The proposal for a single law of physics, the fundamental principle:

Nature consists of invisible fluctuating flexible strands of Planck radius, reaching to the cosmological horizon, and whose crossing switches define Planck’s quantum of action, the Boltzmann constant, and the (double) Planck time:

In nature, space is a collection of untangled fluctuating strands, tangles of strands are particles, and tight weaves of strands are horizons. The universe consists of one or a few strands criss-crossing it.

Strands are unobservable and invisible - only their crossing switches are observable.
All physical quantities are defined with crossing switches. For this reason, particles appear localized, space appears empty, and horizons appear to be surfaces.

The evidence

Only strands agree with all observations.

Furthermore, only strands explain the spin 1/2 of fermions (as Dirac already implied in 1929). Only strands explain the spin-statistics theorem. Only strands explain the Dirac equation (Battey-Pratt and Racey, 1980). Only strands explain the quantization of action and of angular momentum. Only strands explain the maximum speed in nature. Only strands explain the observed particles and their three generations (by classifying the possible tangles). Only strands explain the three observed gauge interactions (with Reidemeister’s theorem) and the Higgs mechanism. Only strands explain the emergence of wave functions. Only strands explain virtual particles. Only strands explain the masses of elementary particles (also for neutrinos), their quantum numbers, their mixing, and the three coupling constants. Only strands explain the Lagrangian of the standard model, excluding any deviation. Only strands explain the three dimensions of space. Strands (like other approaches) explain the entropy of black holes (Weber 1993) and yield thermodynamics, maximum force (Gary Gibbons 2000, Schiller 2000), and therefore general relativity, with its Lagrangian. Only strands explain both fundamental Lagrangians. Only strands explain the equivalence principle. Only strands explain the principle of least action. Only strands explain the smallest system entropy.

Objections

What about dark matter? Strands predict that there are no additional elementary particles. Find one, and the strand model is falsified and collapses. More precisely, strands predict that dark matter consists of known matter and of black holes.

What about dark energy? Strands seem to predict (more research is needed) that dark energy is an apparent effect, as David Wiltshire proposes.

More objections are discussed in the longer article linked below. None is a real issue.

Predictions

No new physics. No additional elementary particles. No elementary dark matter. No additional interactions. No supersymmetry. No GUTs. No MOND. No quadratic gravity. No wormholes. No cosmic strings. No additional dimensions. No time-like curves. No additional observable relativistic quantum gravity effects. No unified equation or Lagrangian. No points in space and no instants in time. No effects of quantum gravity.

All fundamental constants can be calculated.

Status of checks and falsification attempts

No known evidence disagrees with the predictions. Elementary dark matter has not been found, and David Wiltshire’s work shows that it is unnecessary. Dark energy is in contradiction with the most recent data, as Subir Sarkar and his coworkers, David Wiltshire and his coworkers, and many others have shown in recent years.

But: precise strand calculations are available only for the fine structure constant. They still are needed for the other constants.

History

The idea of strands is due to Paul Dirac (1929). It inspired Peter Battey-Pratt and Thomas Racey (1980). The fundamental principle is an extended version of a proposal by Kauffman (1987) that was taken up only much later.

A public prediction

The public prediction for the validity of this single law of nature is online at https://longbets.org/975/. Feel free to discuss it with everybody you know.

A dinner invitation

As usual, any interesting counter-argument - even if wrong - will yield a dinner invitation.

Uniqueness

Strands are the only approach that explains all laws of physics - not because no other published approach achieves these results, but because no other, inequivalent approach can explain all laws. No other approach explains space, particles and black holes, with all their properties. Not strands of different radius, nor fluctuating sheets, nor fluctuating little spheres, nor fluctuating rings or knots, etc. Only fluctuating strands with Planck radius and no ends describe nature.

Summary

In summary, strands explain all laws of physics.

The proposal of a single law of nature is crazy. But it has been tested successfully for a long time. The proposal is also easily falsified: just find a single observation that disagrees with the strand model.

Finally, the proposal is beguilingly simple and beautiful. As long as the proposal remains unfalsified, it contains all known laws of physics and, thus, all known laws of nature.

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References

A simple introduction is available on the web pages motionmountain.net/9lines and motionmountain.net/tiny. A detailed list of predictions is motionmountain.net/bet.

Two physics articles, a shorter one and a longer one, provide the present status:

• Testing a single fundamental principle yielding general relativity, particle physics, and gauge anomaly cancellation, https://doi.org/10.5281/zenodo.19479823.

