Introduction

Atomic Geometry is a geometric model of the atom that maps the four orbital types — S, P, D, and F — onto Platonic and Archimedean Solids nested perfectly inside one another. It produces a 3D model of every stable element on the periodic table that is significantly more accurate than the Bohr model — over 500% more accurate for helium, for example — and requires no equations, only geometry.

Where all previous models of the atom have been built from energetic data, Atomic Geometry asks what geometry underlies those patterns. The results reveal that the electron cloud is not a probabilistic blur but a highly ordered geometric structure.

Key takeaways

  • The four orbital types (S, P, D, F) map directly onto Platonic and Archimedean Solids — a geometric sequence that predicts atomic radii more accurately than the Bohr model and explains electron configuration without probability theory.
  • The electron cloud is not a probabilistic particle but a 4D toroidal field: the up/down spin of electron pairs, the octahedral symmetry of P-orbitals, and the cubic symmetry of D-orbitals all emerge naturally from 4D geometric principles.
  • Atomic Geometry can be physically constructed — using nothing more than a compass, ruler, card, scissors, and glue — making advanced quantum concepts accessible to learners of any age.

Watch: Atomic Geometry Overview

Atomic Geometry — 3D animation showing how the four orbital types combine into an interlocking structure based on Platonic and Archimedean Solids.

The Atom

Almost everyone knows that matter is made of atoms. Each atom has a nucleus of protons and neutrons accounting for over 99.99% of the atomic mass, surrounded by an electron cloud that falls into distinct shells at specific distances from the nucleus. There are only 6 main orbital shells that make up all stable atoms — the 7th being naturally radioactive.

Diagram of an atom showing the nucleus of protons and neutrons surrounded by electron orbital shells — the Bohr model
The standard Bohr model of the atom: a nucleus of protons and neutrons surrounded by circular electron orbits. Atomic Geometry replaces this with a 3D geometric model built from Platonic and Archimedean Solids.

The Electron Cloud

The Bohr model was overturned in the 1920s when Louis de Broglie discovered that electrons exhibit wave-like characteristics. This led Werner Heisenberg to propose that electrons exist as a wave of probability — a notion supported by the fact that the electron's exact location defies measurement. The act of measurement itself was said to collapse the wave into a particle. This established the electron cloud as a function of probability: the Copenhagen Interpretation, which remains the dominant model today.

The quantum probability cloud model of the electron — a diffuse cloud of probability density around the atomic nucleus
The Copenhagen interpretation: the electron exists as a probability cloud rather than a definite orbit. Atomic Geometry offers a deterministic geometric alternative.

The idea that the electron might not be a particle has rarely been explored by the mainstream. Yet our wave solutions to the Ultraviolet Catastrophe and the Photoelectric Effect demonstrate that the photon particle proposed by Einstein is not required — both phenomena can be resolved with a pure wave model. Applying the same logic to the electron cloud, it can be understood not as a particle but as a 4D field of energy — dissolving the wave-particle paradox and removing the need for probability theory.

For more on the wave-based alternative, see Is there an alternative to wave-particle duality?

Electron Orbitals

The S, P, D and F electron orbitals shown as 3D probability cloud shapes around an atomic nucleus
The four orbital types — S, P, D, and F — define the precise regions around the nucleus where electrons appear with highest probability. Atomic Geometry maps each type to a specific geometric form.

The S, P, D, and F orbitals define precise spaces around the nucleus. Current quantum wave mechanics has noted their relationship to spherical harmonics, but despite consistent experimental evidence no clear geometric conclusion has ever been drawn — until now.

Electron Configuration

The sub-orbitals appear in a specific order and number — the Electron Configuration. S-orbitals appear once per shell; P-orbitals in sets of 3; D-orbitals in sets of 5; F-orbitals in sets of 7. The pattern 1, 3, 5, 7 is an odd-number series that falls naturally into a triangular formation — the first geometric insight into atomic structure.

When the orbital pairs per shell are totalled, the pattern 1, 4, 9, 16 emerges — a square-number series (1², 2², 3², 4²). The maximum number of electron pairs in a single shell is 16.

