A 4D geometric wave model of matter without wave particle duality

Introduction

Ask any physicist what an electron really is and you will get an uncomfortable pause. It behaves like a wave when no one is watching, and like a particle when measured. This contradiction — called wave-particle duality — has sat at the heart of physics for a century, and no one has ever fully resolved it. Most textbooks quietly advise students to accept it and move on.

This article proposes something more satisfying: that the contradiction disappears entirely when we extend our geometry into four dimensions. What we call a "particle" is not a tiny ball of matter at all — it is the boundary of a four-dimensional wave structure called a torus (think of a doughnut shape, or the hollow ring of a smoke ring). The wave is always there. The particle-like behaviour is simply what that geometry looks like when we slice through it in three dimensions.

This is not a minor adjustment to existing theory. It is a fundamentally different picture of reality — one in which mass, charge, gravity, and electromagnetism all emerge from the geometry of how waves rotate in 4D space. If the model is correct, it resolves not just wave-particle duality but several of the deepest open problems in physics simultaneously.

The concept of wave-particle duality for light was first proposed by Albert Einstein as a solution to the photoelectric effect. Shortly afterwards, it was discovered that matter itself exhibits the same wave-like properties, extending the duality to the atom. However, because the Aether had been discarded from physics, the possibility that all matter might be composed purely of waves was never seriously explored.

Our resolution of the photoelectric effect and the ultraviolet catastrophe through the geometry of a 4D Aether removes the need for light to be treated as a particle. The same principles, applied to matter, produce an equally radical result: a completely new picture of the physical world grounded in the geometry of 4D rotation. The electron cloud, the proton, the forces of gravity and electromagnetism, and the nature of mass and charge all emerge naturally — without requiring wave-particle duality.

The history of the particle and the atom

The ancient world

The idea that reality is formed of tiny particles of matter dates back more than 2,500 years. Around 600 BC, the Indian philosopher Acharya Kanada first proposed that matter consists of indivisible particles he called the Anu. A century later, the Greek philosopher Leucippus is believed to have developed the same concept independently. His student Democritus is more widely known today for the doctrine of atomism. None of the original works of either ancient Greek thinker has survived; our knowledge of their ideas derives entirely from secondary sources.

This ancient tradition of atomism — matter as indivisible, irreducible points — proved extraordinarily persistent. It shaped the scientific revolution and reached its modern expression in the Standard Model of particle physics. Yet the wave-based experiments of the twentieth century quietly undermined it. The 4D geometric model presented here takes that experimental evidence at face value: matter is a wave, and the particle-like behaviour we observe is a consequence of the geometry of 4D space.

What is mass?

There are two distinct types of mass, differentiated by the velocity of an object. An object at rest exhibits rest mass — the quantity captured in Einstein's equation E = mc². A moving object also possesses momentum mass — the inertia that must be overcome to accelerate it further. Massless entities such as photons carry no rest mass but do carry momentum mass.

These two definitions map neatly onto the two categories of entity in the 4D model: bounded toroidal structures (which carry rest mass) and free spherical waves (which carry only momentum). Think of it this way: energy trapped inside a closed ring has nowhere to go — that confinement is what we experience as rest mass. Energy that breaks free of the ring spreads outward as a spherical wave — that is what we experience as light.

The speed of light through a vacuum is limited by two fundamental constants: the electrical permittivity ε₀ and the magnetic permeability μ₀. Together these set the electromagnetic resistance of the vacuum — the "drag" that determines how fast a free wave can travel. When energy is instead confined within a geometric boundary — such as the proton or the electron cloud — it no longer propagates; it resides as rest mass.

In geometric terms: energy constrained within a boundary exhibits rest mass; energy free of that constraint radiates as a spherical wave. The resistance of the vacuum determines the rate of expansion of that wave, originating at the electron cloud and deriving from the spin relationship between the electron and the proton. This begins to unify the nature of light with the structure of the atom itself.