• How come the quantum? From the topological origin of Planck’s quantum of action to the fine-structure constant, https://doi.org/10.5281/zenodo.19628906.

Enjoy.

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But already measuring 𝜶 is a challenge by itself. At present, the best measurements yield the average value 𝜶=1/137.0359991(1). Measuring the value to higher precision is a challenge in itself, a challenge that is taken up by several groups across the world.

The Sorbonne group led by Guellati-Khélifa found 137.035999206(11).
The Northwestern group, which is led by Gabrielse, found 137.035999166(15).
Other groups found values around 137.0359990…, as told in the article by K. A. Bronnikov, V. D. Ivashchuk, and V. V. Khruschov, The fine-structure constant: a review of measurement results and possible space-time variations, Measurement Techniques 68, 125 (2025). In simple words, the digit after the three nines is not settled yet!

The 2025 research conference in Mainz, shown hereunder, explored all the present approaches to increase measurement precision: https://indico.mitp.uni-mainz.de/event/417/timetable/#all.detailed.

Everybody at that conference aims to achieve higher measurement precision. The coming years should bring progress and clarity, because the results of the different groups do not overlap.

On the other front, in the efforts for the calculation of 𝜶 ab initio, something interesting happened already in 2004. In that year, Hans de Vries published a formula for alpha that agrees with measurements so far. He was writing a book on quantum electrodynamics. While doing so, he guessed an expression that reproduces all measurement averages to this day. De Vries’ expression

predicts

𝜶_predicted = 1/137.035 999 095 829 70…

Note that the expression only contains e and 𝞹, exactly as many researchers demanded. De Vries’ guess was the result of an amazing and refreshing intuition.

When ou Kauffman and I saw de Vries’ fomula about a year ago, we thought directly that we should be able to derive it from the strand model. And we managed to do so. Our article, linked below, deduces the formula from the strand tangle model for the electron and for the photon.

One notes directly that the two measurements cited above appear to disagree with the expression by de Vries; one disagrees by 10 sigma, the other by 5 sigma. However, the de Vries prediction does not disagree with other measurements, and it agrees with the world average 137,035 9991(1).

In short, we have a numeric prediction that can be checked with experiments. Future measurements should settle the issue. We live in interesting times. (And yes, I believe that spending money on several experiments measuring the fine-structure constant to high precision should have top priority in research spending about the foundation of physics – all over the world.)

In future, a further point that needs to be explored is the energy dependence of the fine structure constant, its so-called running. Strands predict that the value should increase with energy. We are working on this topic. Also many other aspects of the strand model are fascinating, and we still have to decide which one to explore first.

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The way to calculate the fine structure constants from strands is told this new article, from July 2026:
Louis Kauffman & Christoph Schiller, How come the quantum? From the topological origin of Planck’s quantum of action to the fine-structure constant, https://zenodo.org/records/21467195. It derives the above formula by de Vries from strand geometry. The paper contains many additional explorations.

Note that only a few proposals to calculate the fine structure constant from a model that describes nature are available in the research literature. And the other approaches all predict new effects, or new particles, or new forces – none of which have been observed yet. Future will tell.

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Understanding the value of the fine-structure constant requires going beyond the standard model of particle physics. As told in this blog already a few times (and in the preprint linked at the end), there exists a single fundamental principle, based on fluctuating strands of Planck radius, that reproduces quantum theory, quantum electrodynamics, quantum chromodynamics, the standard model with massive neutrinos, and general relativity, all without measurable deviations. Strands thus go beyond the standard model, but, at the same time, agree with all experiments.

Strands have the advantage that, in contrast to the standard model, they determine unique particle masses and coupling constants, of which 𝛼 is one. The fine structure constant 𝛼 describes the strength of electromagnetism. Its square root also describes, as Feynman tells, the probability that an electron absorbs a photon. The square root also describes the phase change that an electron acquires when it absorbs a photon. These statements are sufficient to determine 𝛼 with the strand model, as shown now.

Electrons have a phase because they rotate. The rotation of electrons, their spin, is proven by the Einstein-de Haas effect. If you do not know it, look it up. If you are bombarded with stupid internet memes claiming that electrons do not rotate, read about this effect to know how wrong they are.