Diagram showing the square number pattern of electron pairs across atomic shells: 1, 4, 9, 16
The squaring pattern of orbital shells: the number of electron pairs per shell follows the sequence 1², 2², 3², 4² — a geometric mechanism encoded in the electron cloud.

The four quantum numbers — a geometric reading

Each electron's state is defined by four quantum numbers. Atomic Geometry gives each a geometric meaning:

  • N = 1–7  (shell / energy level) → geometric nesting depth
  • L = 0, 1, 2, 3  (S, P, D, F) → Platonic or Archimedean Solid type
  • Mₗ = −L…L  (orbital orientation) → face or axis of the containing solid
  • Mₛ = ±½  (spin) → north/south pole of the 4D torus field
Diagram of the four quantum numbers N, L, M and S that define each electron's state in the atom
The four quantum numbers visualised. In Atomic Geometry each maps directly to a geometric property rather than an abstract label.

2D Orbital Geometry

When we examine the shapes of each sub-orbital, the geometry deepens. S-orbitals are spherical. P-orbitals divide the sphere into two lobes — one up, one down. D-orbitals divide again to form a cross. When it comes to F-orbitals, the pattern shifts to a hexagonal arrangement. This progression — sphere, line, cross, hexagon — corresponds exactly to the four geometric primitives constructible from regular 2D tessellation. For more on this, see 2D Orbital Geometry of the Electron Cloud.

The shapes of the S, P, D and F electron orbitals shown side by side — sphere, dumbbell, cross and hexagonal
S (sphere), P (dumbbell), D (cross), and F (hexagonal) orbital shapes — a geometric sequence that maps onto the four primitives of 2D tessellation.

3D Orbital Geometry — The Platonic and Archimedean Solids

Each sub-orbital comprises a specific number of lobes to form a complete 3D set. In 3D, Atomic Geometry maps these sets onto the five Platonic Solids and thirteen Archimedean Solids — the only regular and semi-regular polyhedra in three dimensions.

The five Platonic Solids — the only regular polyhedra in 3D space. Three (Tetrahedron, Octahedron, Icosahedron) have triangular faces; the Cube has square faces; the Dodecahedron is pentagonal. Drag to rotate.
The five Platonic Solids alongside the thirteen Archimedean Solids — all the regular and semi-regular polyhedra
The Platonic Solids transform into the 13 Archimedean Solids through truncation, expansion, and twisting. Together these 18 forms provide the complete geometric vocabulary for Atomic Geometry.

Comparison with Other Atomic Models

Comparison table of Atomic Geometry versus the Copenhagen Interpretation, Pilot Wave model, and MCAS model
A side-by-side comparison of Atomic Geometry with the Copenhagen Interpretation, the Pilot Wave (de Broglie–Bohm) model, and the MCAS model — across criteria including geometric structure, orbital prediction, and quantum number interpretation.

Atomic Geometry shares common ground with existing models while resolving several longstanding problems. Unlike the Copenhagen Interpretation, it treats the orbital as a definite geometric form rather than a probability cloud. Unlike Pilot Wave theory, it does not require a non-local guiding field. Unlike the MCAS model, it grounds orbital shapes directly in the Platonic and Archimedean Solids rather than algebraic curve-fitting.

S-Orbitals

The spherical S-orbitals are the first to appear in every atom. Each consists of two electrons oriented in opposite directions — up and down. S-orbitals play a key role in simple atomic bonds, appearing first on the outermost valence shell.

S-orbital electron pairs mapped geometrically, showing the up and down spin orientation
S-orbital electron pairs: the two electrons sit at opposite poles, corresponding to up and down spin. In Atomic Geometry this is understood as the north and south poles of a 4D toroidal field.

S-Orbitals and the Torus

S-orbitals can be perceived as a circle (2D), sphere (3D), or torus (4D). In 4D, the two electrons follow the flow of the toroidal field — a dynamic structure in continuous motion, exhibiting a unidirectional energy flow from the down-oriented electron, through the nucleus, to the up-oriented electron. This is why electron pairs always appear as opposites: they are the two poles of a 4D torus field.