Energy contained within the proton's toroidal shell (with its 1/3 spin) produces a concentration of energy from the vacuum proportional to the energy of the electron cloud. Atoms continuously absorb and emit electromagnetic radiation. When excited with additional energy, the emitted light shifts toward the ultraviolet — the object begins to glow. This is the displacement of the energy balance between the proton and the electron cloud overcoming the vacuum's resistance and expanding to infinity at the speed of light.

Supporting evidence comes from antenna physics. The electromagnetic field surrounding a transmitting antenna forms a near-field double-layered torus — a doughnut-shaped region that contains and amplifies the electric field before the wave escapes. Beyond the torus, the far field begins: a spherical wave obeying the inverse square law. The length of the antenna determines the size of the toroidal near-field region; the energy input determines the signal strength. This is precisely the geometry predicted by the 4D wave model. The antenna is, in effect, a macroscopic demonstration of what every atom does at the quantum scale.

The energy of the vacuum has been described under many names: quantum fluctuations, vacuum energy density, zero-point energy. Quantum physics portrays it as a kind of foam of virtual particles that appear to violate the laws of physics. Examined through the lens of the 4th dimension, the reason for these apparent anomalies becomes clear.

What is energy?

The direct equivalence of energy and mass was one of Einstein's great discoveries. E = mc² establishes the conversion between Joules and kilograms, first set out in Einstein's paper "Does the inertia of a body depend upon its energy content?". However, when Einstein drew his conclusions, he expressly excluded the nature of the 4th dimension.

The theory implies that a moving object loses kinetic energy by emitting electromagnetic radiation — unifying motion, weight, and energy. In the Bohr model this created a fatal problem: the "motion" of the electron should add to its energy, causing it to spiral into the nucleus. An object with mass will naturally decelerate until it resides at its rest mass, which implies an intimate relationship between vacuum energy, its resistance, and the speed of light.

When the 4th dimension is reintroduced, the speed-of-light limit is seen as a consequence of 4D rotation — specifically, the rotation of the torus. Imagine a wave travelling endlessly around the inside of a ring: the geometry of the ring itself limits how fast the wave can circulate. A particle that receives energy is propelled through space, losing some energy to the vacuum field as it moves. That energy radiates away as a spherical wave carried through the medium of the 4D quantum field (the Aether). Rather than dissipating formlessly, the energy drives the rotation of a 4D object. The particle is accelerated — its rate of 4D spin increases — and moves out of energetic balance from its rest mass. Excess radiation propagates through the Aether as a set of nested toroidal shells, like concentric smoke rings expanding outward, whose spacing determines the wavelength and frequency of that particular EM emission.

This 4D model of EM waves and mass connects directly to the inverse square law: as a spherical wavefront propagates outward, its energy is distributed over an ever-growing surface area (4πr²). The magnetic permeability of the vacuum is 4π, representing precisely the resistance an EM wave must overcome to propagate. The speed of light squared equals the energy per unit of mass emitted into the vacuum — a direct geometric expression of E = mc².

Gravity and the Aether

The Aether was originally proposed as the medium that transports light through the vacuum. It was abandoned when the photon was assumed to travel through empty space without a medium. Quantum field theory has since readopted the concept in a different guise, but the term Aether itself remains taboo. The discovery of the Cosmic Microwave Background (CMB) in 1964 established that the entire Universe is immersed in an almost uniform energy field — whether we call it a quantum field, the CMB, or the Aether, the physical reality is the same.

It is unfortunate that science has chosen to interpret the CMB as the remnant of the Big Bang. In the 4D model, the rest mass of particles arises from the spherical boundary of a 4D torus, created by the difference in quantum spin states. The proton and electron cloud occupy very specific energy volumes that are exactly proportional — this uniformity is encoded in the electrical charge constant e. The proton's radius is around 0.84–0.87 femtometres, yet it contains over 99% of the atom's mass. This stark asymmetry becomes intelligible when the energy of the 4D Aether is taken into account.