Electrons rotate, but they are not spheres. Their spin 1/2 proves it. (Spheres have spin 0.) But also 𝛼 proves it: if electrons were spherical, they would not have such a crazy number as absorption probability. Or as a phase change. In short, we know that electrons are rotating, but also that they are not spheres. What structure do they have? Well, Dirac told us since 1929 that electrons are fermions, and that therefore they are tethered.

Particles are indeed tethered, as confirmed by Battey-Pratt & Racey in 1980 for matter particles. (Click for reading their paper.) Indeed, Battey-Pratt and Racey showed that a tethered matter particle automatically follows the Dirac equation. In simple words, the Dirac equation is due to tethers!

But what about the central part of the particle, the part that spins? What structure does it have? Is it a golf ball, like Battey-Pratt and Racey imagined? The answer given by the strand tangle model is simple: particles, space, and horizons are all made of strands/tethers. Nature consists of strands. The strands have Planck radius. As a result, the strands are unobservable: they are invisible. They have no mass, no tension, and no elasticity. But the effects of the strands are observable: the three dimensions of empty space, and the curvature of space is due to the many strands making up space. In addition, particles are tangles of strands. This explains both spin 1/2 and spin 1, depending on the type of their tangling. Furthermore, horizons are tight weaves of strands; this explains the entropy, the temperature, and the mass of black holes.

It turns out that the tangle geometry of electrons completely determines the value of 𝛼. More precisely, the fine-structure constant follows from the geometry of the following process:

Taking into account the fluctuations of the strands yields the fine structure constant to full measurement precision, as will be shown in this and in the next part of the calculation.

The electron is assumed to be at rest. As a consequence, its 6 tethers are perpendicular. Each of the three outer crossings yields a charge e/3. (This also explains the charge of the quarks, which have a different structure, not shown here, and the charge of the W boson. Details are found in the preprint on the tiny theory linked at the end.) Given that the two images differ only by a few Planck times, spin and precession play no role.

The photon absorption probability p will depend on the incoming angle 𝝑. Because the full cone angle is a right angle on average, and because a photon arriving along the side of the cone does not affect the electron, whereas a photon along the cone axis has maximum effect, the general probability dependence for a fixed cone is

At the same time, the strands fluctuate. The cone will fluctuate in its cone angle and in its orientation. This yields a folding of the previous expression with a Gaussian distribution, leading to the probability

The cone orientation can have infinitely large angles with respect to the incoming photon, because the strands can wind up without limit.

The variance of the angle in one direction is the square of the average cone angle

Given that the incoming photon is described by an azimuthal and a polar angle, the total variance is twice this value. Finally, we need Feynman’s definition that the fine-structure constant is the square of the probability p. Together, this yields an initial approximation for the fine structure constant given by

This initial approximation differs from the measured value - 1/137.0359991(1) - by about 1.5%. This is not a good approximation, but it will get better soon, in the second part of this post series.

Note that this initial approximation was already proposed in 2004 by de Vries! He found it intuitively, without an underlying model. No other author ever thought about this approximation. Only de Vries, and nobody else, had the intuition to use it as the staring point for further improvements. As we will see in the next part of this calculation, de Vries also deduced, already in 2004, an improved expression for the fine-structure constant that still agrees with all observations today, 22 years later.

Note that fluctuating strands provide a model explaining de Vries’ initial approximation. The strand fluctuations lead to an expression for the fine structure constant that only depends on e (because of the Gaussian) and on 𝜋 (because of the 3D structure of the electron tangle).

As we will see in the next part, strands also provide the underlying model for all the improved expressions by de Vries: not only for his initial approximation but also for all the following ones. The fluctuations of the strands and the geometry of the electron tangle explain them. A literature search confirms that no other attempts to derive the fine structure constant from a Planck-scale model of particles in 3D exists. (However, there is one different, competing model: Singh’s octonion model also derives the fine-structure constant, though with a more involved and more abstract geometry, not in 3D.)

De Vries’ final formula is unique across the literature. The next part of the strand calculation of the fine-structure constant will increase the precision by taking into account virtual particles, which were neglected so far, and will yield a result for the fine structure constant that agrees with all experiments and that provides a test for all future measurements.

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A technical reference is this new paper, from July 2026: https://zenodo.org/records/21467195

Strands allow to classify tangles and interactions with theorems from topology. To read more about the classification leading to the interactions with their gauge groups, as well as the classification leading to the elementary particles with their masses and quantum numbers, see the preprint https://www.researchgate.net/publication/397264142 on the tiny theory. Enjoy.

