The Electron Is a Wave
Williamson, van der Mark, and the Idea That Changed Everything
Stephen Horton | Independent Researcher | February 2026
The Wave Coherence Blog Series — Bridge Post #2
In 1997, two physicists at Philips Research in Eindhoven published a paper that should have detonated across physics departments worldwide. It didn’t. Almost nobody noticed. But the idea is simple enough to state in one sentence, and radical enough to restructure everything we think we know about matter:
The electron is a photon, confined to a toroidal loop by its own topology.
That’s the Williamson-van der Mark model. If you’ve been reading this series and hit phrases like “toroidal electron,” “topological twist,” or “720 degree rotation” without context — this is the piece that was missing. Let’s unpack it.
The Problem It Solves
Physics has an uncomfortable secret: nobody knows what an electron is.
We know what it does. It has a mass of 9.109 x 10^-31 kg. It carries a charge of -1.602 x 10^-19 coulombs. It has spin-1/2, meaning it must rotate 720 degrees — two full turns — before it returns to its original state. We can predict its behavior to twelve decimal places using quantum electrodynamics.
But ask a physicist what the electron is made of, what its internal structure looks like, why it has the mass it has, why spin-1/2 requires that bizarre double rotation — and you get silence, or a deflection to “it’s a point particle” or “it’s an excitation of the electron field.” These are descriptions of behavior, not explanations of structure.
Williamson and van der Mark asked a different question: what if the electron isn’t a thing at all? What if it’s a pattern?
A Photon That Got Stuck
Start with something we understand well: a photon. Light. An electromagnetic wave propagating through space at c. It has energy, frequency, and momentum. It’s a boson — integer spin, no exclusion principle, waves can pile on top of each other without limit.
Now imagine that photon bending. Not scattering off something, but curving back on itself — circling into a closed loop. If the loop closes perfectly, the electromagnetic wave is now chasing its own tail. It’s not propagating through space anymore. It’s trapped in a self-sustaining circuit.
A standing wave. (If you read the primer, you saw this coming.)
But here’s the twist — literally. A simple loop, like a circle, would give you a boson. The wave goes around once, returns to its starting phase, done. That’s not an electron. Electrons are fermions. They need 720 degrees to complete a cycle, not 360.
So Williamson and van der Mark proposed that the loop isn’t a simple circle. It’s a torus — a donut shape — and the wave follows a path that includes a topological twist. Think of it like a Mobius strip wrapped around a donut. The wave travels around the donut, but because of the twist, after one full trip around the ring (360 degrees), its phase has flipped. It needs to go around again to get back to where it started. Two trips. 720 degrees.
That’s spin-1/2. Not a mysterious intrinsic property. Not an abstract quantum number with no classical analog. It’s the geometry of the path the trapped light follows. The double rotation is a direct consequence of the toroidal topology with a twist.
Why a Torus?
The torus isn’t arbitrary. It’s the simplest three-dimensional topology that can sustain a self-reinforcing electromagnetic standing wave.
A sphere can’t do it — there’s no stable path for a wave to circulate without destructive interference killing it. A simple loop can’t do it — the wave would radiate away, losing energy on every cycle. But a torus creates a closed resonant cavity with two independent circulation paths: around the ring (the long way around the donut) and through the hole (the short way). These two modes can sustain each other through continuous constructive interference.
The key insight is that this isn’t hand-waving. The electromagnetic field equations — Maxwell’s equations — permit self-sustaining toroidal solutions. The photon isn’t being confined by an external boundary (like a microwave in an oven). It’s confined by its own field structure. The topology is the boundary.
Where Mass Comes From
This is where the model becomes genuinely powerful.
A photon in free space is massless. It always travels at c. But a photon trapped in a toroidal loop appears to have mass. Why? Because it has energy (E = hf), and it isn’t going anywhere. It’s localized. Einstein told us that energy and mass are the same thing (E = mc^2). A confined photon with energy E has an effective mass of E/c^2.
The electron’s mass, in this model, is simply the energy of the trapped photon divided by c^2. It’s not a separate property bolted onto the particle. It is the confinement.
This also explains something beautiful about the electron’s relationship to the photon: when an electron and a positron annihilate, they produce two photons. In standard physics, this is described as “mass converting to energy.” In the Williamson-van der Mark model, nothing converts. The toroidal knots simply untie, and the trapped light is released. The photons were always there. They were just circling.
Charge as Topology
The electron has charge -1. The positron has charge +1. In the toroidal model, the difference between them is the direction of the twist.
If the wave circulates through the torus in one rotational sense, you get one sign of charge. Reverse the twist, you get the opposite sign. Charge isn’t a mysterious label assigned to particles by nature. It’s the handedness of a topological knot.
This gives charge quantization for free. You can’t have half a twist. The topology is either wound one way or the other, or it’s unwound entirely (a photon). That’s why charge always comes in integer multiples of the electron charge — there’s no continuous spectrum of possibilities, only discrete topological states.
Why This Matters for Wave Coherence
If Williamson and van der Mark are right, the implications cascade through the rest of this framework:
Matter is standing light. Every electron in every atom is a photon that topology locked into a stable resonant pattern. The “material world” is electromagnetic waves all the way down, organized by geometry into persistent structures. The distinction between “matter” and “energy” becomes a distinction between confined and unconfined wave patterns.
Spin is geometry. The 720 degree rotation that makes fermions weird isn’t abstract — it’s the physical consequence of a twisted toroidal path. And when two electrons form a Cooper pair in a superconductor, their opposite twists cancel. Two fermions (720 degrees each, opposite handedness) become one boson (360 degrees). The topological obstruction to overlap disappears, and the pair can condense into a single coherent quantum state. That’s superconductivity — not despite the twist, but because the twists can cancel.
Benzene is a molecular echo. The pi electron cloud of benzene is toroidal — a donut of electron density above and below the hexagonal ring. The ring current that circulates through it behaves like a superconducting loop. The hexagonal geometry is the molecular-scale platform for sustaining toroidal standing wave patterns. This is the Williamson-van der Mark electron, expressed at the molecular level.
Scale invariance follows. If the fundamental unit of matter is a self-sustaining wave pattern stabilized by topology, then the principle doesn’t care about scale. The same pattern — toroidal standing wave, stabilized by geometry, sustained by constructive interference — shows up in the electron, in benzene’s pi cloud, in graphene’s lattice, in the Earth’s magnetosphere. Not as metaphor. As repetition of the same mechanism at different scales.
The Paper Nobody Read
Williamson and van der Mark published “Is the electron a photon with toroidal topology?” in the Annales de la Fondation Louis de Broglie in 1997. It has accumulated modest citations. It’s never been experimentally refuted. It’s never been widely adopted. It sits in a strange limbo — too elegant to dismiss, too radical to absorb into the Standard Model without rethinking what “particle” means.
For this series, it’s foundational. When we derive hydrogen spectral lines from toroidal wave geometry, when we explain Cooper pairing as topological untwisting, when we trace hexagonal coherence from benzene to the Great Pyramid — the Williamson-van der Mark electron is the bottom of the stack. The thing that makes “waves all the way down” more than a slogan.
If the electron is a wave, then everything made of electrons is a wave. And if everything is a wave, then coherence — the quality of waves being organized, in phase, reinforcing each other — isn’t a special state. It’s the natural state. Decoherence is what requires explanation.
That’s the foundation. Everything in this series builds on it.
Previous: What Is a Standing Wave? — The vocabulary primer. Next: The Bridge — From ancient power to modern physics, and why they’re the same story.
The Wave Coherence Blog Series Stephen Horton — Independent Researcher — February 2026