In the 4D Aether theory, protons and neutrons are formed by toroidal shells acting as resonant chambers — analogous to the mirrors in a laser cavity that amplify light through geometry. The principles of geometry also explain quantum spin directly. A light wave must traverse two half-cycles to complete one full wavelength, mirroring the electron's spin value of 1/2. Protons are composed of quarks carrying spins of 1/3 and 2/3. The ratio of proton spin to electron spin yields a value spectacularly close to the gravitational constant G:

(1/3 proton spin) ÷ (1/2 electron spin) = 2/3 ≈ Gravitational constant (G)

This is a striking result. The unification of quantum spin numbers with gravity is one of the central unsolved problems in physics — and here a simple ratio of two spin fractions produces the gravitational constant directly. The 4D nature of spin begins to illuminate this puzzle. Newton's gravitational equation shares a structural similarity with the inverse square law for light intensity: both describe the dispersal of a field over a growing spherical surface. In the 4D model, gravitational waves arise from the interaction between the 1/3 spin of the proton and the 1/2 spin of the electron — a different harmonic of the same underlying toroidal geometry that produces electromagnetic waves.

If we reverse the ratio — electron spin divided by proton spin — the result is 3/2. The number 3 can be associated with the speed of light; divided by 2, it produces an electromagnetic wave with a wavelength of approximately 150 nm. As shown in our analysis of the photoelectric effect, the work function of a material multiplied by the threshold wavelength, when squared, equals approximately half the speed of light — the same value.

Gravity is therefore the force that contains the energy of the proton through its spin relationship with the electron. The energy extracted from the vacuum is explained by the same relationship.

In dimensionless science, the gravitational constant is assigned the value 2/3, with the speed of light given the value 3 and space the value 2.

Unified field theory

Quantum field theory has succeeded in unifying three of the four fundamental forces, but gravity has resisted all attempts at unification for over a century. Even today, no theory fully accounts for gravitational behaviour at galactic scales. The 4D geometric model offers a different route — and the answer turns out to be hiding in plain sight.

An electromagnet is made by coiling a wire in concentric circles; an electrical generator passes a magnet through a coil. In both cases, it is geometry — the shape of the conductor — that creates the effect. If shape produces electromagnetism, it is natural to ask whether shape also produces gravity.

It is a remarkable fact that the number of protons and electrons in the Universe appears to be equal. This equality points to a deep unified cause. Consider proton and electron spins represented on the circumference of a single circle divided into two halves. The electron half (spin 1/2) is divided into two sections, each subtending 1/4π. This is the reciprocal of the magnetic permeability constant μ₀. The proton half (spin 1/3) is divided into three sections, each subtending 1/6π. Multiplied by 1/6, this gives the electrical permittivity constant ε₀.

The dividing line between the two halves represents the boundary between the proton and the electron cloud. From this geometry, the gravitational constant emerges directly:

(6ε₀) ÷ (1/μ₀) ≈ G × 10⁻⁶

(6 × 1/36π) ÷ (1/4π) = 2/3

Conversely, the electrical circuit does not draw its energy along the wire in the way commonly assumed — it draws energy from the vacuum by overcoming the magnetic resistance μ₀. When the electron sector (1/4π) is divided by the proton sector (1/6π), the result is the threshold wavelength that solves the photoelectric effect — the point of harmonic resonance at which energy can be extracted from the vacuum:

(1/μ₀) ÷ (6ε₀) ≈ c/2

(1/4π) ÷ (6 × 1/36π) = 3/2

This makes a clear distinction between the E and B fields of an electromagnetic wave as two aspects of spin, each expressible on a different type of 2D plane. In geometry, there are only two regular shapes that can tile an infinite plane with two colours: the triangle and the square. Triangles join to form a hexagonal tiling, dividing the plane into sixfold and fourfold symmetry — precisely the ratios of ε₀ and μ₀.