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(NASA/CXC/M.Weiss)

Now, we are used to saying that space is made of points since Euclid said so in the first sentence of his great book, the Elements, over 2200 years ago. (It is still worth reading.) We are also used to saying that wave functions are functions of space, thus with a value at every point. And the idea of the point was indeed successful: we deduce quantum theory and general relativity.

But the idea of point has two problems. It is approximate and thus wrong. But worse, it prevents progress. And I will concentrate on the latter aspect. (For the former, see this short page that explains the existence of a minimum length in nature, or a previous post.)

Simply said, the idea that space consists of points is the opposite of saying that it is emergent. And we know that space is emergent, because it fluctuates. This latter aspect is proven by the Unruh effect and also by the black hole entropy. In other words, if we want to understand what space is, we have to drop the idea that it consists of points. And this is a radical cut. Sometimes, pictures of space like the above are provided. But that picture cannot be correct: it shows fluctuations, but it is still made of points!

In other words, such pictures do not let space (or wave functions) emerge from some deeper substrate. Such pictures continue the old habit of thought of a continuous space - and prevent the understanding of emergence.

“Emergence” means that space (and wave functions) are approximations of something else. As long as points are used, emergence cannot be described, cannot be deciphered. Continuing to use points as fundamental constituents of space is the reason that the unification of physics took so long.

So the challenge of unification is clear: describe nature without points. This means to find arguments without points to explain why space has three dimensions, why there are three gauge interactions, why there are three generations of fermions, and where the masses and coupling constants come from.

The last topic is the final nail in the coffin of the concept of point: if one bases the laws of nature on points, one cannot explain masses and coupling constants. To explain them (and mixing angles), a different concept is needed.

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Soon, we will put online our latest work on the strand model, which describes the standard model of particle physics with massive neutrinos and general relativity, and predicts the lack of measurable deviations from the two. It will also deduce the value of the fine structure constant and provide estimates for particle masses, all ab initio. An introduction to the strand model is found here.

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All answer that electrons are point-like and that spin is not rotation. Both answers are wrong. As long as they are wrong, there is no danger that Ai will be better than physics researchers.

The following human image is half correct and half wrong:

A nonsense image circulating on the web - no author can be found.

Here is what nature tells us.

All experiments show that electrons are extended but that they are not balls.

If electrons were balls, they would have spin 0. But all experiments show that they have spin 1/2. If electrons were balls, they would not generate the fine-structure constant. If electrons were balls, there would not be anti-electrons (positrons) in nature. Or, equivalently, if electrons were balls, they would not be chiral. Electrons are not balls.

If electrons were point-like, they would be smaller than the minimum length in nature. The minimum length arises by combining the highest speed c, the maximum force c^4/4G, and the minimum action h-bar. Combining these three statements yields the minimum length

This is the minimum length and minimum length measurement error in nature. An electron cannot be smaller than the minimum length. An electron cannot be point-like.

(1) Experiments thus confirm: electrons are extended but not spherical (thus not balls).

Now about spin. Is it rotation? Yes, and experiments show this! The simplest way to see this called the Einstein-de Haas effect, as told in physics books and also in Wikipedia.

If the magnetic field around a metal bar is reversed, the metal bar starts to rotate. This is due to the conservation of angular momentum. The angular momentum is due to the spin of the electrons. Electrons can thus transfer their rotation to a metal bar!

(2) Experiments thus confirm: electrons rotate.

These two statements are clear and well-known. True, present Ai systems make wrong statements. They do not make misleading statements; they make false statements. Given that all the above is known for more than 50 years, this is quite disappointing. (But not surprising, as shown in my list of Ai physics challenges at https://www.motionmountain.net/prizes.html#Ai - Ai is really bad at physics…)

How can one imagine electrons then? This a topic of research. At present, the only model of electrons in the research literature that shows how they rotate, that reproduces their spin 1/2, and reproduces their chirality and their extension, is the strand tangle model. You can have a first impression on this web page: https://www.motionmountain.net/strandsquantum.html. The strand tangle model is also the only model explaining why different particles all have the same spin value 1/2.

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More details about the strand tangle model are found in this text: https://www.researchgate.net/publication/403963214_How_come_the_quantum_Testing_the_topological_origin_of_Planck’s_quantum_of_action

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Addition: spin 1/2 also implies that electrons are not round.