S-orbitals viewed in 2D as a circle, in 3D as a sphere, and in 4D as a torus
S-orbitals across dimensions: circle (2D), sphere (3D), torus (4D). The torus model explains electron pairing as a consequence of 4D geometry rather than an arbitrary quantum rule.
Toroidal fields at planetary, solar, and galactic scales — the same geometry as the atomic S-orbital
The toroidal field appears at every scale: electron cloud, planet, solar system, galaxy. Atomic Geometry suggests this is not coincidence but a consequence of 4D geometry operating at all scales of reality.

For the detailed treatment, see S-Orbital Geometry.

P-Orbitals

After the first four elements, the first P-orbitals appear. Starting with boron (5), this set includes the building blocks of biological life — carbon (6), nitrogen (7), and oxygen (8). All noble gases (except helium) occur when a full set of six P-orbital electrons completes the shell.

P-Orbitals and the Octahedron

P-orbitals always appear in sets of three, each orientated at 90° to the others, forming a 3D cross spanning x, y, and z. Superimpose an Octahedron over the centres of the three P-orbitals and it fits perfectly — making the Octahedron the defining solid of the noble gas configuration and the foundation for the octet rule.

Three P-orbitals arranged at 90° to each other, with an Octahedron superimposed to show the geometric correspondence
Three P-orbitals at 90° define the six faces of an Octahedron. This geometric relationship explains the octet rule — why atoms bond to complete a set of eight electrons — without any additional assumptions.
Interactive 3D model: three P-orbital pairs mapped onto an Octahedron. Drag to rotate, scroll to zoom. Use the buttons to toggle individual orbitals and the Octahedron wireframe.

4D P-Orbitals

In 4D, each P-orbital lobe connects to the other through a torus ring, so the complete set of three P-orbitals resembles three interlocking torus fields mapped onto an Octahedron rather than the simple dumbbell shape of the 3D model.

P-orbitals shown as three interlocking torus fields mapped onto an Octahedron in 4D
4D P-orbitals: three interlocking torus fields rather than three separate dumbbells. The Octahedron remains the container, but in 4D the lobes are connected — eliminating the mysterious gap between them.

Watch: P-Orbitals as an Octahedron

Dr. Heike Bielek demonstrates the octahedral nature of the P-orbitals.

The octahedral geometry of the P-orbitals also provides a framework for understanding the noble gases — the non-reactive elements that complete each orbital shell. Through the jitterbug transformation and dodecahedral nesting, the geometric model predicts noble gas radii and offers an explanation for their chemical inertness based on the inability of pentagonal geometry to fill space. For the detailed treatment, see P-Orbital Geometry.

D-Orbitals

Building on the P-orbital Octahedron, D-orbitals extend the geometry into cubic space. Three of the five D-orbitals share the same x, y, z axis as the P-orbitals and divide a cube of space into eight equal parts. The remaining two map onto the Rhombic-Cuboctahedron — the only Archimedean Solid whose midsection can rotate freely, matching the toroidal dynamic of the final D-orbital pair.

Three D-orbitals arranged on x, y, z axes dividing a cube of space into eight smaller cubes
Three cross-shaped D-orbitals divide a cube of space into eight equal parts — modelled as eight small cubes forming a larger Cube, the only Platonic Solid that fills space uniformly.
Interactive 3D model: the three cross-shaped D-orbitals (dxy, dxz, dyz) mapped onto a Cube. Drag to rotate, scroll to zoom. Toggle individual orbitals and adjust transparency.

D-Orbitals — dx²-y² and dz²

The remaining two D-orbitals complete the set of five. The dx²-y² orbital places four lobes directly along the x and y axes — rotated 45° relative to the dxy orbital. The dz² orbital is unique: two lobes along the z-axis plus an equatorial torus ring, mapping onto the Rhombic-Cuboctahedron — the only Archimedean Solid with a freely rotating midsection, matching the toroidal dynamic of this orbital.

Interactive 3D model: the axial D-orbitals — dx²-y² (four lobes along x/y) and dz² (two lobes plus equatorial torus ring). Together with the three cross-shaped orbitals, these five sets define the cubic geometry of the D-orbital shell.
P and D-orbitals mapped onto a cube of space, with an Octahedron and Cuboctahedron nested inside
P and D-orbitals together: the Octahedron (P) and Cuboctahedron (D) nest inside a Cube of space, explaining the geometric basis of the transition metals.

