic-Cuboctahedron. The ‘Extended Jitterbug’ contains five polyhedra, with the Cuboctahedron located at the centre of the set. With the exception of the Snub Cube, all of these geometries have been identified as having a particular relationship to the different orbital types. The pattern terminates with the Rhombic-Cuboctahedron, which is also formed by the arrangement of the hexagonal F-orbitals. What follows is a small introduction to Geoquantum Mechanics, that we believe is enough to prove that validity of Atomic Geometry.

In2Infinity Atomic Geometry Extended Jitterbug Buckminster Fuller

Geometric Truncation

The second important geometric transformation used in Geoquantum mechanics is ‘Truncation’. This is enacted by the division of the side length of a Platonic Solid into two or three. By dividing the side of a Cube or Octahedron in half, the Cuboctahedron is formed. By dividing the sides into three, the corners of each polyhedra can be removed, forming the truncated Cube or Octahedron.

We can place these objects in a row, with the Octahedron at one end and the Cube at the other. Comparing this to the ‘Extended Jitterbug’ both begin with an Octahedron (P-orbital). The truncated series collapses the Octahedron into a Cuboctahedron, whereas the Jitterbug expands into the Cuboctahedron. The first diminishes in size and the latter grows in size, meaning that the two Cuboctahedra exhibit side lengths that are double in size.

In2Infinity - Theory - Atomic Geometry - Truncation of Octahedron and Cube into Archimedean Solids

Compound Solids

The final geometric concept utilised by Geoquantum mechanics is not a process, rather a combination of platonic duels scaled to a specific ratio. A compound solid is one where the mid-sphere of the dual pair is exactly the same size. The Star-Tetrahedron is an example of a compound of two Tetrahedra. The mid-sphere, in this case, contains an Octahedron. The Cube and Octahedron also form a perfect compound, and this time we find a Cuboctahedron occupying the central space. The last of the simple Platonic Compound involves the combination of a Dodecahedron and Icosahedron.

In all of the examples above, we see that the intersection each solid is located at the halfway point on its sides. Previously we identified the same mathematical process to define the position of each orbital type. The division of a side into two equal parts.

Geoquantum Blueprint

We can combine the three previous geometric concepts, the Jitterbug, Truncation, and Compounds, to produce the ‘Geoquantum Blueprint’ that we can use to map the atomic structure.

The compound of the Cube and Octahedron, share the same mid-sphere, which  each defines a Cuboctahedron at its center. Both the Octahedron and Cube can have a portion of their corners removed to produce their truncated versions.

The Octahedron can explode through the Jitterbug process, to create a larger Icosahedron and Cuboctahedron (double in size), and then on to create the Snub Cube and Rhombi-Cuboctahedron.

The Cuboctahedron in the centre of the cube/octahedron compound can also expand and shrink through the same jitterbug process. As the Cuboctahedron collapses into a smaller octahedron, so new compounds can be formed, firstly from the Icosahedron/dodecahedron pair and then another Octahedron/Cube. This can also be truncated and als contains a smaller Cuboctahedron as its centre. Again it can expand and contact in size, through the extended jitterbug process.

Each of these polyhedra has an IN, MID, and OUT sphere, most of which exhibit different sizes. However, some of these spheres do fall in exactly the same space. The mid-sphere of the Cube/Octahedron compound, which also form the out-sphere of the Cuboctahedron, for example.

This Geo-mechanical scaling is what we believe structures, the not just the radii of the atomic shells, but more incredibly, the variation of atomic radii for every element on the periodic table.

Predicting Atomic Radii

When we think of an atom, we often imagine that the size must be increasing in incremental steps as more electrons begin to fill each shell. However, this is far from the truth. The atomic radius seems to vary widely for each type of atom. Additionally the different sizes seem not to follow any kind of predictable pattern. Sometime we see noble gases like Helium (2) and Neon (10) expand in size, whereas others, such a Argon (18), contract in size. At the same time we find certain element exhibiting the exact same radius. Why?

Using the Geoquantum Mechanical model we are able to explain this large variation, whilst maintaining a logical geometrical process. In this way we can produce a curvature that almost exactly matches the experimentally measured values for all the atomic radii on the periodic table.

Currently, there are two sets of data that are used to try and calculate the atomic radii. The first is the Bohr Radius, which is a theoretical value based on an infinitely dense atomic nucleus. The second is the Van da Waal radius, which considers the atom as a hard shelled sphere.

However, for the majority of elements we see that the predicted values do not even come close to the data extrapolated by experimental results, that are accurate within a tolerance of about 5 picometers.

The table below shows the results generated for each atomic radii for the first 18 elements. The blue line provides the results generated by experimental data. the dotted green line might be hard to see, as it is so close to the measured data, whereas the red dotted line depicted the data derived from the Bohr radius, and the Yellow line from the Van da Waal radius. Clearly Geoquantum mechanics is a more accurate representation of experimental results.

All of these radii are derived from the Geoquantum Blueprint. Two Cube/Octahedron Compounds are scaled through the extended jitterbug, so that the fist exhibits a mid-sphere radius of 50pm and the second a radius 100pm

In the graph above the green sections produce radii based on the Cube, Truncated Cube, Snub Cube, or Rhombi-Cuboctahedron, the red section is based solely on the Octahedron, and the blue section is based on the Icosahedron/dodecahedron compound.

From this we can see that the first 3 element expand through a Cube and then an Octahedron, after which the 4th element collapses back onto a truncated version of the cube. The atoms diminish in size, until 50pm, which is the mid-sphere of the first compound. Then Neon (10) grows dramatically in size, filling the pentagonal mid-sphere of the Icosahedron. The 10 electrons of this noble gas, are a perfect fit for the rotation midsection of an icosahedron.

As more electrons are added, the radii decreases through an Icosahedron, compound with a Dodecahedron, before collapsing into the Snub Cube. The atom continues to decrease in size, levelling out at element number 15-17. This occurs in the mid-sphere of the second Cube/Octahedron compound, which also happens to contain a third polyhedra, the Cuboctahedron. Having fill each of these forms that occupy the same space, the atom  suddenly collapses in size by a factor of √2, as it falls into the in-sphere of the Cube or onto the out-sphere of an octahedron nested inside. this completed octahedra now forms the 3rd noble gas. Argon (18).

This explains why Argon (18) has a radius of around 71pm, whereas Helium (2) has a much larger radius of 120pm. It all follows a logical geometric progression. In this way we are able, not just to construct the most accurate model of the Atomic radii so far produced by any atomic theory. We are also able to show the logic that produces it.

The implication it that the space around the atom is geometric, and that is why the electrons are held in such stable ‘orbits’. This insight sheds a completely new light on the functioning of the atom, which validates the theory of Atomic Geometry, and introduces another new filed of science to the world. Geoquantum Mechanics.

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.

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In 2 Infinity - established in 2015 -