This offers the first geometric explanation for the origin of charged particles: charge arises from the ratio of energy derived from two different types of 2D plane. The structure of the visible light spectrum reflects the same ratios — different frequencies of colour arising from the mixing of two wave types (red-green-red and yellow-blue). This geometric view of colour provides a wave-based solution to the photoelectric effect without requiring the photon.

A single geometric framework now accounts for light, gravitation, and charge through the interplay of 4D toroidal spin.

The geometric Universe

One of the key differences between standard quantum physics and the 4D geometric model is the treatment of the Aether. Quantum science alludes to a quantum field but does not clearly connect the Cosmic Microwave Background with the nature of quantum foam. The 4D Aether theory describes the quantum field in terms of 4D geometry, providing a medium for the propagation of light waves, removing the need for the photon particle, and eliminating much of the quantum weirdness that accompanies the particle model.

One consequence of this view is that the CMB is no longer associated with the Big Bang, which brings into question the conventional account of the Universe's origin. According to present theory, the observable Universe has a diameter of around 93 billion light years — a sphere whose boundary is set not by the limits of our instruments but by the speed of light and the predicted age of the Universe.

Regardless of the actual age of the Universe, the energy of the Cosmic Microwave Background is extraordinarily flat and uniform. If the protons and electrons of the Universe arose from this field, the Universe itself may be in a state of equilibrium — just as the proton's toroidal boundary contains its quarks, the observable Universe itself may have a geometric limit of the same kind.

The radius of the observable Universe (0.44 × 10²⁴ km) yields an interesting geometric chain. Dividing by the speed of light c produces a smaller sphere of radius 14.67 × 10⁻⁹ km. Subtracting Newton's gravitational constant G reduces this to 14.009 × 10⁻⁹ km — close to the whole number 14. Dividing the original radius by this smaller one gives a value close to π × 10⁻⁸, suggesting a geometric relationship linking the predicted size of the observable Universe, the speed of light, and gravity.

This may appear speculative, and the actual size and age of the Universe remain genuinely open questions — a point of active debate captured in the so-called Cosmological Crisis. Nevertheless, large-number coincidences have attracted serious scientific attention. The Large Numbers Hypothesis (LNH), proposed in 1937 by Paul Dirac, noted that the gravitational constant G appeared inversely proportional to the age of the Universe, and that the mass of the Universe seemed proportional to the square of its age.

Arthur Eddington pursued a related line of inquiry, attempting to calculate the number of protons in the Universe from the fine structure constant α (approximately 1/137) rather than from galactic mass surveys. His estimate was:

α × 2²⁵⁶ ≈ 1.58676 846 249 327 73 × 10⁷⁹

(α × 2²⁵⁶ × 2) ÷ π × 10⁷⁹ ≈ 1.0101 ≈ 1

This value is close to 1/2π — the same number used above to represent the proton and electron spins that yield the electrical and magnetic resistance constants of the Universe. In Dimensionless Science, the electrical charge constant e is also assigned the value of 1/2π. It relates to a single unit of energy E via the photon energy Ep, giving a photon energy of 2/π — the ratio of the circumference of a circle to its diameter. The transformation between the line and the circle is a mathematical enigma most eloquently expressed by a drawing compass.

What is the 4D wave model of matter?

This model rests on the identification of the CMB, quantum foam, and quantum field as different names for the 4D nature of space. The 4D Aether creates the particle boundaries of the proton and the electron cloud. Electrical and magnetic components of the EM field arise from quantum spin: the 1/2 spin of the electron and the 1/3 spin of the quark determine the relationship between gravitational and electrical phenomena. The concept of a particle is resolved as the boundary between these different quantised spin states, forming an energy density that maintains equilibrium. Any excess energy not required by the rest mass is radiated outward through the cosmos as an EM wave.

This picture dissolves the need for wave-particle duality. Electrons do not sometimes behave as particles and sometimes as waves — they are always waves. The geometry of their 4D toroidal boundaries produces the discrete, localised properties we associate with particles. The same geometry, applied at larger scales, produces gravity and, at the largest scales, may define the boundary of the observable Universe itself.