(Photo by Postdlf, via Wikipedia)

Some people fear alien spaceships. Alas, spaceships exist as much as pink flying elephants. The term “spaceship” is a lie in one word.

Other people fear that some evil energy may appear and suddenly take over the world. No, this will not happen. Energy is conserved. Nature follows rules.

Other people sell shares in companies that extract energy from the vacuum. But the vacuum is defined as the state of lowest possible energy. Such energy extraction is impossible. These people are lying.

Still other people say that nature has secrets, that there are mysterious phenomena in nature, or that there are hidden powers. Instead, a bit of physics will make you understand that this is not the case. Many cults spread such fears, all based on lies. Physics, your own experience, and your own reason will reject these fears and lies.

Physics does so by explaining that every change in nature is due to motion. Therefore, every change in nature has conserved quantities. In particular, in nature, things cannot appear or disappear suddenly. Not even lightning. Such fears are unfounded.

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But physics also helps to push curiosity to the limit. One learns what matter is made of, how light moves, how computers work, how lasers function, how atoms and molecules are constructed, and how stars shine and move. One explores and learns to explain chemistry, biology, machines, and the universe, with its size and its history.

Physics also tells us that everything that moves in nature is made of smallest countable constituents. This applies to people, animals, stars, black holes, and even to the curvature of empty space.

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Physics also tells us how we can change the world. When we dream about the future of medicine, of agriculture, of communication, of computing, of transport, of space travel, or of serving others in any other way, it makes sense to check what is possible and what is not by using all knowledge about how change and motion occur in nature. Physics is the science of motion.

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For a simple start, enjoy the free physics textbook in 5 volumes, downloadable at motionmountain.net, written to be entertaining and motivating on every page, for readers between 16 and 106 years of age.

This is satisfying, but one thing is missing: strands provide no unified equation.

How can a theory describe both quantum theory and general relativity, thus unify physics, without providing a unified equation?

(Photo by Parker Bossier, via Wikimedia)

It turns out that unifying physics eliminates the possibility of a unified equation. Though strands make this point especially clear, the arguments are general, and they are independent of the details of the unified theory.

Any equation describing all of nature requires an outside observer. This is impossible. We are part of nature. A global equation describing all of nature is impossible. But what about the fundamental constituents of nature, whatever they might be? Is there a unified equation for these constituents? Is a local equation possible?

In textbook physics, space and particles consist of points. In textbook physics, points are the fundamental constituents of nature. We use points to deduce local descriptions. But points are not observable. Besides that, points have very few specific properties. Therefore, one can invent many theories based on points. Indeed, there are hundreds of possible physical theories. Alas, none of them explains the number of dimensions, the number of interactions, the number of elementary particles, or the thermodynamics of black holes.

In full contrast, strands explain the properties that are unexplained by points. But we pay a price.

1. Strands, like points, are fundamental constituents. And all fundamental constituents are unobservable. It is easy to check that any fundamental constituent must be unobservable: after all, it must yield empty space, and no experiment has ever observed constituents of empty space. As a result, no equation for fundamental constituents can be tested against observations. If one cannot test such an equation, one cannot state that such an equation is possible.

2. But there is an even stronger objection. Any fundamental evolution equation for a strand or any other fundamental constituent would need to describe the detailed influence of all of nature, the rest of the universe, on the constituent. This is impossible. In contrast to everyday life, strands, like all alternative fundamental constituents, do not have an environment that can be ignored. No fundamental constituent can be isolated from its surroundings. Fundamental constituents are never alone. Therefore, one cannot find an equation that describes their evolution.

3. In quantum theory, the lack of local (or non-contextual) hidden variables also prevents a unified equation. But this is a highly technical argument. In simple words, nature is probabilistic, and local equations reproducing this property are impossible, because they contradict observations.

4. In physics, all equations of motion are based on physical observables., such as force, magnetic fields, probability denisties, etc. Because the fundamental constituents of nature are not observable, there is no way to even think about a fundamental equation of motion. But we can say more. All physical observables are emergent. Textbook physics does not realize this aim, because no evolution equation does so. In textbook physics, all evolution equations are based on the distinction between points and observables. To put it in a provoking manner, mathematics invented points and physics invented the observables that describe nature. This works very well, as we all know (there is no deviation between theory and practice in modern physics), but the approach cannot explain why nature is as it is. Points are not specific. A full explanation of nature needs to explain physical observables as emerging concepts. Points prevent this. Strands solve the problem.