There are three stable sets of D-orbital elements. Each set maps to a progressively higher-dimensional polytope, explaining Aufbau anomalies, ferromagnetism, and electrical conductivity through geometry rather than energy rules alone. The reciprocal space of the atomic lattice — known as Brillouin Zones — reveals that magnetic elements (iron, cobalt, nickel) express a Rhombic-Dodecahedron, while conductive elements (copper, silver, gold) express a Truncated Octahedron. For more on this geometric connection, see Brillouin Zones and the Geometry of Ferromagnetic and Conductive Waves.

For the full treatment of D-orbital geometry see D-Orbital Geometry Part 1, Part 2, and Part 3.

F-Orbitals

F-orbitals appear spatially in the 4th shell, though the periodic table places them in the 5th and 6th rows because elements are ordered by energy level. Only one set is stable (elements 57–71); the second set is radioactive.

The periodic table highlighting the F-orbital elements in the lanthanide and actinide rows
The F-orbital elements (lanthanides and actinides): one stable set and one radioactive. The geometric model places them correctly in the 4th shell, resolving their apparently anomalous position in the periodic table.

F-Orbitals and the Cuboctahedron

The most common F-orbital configuration is hexagonal. Four hexagonal rings orientated along x, y, and z fit the Cuboctahedron perfectly — which has both square and triangular faces, with an internal geometry of four hexagons. The full orbital sequence is: Octahedron (P) → Cube (D) → Cuboctahedron (F).

Four hexagonal F-orbitals mapped onto a Cuboctahedron
F-orbitals and the Cuboctahedron: the four hexagonal rings of the F-orbitals fit the Cuboctahedron's four internal hexagons exactly, completing the geometric progression from Octahedron through Cube.
Interactive 3D model: four hexagonal F-orbital rings mapped onto the Cuboctahedron's internal hexagons. Drag to rotate, scroll to zoom. Toggle individual rings and adjust transparency.

F-Orbitals and the Star-Tetrahedron

Within the F-orbitals there is one unique cubic-shaped pair: each electron contained within a Tetrahedron, the two interlocking at 180° to define the corners of a Cube. This is the Star-Tetrahedron — an Octahedron with eight tetrahedra on each face. Just as P-orbitals begin atomic structure with an Octahedron, F-orbitals terminate it with the Star-Tetrahedron.

The Star-Tetrahedron formed by two interlocking tetrahedra at 180°, corresponding to the cubic F-orbital pair
The Star-Tetrahedron: two tetrahedra interlocked at 180°, containing an Octahedron at their centre. This form closes the orbital sequence — P-orbitals begin with the Octahedron; F-orbitals end with the Star-Tetrahedron.
Interactive 3D model: the cubic F-orbital lobes at the 8 corners of a cube, forming two interlocking tetrahedra (Star-Tetrahedron / Merkaba) with the inner Octahedron at the centre. Toggle lobes, wireframes, and solids to explore the geometry.
P, D and F orbitals all mapped together onto a cube of space showing the complete nested geometric structure
The complete P, D, and F orbital set mapped onto a single cube of space. The nested geometric structure — Octahedron, Cuboctahedron, Rhombic-Cuboctahedron — reveals the atom as a fractal geometric system.

F-Orbitals and the √2 Fractal

The star-tetrahedral orbitals locate at each corner of the cube. Viewed from the centre face, the electron distribution is limited to the corner points of nested squares, each successive square differing in size by a factor of 1:√2. This √2 fractal is also found in the geometry of electromagnetic waves — and explains why one set of D-orbitals is offset by 45°.

Hexagonal and Star-Tetrahedral F-orbitals showing the √2 fractal nested square pattern
The hexagonal and star-tetrahedral F-orbitals from the centre face: the electron positions form nested squares in the 1:√2 ratio — the same ratio that governs the magnetic field geometry of EM waves.