A new vision of reality

The idea that matter has wave-like properties has been experimentally established for nearly a century. What has never been clearly resolved are the precise scientific definitions of mass, charge, and matter itself. The 4D wave model of matter offers a concise account of all three: it outlines the geometric origin of charge, unifies the laws of electromagnetism and gravitation, and connects the structure of the atom to the large-scale structure of the Universe — without requiring a single fundamental particle.

For further reading on the medium through which these waves propagate, see 4D Aether. For the wave-only resolution of the photoelectric effect that underpins this model, see Can the photoelectric effect be explained without the photon?. For the implications of this model for cosmology, see Did the Big Bang happen?.

Conclusion

Wave-particle duality has never been a satisfying answer — it is, at best, a description of our confusion. This article has laid out a model in which that confusion dissolves: matter is always a wave, and the "particle" is always a consequence of geometry.

The 4D geometric wave model of matter is a coherent alternative to the particle-based Standard Model. By treating the 4D Aether as a real geometric medium, it resolves wave-particle duality, explains the origin of mass and charge from first principles, and unifies gravity with electromagnetism through a single spin-ratio relationship. It situates all of this within a framework rooted in the properties of the torus — a shape so simple a child can draw it, yet rich enough to encode the structure of the atom, the behaviour of light, and potentially the boundary of the observable Universe.

The model does not merely describe these phenomena — it derives them from geometry. No new particles are required. No extra dimensions beyond the fourth are invoked. The same doughnut-shaped wave structure that explains the electron cloud also predicts the gravitational constant, the electromagnetic vacuum constants, and the threshold wavelength of the photoelectric effect. When a single geometric idea connects so many apparently unrelated phenomena, that is not coincidence — it is a signal pointing toward something real.

FAQ

Does this model mean the Universe is not expanding?

The 4D Aether theory does not explicitly prohibit the expanding Universe theory, but it seriously challenges the prevailing framework. We identify the Cosmic Microwave Background as the energy source that forms a specific number of protons — equal in number to the electrons in the Universe, each carrying a single unit of electrical charge. This uniformity suggests a state of equilibrium, whereby the force of gravity defines the boundary of the proton. This interpretation of quantum spin radically alters concepts such as the strong nuclear force: the binding agent becomes a property of a 4D spin ratio rather than a discrete particle exchange.

How does the ratio 3/2 indicate that energy is being extracted from the vacuum?

In our solution to the photoelectric effect, the energy created in an electrical circuit by ultraviolet light is not stored within the light itself. Instead, the wavelength of light creates a resonance in the material at a specific frequency, triggering a release of energy at the surface boundary and drawing energy from the vacuum. Different materials release energy at frequencies determined by their structure and the incident wavelength. When the work function and threshold wavelength are multiplied together and the result squared, the value is consistently close to 150 nm — half the speed of light.

How does 4D spin unify gravity and electromagnetism?

The 1/2 quantum spin of the electron and the 1/3 spin of the quark (constituent of the proton) map directly onto the magnetic and electrical resistance constants of the vacuum. When the proton spin ratio is divided by the electron spin ratio, the result closely approximates the gravitational constant G. Conversely, dividing the electron spin ratio by the proton spin ratio produces 3/2, which relates to the threshold wavelength in the photoelectric effect. This single geometric relationship therefore connects gravitation, electromagnetism, and charge within one unified framework.

What replaces the photon in this model?

Rather than a discrete photon particle, electromagnetic radiation is modelled as a spherical wave expanding through the 4D Aether. The wave originates at the electron cloud boundary and propagates outward at the speed of light, limited by the electrical permittivity (ε₀) and magnetic permeability (μ₀) of the vacuum. Quantisation arises not from particle-like energy packets but from the geometry of nested toroidal shells, whose spacing sets the wavelength and frequency of each EM emission.