5. When Euclid defined a point as “that which has no parts”, he made clear that the concept of point is a reductionist concept. But this approach cannot be correct. At the foundations, the constituents of nature cannot be isolated. And observations confirm this directly. The minimum action in nature, the highest speed in nature, and the maximum force in nature, when combined, imply that points do not exist. (Equivalently, the entropy and the temperature of black holes imply that points do not exist.) These Planck limits, when combined, imply that nature has a minimum length. With a minimum length and without points, there are no equations.

In simple words, a unified equation, thus a unified local description, is impossible because everything is related to everything else. Now, this may sound familiar. We all have read arguments claiming that a unified theory of nature is impossible because we are, ourselves, part of nature. But there is a difference: the arguments just given prevent the existence of a unified equation. The arguments just given do not prevent unification itself.

In short, whenever people talk about a unified equation, or a core equation, or a God equation, or a “world formula”, you directly know: they have not thought about the topic and are just drawing attention to themselves by talking nonsense. The Planck limits of nature also limit reductionism, and imply:

A falsifiable, unified equation is impossible.

In contrast, a falsifiable, unified theory is possible.

A unified theory needs to abandon points and equations. It needs to take into account the minimum length in nature. And such a unified theory appears to be available, as suggested by the tiny theory, also called the strand tangle model.

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References:

How come the quantum? Testing the topological origin of Planck’s quantum of action, by L.H. Kauffman and C. Schiller, available at zenodo.org/records/19628907

Testing the tiny theory: a single fundamental principle yielding general relativity, particle physics, and gauge anomaly cancellation, by C. Schiller, available at https://www.researchgate.net/publication/397264142.

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The following image, showing the platinum atoms on a needle tip, captures the fascination of atoms making up a metal. We are also made of atoms - just different ones, many of them, and always moving around.

Tatsuo Iwata, via wikimedia

Why do atoms exist? The first answer you will learn at school is: atoms exist because of Planck’s quantum of action ħ. In my view, Planck’s discovery was one of the most influential ideas of modern science. But then we ask: where does the quantum of action come from? There was no answer for a long time. But finally, in 1987, Louis Kauffman provided the simple and fascinating answer: Planck’s quantum of action ħ is a topological effect.

This is the basis of the strand model of nature, in which particles are tangles of unobservable strands whose crossing switches ħ and are observable.

Why is this answer interesting? It enables everybody to understand and to visualize quantum theory. The definition allows everybody to visualize wave functions, to visualize spin, to visualize statistics, to visualize multi-particle states, to visualize superpositions and interference, to visualize wave function collapse, to visualize entanglement and to visualize measurements and probabilities.

Recently, Louis Kauffman and I wrote a text together that realizes this promise. The text is pedagogical, thus somewhat long, but you can start by just looking at the pictures.

By imagining nature - particles, vacuum, and horizons - as built of strands, it is possible to deduce, to visualize and to understand quantum theory. But it is also possible to deduce and to visualize the three gauge interactions, general relativity and the various elementary particles found in nature. The text dramatically simplifies our description of nature. I hope you enjoy it.

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The full article is
How come the quantum? Testing the topological origin of Planck’s quantum of action, by L.H. Kauffman and C. Schiller, available at zenodo.org/records/19628907

When you look at it for the first time, only look at the pictures.

Then, only read the last paragraph of each section, which starts with “In short”.

Enjoy.

And as usual, every good remark about the text is rewarded with an invitation to dinner - or more.

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(photograph by AnRo0002, via wikimedia)

Since 1916, the origin of this number, usually abbreviated as 𝜶, has been unknown. Many people have tried to understand it. Dirac tried. Pauli tried dreams and psychotherapy. Feynman tried. For almost a century, nobody succeeded.

Then, in 2004, Hans de Vries, in the Netherlands, published a fascinating formula for 𝜶. He published it on his website, where it is still found today: http://www.chip-architect.com/news/2004_10_04_The_Electro_Magnetic_coupling_constant.html De Vries was exploring quantum electrodynamics and found an astonishing formula for the fine-structure constant that looks like many others in quantum electrodynamics, the domain of theoretical physics exploring everything electric. His formula is the implicit expression

and predicts the value

𝜶_predicted = 1/137.035 999 095 829 70…

Note that the expression only contains e and pi, the integers, and nothing else.

