# 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.

## Overview

Our innovative solution to the nature of infinity is based in the concept that numbers do not only exist on a line. The numerical theory of modern mathematics has been confined to the concept of a number line, with the limitation of number only extending to the complex plain. Yet we suggest this is only the beginning. Our new Geometric Mathematical Theory exhibits a numerical number space that can be enacted in much higher dimensional spaces that is traditionally considered possible. In this way we are able to explore the nature of infinity is a completely new way. Based on the foundational principles of Euclidean Geometry, extended into higher dimensions we are able to  transform, reflect, rotate, and fold number space, transforming many of the assumed law of present number 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 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.

## Geo Quantum Mechanics

From the concepts of Atomic Geometry we will also dive into what we call Geo-quantum 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.

## Watch the 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.

## Overview

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 recognise complex ideas such as the electron configuration, orbital shells S, P, D and F, the atomic nucleus, and electron pairing.

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.

## Geo Quantum Mechanics

From the concepts of Atomic Geometry we will also dive into what we call Geo-quantum 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.

## Watch the 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.

## Key Points

• 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

THE

# Concept

## A Geometry perspective of the Atom

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!

## 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.

## 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.

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.

## 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.

## 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.

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.

## Key point

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.

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.

## 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.

## 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.

## Key point

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.

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.

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.

# S-Orbitals

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.

## 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.

# P-Orbitals

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.

## Key point

The set of p-orbitals is 3 interlocking torus fields, mapped onto 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.

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.

# D-Orbitals

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.

## 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.

## Key point

The P and D-orbital configuration of the atom is exactly modelled by The Corners of an Octahedron Nested in a Cuboctahedron.

## 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.

# F-Orbitals

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.

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.

## 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.

## Key point

Just as the P-orbitals begin the atomic structure with an Octahedron, the F-orbitals terminate at the Star-Tetrahedron.

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## D-Orbital Torus and the Rhombi-Cuboctahedron

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.

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.

## 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.

## 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.

THE

# Conclusion

## What does this tell us about the Atom?

Unlike other models of the atom, Atomic Geometry provides a clear description, that can be modelled using simple geometry. This makes understanding the sub-orbit structure far more simple than any other model of the atom.

## Atomic Geometry: a fresh perspective.

Atomic geometry is applicable to the electron cloud that surrounds the hydrogen atom, from which the scientific data of the S, P D and F orbitals have been collated. It offers a clear view of the fractal nature of space. The same structures can also be seen to organise other physical phenomena too, such as the planets of our solar system. This in turn lends us to a new geometric model of the universe that begins to solve some of the most perplexing problems facing traditional atomic modles.

### Why does light not have any mass?

Dimensionless Science and Geometric Maths that provides a completely new perspective on why light does not have any mass. When the standard model of science suggests that light has a mass of zero, many people will look at the equation E=mc2 and, replacing the value of mass for a zero,

### From 2D to 3D through rotation

Traditionally we see cube as the assumed shape of space. Yet we propose a very different model of how space gets transformed from 2D to 3D.

### Circles of Infinity

We can demonstrate that circles of infinity have an infinite number of sides, but what can that teach us about the nature of infinity?

###### Question?

This post seem quite striaght forwards, but why hasn’t science recognised this geometric nature?

We have tried to approach the scientific community with our ideas. Unfortunately, we have not had much success. This might be because QM is generally involved with quite complicated mathematics, and so the geometric model, and its simplicity, may not be taken seriously, simple bacause it is so easy to grasp