F-Orbital Torus and the Icosahedron

The final F-orbital is a double torus — a 5D hypersphere assigned to the Icosahedron. Like the Rhombic-Cuboctahedron among the Archimedean Solids, the Icosahedron has a rotational property: deconstructed into three sections it reveals a pentagonal middle prism containing the Golden Ratio (1:1.618, related to √5).

The double torus F-orbital mapped onto an Icosahedron, showing the pentagonal middle prism and Golden Ratio proportions
The double-torus F-orbital and the Icosahedron: the pentagonal middle prism contains the Golden Ratio — the same proportion that governs the Silver Ratio (√2) of the magnetic field and the structure of the D-orbital elements.

The Jitterbug Transformation

The geometric progression from S-orbitals through to F-orbitals is not a coincidence — it is driven by a single geometric mechanism. The jitterbug transformation, first described by Buckminster Fuller, shows how the cuboctahedron can collapse through the icosahedron and octahedron to the tetrahedron. This same transformation connects each orbital type to the next, explaining why the electron cloud builds in the specific sequence it does.

The Jitterbug transformation — the Cuboctahedron collapsing through the Icosahedron and Octahedron to the Tetrahedron — the same geometric transformation that underlies the orbital sequence in Atomic Geometry.
An introduction to the complete geometric model of the atom — how the S, P, D, and F orbitals map onto three-dimensional geometric forms to produce a model more accurate than the Bohr model.

Conclusion

Atomic Geometry provides a clear, geometrically rigorous description of the atom. Where standard models invoke probability and wave-particle duality, it offers a deterministic geometric alternative — one that is more accurate, simpler to grasp, and physically constructible.

The orbital sequence S → P → D → F is not arbitrary. It is a geometric progression: torus → Octahedron → Cube → Cuboctahedron — each step adding a dimension, each form nesting inside the last. The geometry is not merely three-dimensional: orbitals exist in the fourth dimension and possibly the fifth, based on the Euclidean axioms of 1D through 5D rather than on string theory. The same toroidal and hypercubic structures that define the electron cloud appear at planetary, stellar, and galactic scales — suggesting a single geometric principle operating at every level of reality.

Atomic Geometry is compatible with the Copenhagen Interpretation and the de Broglie Pilot Wave model, but extends both by offering a mechanism — geometry — where they offer only description. It connects naturally to Geo-Quantum Mechanics, Dimensionless Science, and the 4D Aether framework.

For the detailed article series: History of the Atom · The 4D Electron Cloud · S-Orbital Geometry · P-Orbital Geometry · D-Orbital Geometry (parts 1, 2, 3) · 2D Orbital Geometry · The Atom and the Seed of Life · Brillouin Zones

FAQ

Why hasn't science recognised the geometric nature of the atom?

We have approached the scientific community with these ideas. The geometric model's simplicity may work against it — quantum mechanics is generally associated with complicated mathematics, and a model that can be grasped without equations can seem too easy to take seriously. We continue to develop the quantitative side of the theory alongside the geometric presentation.

Is this model applicable to all elements on the periodic table?

Data about the S, P, D, and F orbital shells has been gathered by energising a hydrogen atom, which science uses as the simplest case. As hydrogen has no neutron, atomic radii change across the periodic table. Our more advanced model, Geo-Quantum Mechanics, produces predictions of atomic radii that more closely match experimental values than the Bohr model across all stable elements.

How accurate is Atomic Geometry compared to the Bohr model?

For elements such as helium, the Atomic Geometry model predicts atomic radii over 500% more accurately than the Bohr model. The geometric approach outperforms energy-based models because it works from the spatial structure of the electron cloud rather than fitting equations to observed energy levels.

Can children understand Atomic Geometry?

Yes — children as young as seven have recognised concepts such as electron configuration, orbital shells, and electron pairing when introduced through the geometric model. Because it can be physically constructed with a compass, ruler, card, scissors, and glue, it becomes a tactile experience rather than an abstract mathematical one.

What does Atomic Geometry say about wave-particle duality?

Atomic Geometry treats the electron not as a particle with a probabilistic location but as a 4D field of energy whose quantised states are defined by the geometry of fourth-dimensional polytopes. This removes the need for probability theory to describe the electron cloud and resolves wave-particle duality at its source — the same resolution proposed in our 4D wave model of matter.