"The fine structure constant is not a mystery. It is an angle."
For over a century, physicists called α one of the deepest unsolved mysteries in science. Feynman called it "one of the greatest damn mysteries of physics." Pauli was haunted by it to his dying day. This book shows that the answer was hiding in plain sight — in two measurable lengths and the definition of an angle.
The fine structure constant α ≈ 1/137 is one of the most precisely measured and least understood numbers in all of science. It governs the strength of electromagnetism, determines the size of atoms, and sets the conditions for chemistry — and therefore for life itself.
Physicists have treated it as a mysterious free parameter for over a hundred years — a number that must be measured and inserted into equations by hand, because no one knows where it comes from. Feynman admitted that physicists must "smuggle this number into their equations." Schwinger carved α/2π on his tombstone without knowing what it was.
This book argues that the mystery was never real. The answer is geometry — hiding in plain sight in equations that every physics student already knows.
The reduced Compton wavelength divided by the Bohr radius. This equation has appeared in textbooks for a century. It is the definition of an angle. No one read it geometrically — until now.
Arthur Compton measured the electron's characteristic length in 1923 and won the Nobel Prize. No one returned to Bohr's model to ask what this meant for an electron sitting on an orbit. The clue sat in the literature for a century.
Place an extended object on a circular orbit and it subtends an angle. The Compton wavelength divided by the Bohr radius gives that angle in radians. That angle is α. This is not a metaphor. It is the definition.
Using only two measurable wavelengths and π, the formula reproduces the CODATA value of α to ten significant figures. No free parameters. No fitting. No adjustment. The reader is invited to verify it.
There is a recurring pattern in the history of science. Breakthroughs often come from outsiders — people not fully embedded in the prevailing paradigm, free to ask the questions that experts have learned not to ask.
The geometric interpretation of α requires connecting ideas from atomic physics, scattering theory, basic geometry, and quantum field theory. Each topic is taught in isolation. The clues remained scattered across textbooks for a hundred years, never assembled into a single picture.
Julian Schwinger carved α/2π on his tombstone. He calculated it from Feynman diagrams. He could have derived it from a picture. He never knew he was doing geometry.
Ask a quantum field theorist to draw a picture of what α physically is. They cannot do it. Here it is. It takes thirty seconds to understand. You will never see it any other way again.
An angle in radians is defined as arc length divided by radius. Every schoolchild learns this. It is one of the first things taught about circles.
The electron has an extent — a characteristic quantum length measured by Compton in 1923. Place that extended electron on the Bohr orbit and it occupies an arc of the circle. Divide that arc by the radius of the orbit and you have the angle it subtends.
That angle is α. That is all α is. That is all it has ever been.
"It has been a mystery ever since it was discovered, and all good theoretical physicists put this number up on their wall and worry about it. We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number present itself."
He was looking at an angle and seeing a mystery.
When an extended electron completes its orbit it must travel slightly further than 2π radians — it must traverse its own angular extent α as well. The fractional correction to the radius is α/2π. Schwinger derived this from Feynman diagrams. It is the geometry of a circle that does not quite close. He took it to his grave without knowing what it was.
"I am not a physicist. Perhaps that is why it dawned on me. The trained eye learns what to search for and what to disregard. I had experienced no indoctrination telling me this question was unanswerable."
David Charles lives in Devon, England. He is a retired Chartered Accountant whose career was spent analysing complex systems — implementing them when they worked, diagnosing them when they failed, identifying root causes, and fixing them.
This forensic discipline led him to one of the most counterintuitive arguments in the history of physics: that the fine structure constant — universally described as one of the deepest mysteries in science — is a simple geometric angle that has been hiding in plain sight for over a century.
David Charles first encountered the fine structure constant in a textbook and noticed something that a century of brilliant physicists had overlooked: the defining equation looked exactly like the definition of an angle. Arc length divided by radius. The most elementary relationship in circular geometry.
He spent weeks convinced he had made an error. When you believe you have found a simple answer to a question that defeated Feynman, Pauli, and Schwinger, the overwhelming probability is that you are wrong. He searched the scientific literature exhaustively for the paper that would explain why this interpretation was incorrect.
The Geometric Truth is the record of what he found. The mathematics are all standard — every equation in the book has been known for a century. What is new is the interpretation: the recognition that these equations, read geometrically, reveal what α actually is.
The argument is transparent and checkable. The formula requires only a scientific calculator and publicly available data from CODATA. Any reader can verify it in five minutes. David Charles invites them to do so.
David Charles discusses his ongoing research and responds to questions and challenges about the ideas presented in The Geometric Truth.
Follow on X — @GeometricTruthThe first volume in the Geometric Truth series. This book presents the geometric interpretation of α — showing that the fine structure constant is an angle, calculable from two measurable wavelengths and π, matching the official CODATA value to ten significant figures. No quantum field theory required. Any reader can verify the central calculation in five minutes with a scientific calculator.
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In 1948 Julian Schwinger calculated the anomalous magnetic moment of the electron. Dirac's equation predicted the electron's magnetic strength should equal exactly 2. Experiment showed it was slightly larger. Schwinger explained the discrepancy with a correction factor:
The agreement with experiment was stunning. It earned him a share of the 1965 Nobel Prize. He requested it be inscribed on his tombstone. When Julian Schwinger died in 1994 they carved it in stone — the crown jewel of quantum electrodynamics, his legacy to physics.
Schwinger derived α/2π from the abstract machinery of quantum field theory. Feynman diagrams. Loop integrals. Renormalisation. The calculation was a tour de force of mathematical physics — pages of formalism producing a number that matched experiment to extraordinary precision.
But the question "what is α/2π?" was never asked. The formalism produced the answer. That was considered sufficient. In the culture of theoretical physics, getting the right number is understanding. It is not.
A point particle closes its orbit perfectly. An extended electron cannot. The gap between the two circles is what Schwinger spent his career calculating.
α/2π is not merely the correction to the electron's magnetic moment. It is the universal geometric correction that appears whenever an extended electron interacts with anything. It is the fundamental consequence of one simple fact: the electron is too big to fit on its own orbit without making the orbit slightly larger.
Every time α appears in a perturbation series — every loop correction, every vertex factor, every higher order term in QED — it is traceable to this geometry. The circle does not close. Every correction is an attempt to account for that fact.
The series converges because the geometry converges. It has to — because there is a precise fact about how big the electron is relative to its orbit. Each term is a more refined attempt to close the same circle. The calculation was always correct. The understanding was always absent.
This is what happens when an institution self-governs and is allowed to mark its own homework. Careers were built on the complexity of perturbation theory. Funding was justified by the difficulty of higher order calculations. Nobel Prizes were awarded for solving what was always a geometry problem.
The tools required to understand α/2π are taught in the first year of secondary school. The problem defeated the greatest physicists of the twentieth century for a hundred years. That is not a tribute to the depth of the problem. It is a measure of how completely an institution can blind itself to a simple truth.
Schwinger's tombstone formula is not a quantum field theory abstraction. It is elementary geometry. The proof is in the book. The numbers are checkable in five minutes. Let them defend.