In 2026, the latest and most precise experiments suggest

1/137.035 999 0… < 𝜶_exp < 1/137.035 999 2…

In other terms, the formula by de Vries from 2004 is in full agreement with the data of 2026! (There are some details here, because the latest experiments disagree among each other in the last digit and, above all, on their (triple) standard deviations, but I leave that for another time.)

De Vries’ formula is the only formula of the fine-structure constant that has withstood the test of time. It is also the only formula that has the chance to agree with quantum electrodynamics. De Vries did not publish how he guessed it. In any case, his intuition for perturbative field theory is extraordinary.

This is already a fascinating story by itself. But the fascination continues: the strand tangle model, about which I have told so much in this blog, can deduce this formula, and does so ab initio! In other words, strands imply that de Vries’ expression is the correct description of nature: de Vries’ expression for the fine-structure constant follows from the single fundamental principle of the strand tangle model that describes general relativity and the standard model of particle physics.

The combined prediction of de Vries and of the strand tangle model is precise and falsifiable. Future experiments can test it further and thus decide whether nature is really made of invisible strands that connect everything with everything else.

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The story of the strand model is told in the papers and presentations found at https://www.researchgate.net/profile/Christoph-Schiller-2/research. Enjoy.

TheA preprint telling how the strand tangle model derives de Vries’ expression is here: https://zenodo.org/records/21467195 .

You can also have a look at the preprint https://www.researchgate.net/publication/397264142 on the tiny theory. Enjoy.

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A naughty comment: Ai missed the connection between strands and de Vries’ formula, even though everything has been available publicly for years. What do we learn from this? In 2026, Ai systems can talk, but cannot think.

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The fundamental principle of the strand model describes Planck’s quantum of action ħ as a crossing switch of unobservable strands of Planck radius:

A literature search shows that no other model describes the quantum of action ħ and its effects, despite various attempts. Likewise, no model for ħ has been used or proposed in the literature about emergent quantum theory, causal sets, loop quantum gravity, superstring theory, causal fermion systems, the octonion model, or any other approach to unification. To allow a precise discussion, it is best to start with a

Definition: A model is a visualization in three-dimensional Euclidean space that uses countable constituents for space and for particles that are subsets of 3d space.

This definition is due to the observation that nature is three-dimensional. This definition of a model allows the comparison of alternative models with the strand model for ħ using crossing switches. This comparison is possible for several aspects of quantum mechanics.

1. As told already several times, tangles of unobservable strands performing both Dirac’s trick and the fermion trick do so in an environment of untangled strands in 3D space. No alternative constituent and no alternative model achieves the visualization and the explanation of spin ħ/2 behavior, of fermion behavior, let alone of the spin-statistics theorem for spin ħ/2 fermions. (Neither the book by Duck and Sudarshan achieves this, nor do the visualizations using the ball B3 in which opposite points are identified, nor the coupled pendula, nor the sphere glowing in different colors.) Above all, no particle model without tethers reproduces the existence of a particle-independent smallest positive value for spin.

The existence of countable elementary particles of different types, including antiparticles and three fermion generations, is a challenge to any model proposal. The challenge is so demanding that only a few models that explain the spectrum of elementary particles using extended constituents have ever been proposed. All such models are based on topology, because only topological particle models have the chance to explain the qualitative differences among particles, such as their different quantum numbers and states, while ensuring the indistinguishability of particles of the same type.

However, topological models for elementary particles face constraints. Because of spin ħ/2, particles must be tethered; they cannot be knots or links of finite size, but must be tangles. In addition, the particle transformations observed in decays and in particle reactions, including particle annihilations, exclude models of particles based on knotted tangles. Therefore, particles must be, topologically speaking, unknotted, i.e., rational tangles. The classification of rational tangles yields that elementary particles can only consist of one, two, or three strands. Tangles of four or more strands are composed of simpler tangles. Elementary fermions of spin ħ/2 can only consist of two or three strands tangled in a non-trivial way. Elementary bosons of spin ħ must be trivial tangles of one strand or specific tangles of two or three strands. One elementary spin-0 boson is also possible.

The topology of tangles determines the quantum numbers. But the cores of rational 3D tangles have, besides their topological properties yielding the quantum numbers, also geometric properties that are due to their shapes. The average geometry of a tangle determines its quantum state; in addition, the average core shape determines mass values, mixings, and couplings. In simple words, among the approaches that agree with all observations, no other model reproduces particles, spin ħ/2, quantum numbers, fermion behavior, unique mass values, mixings, and coupling constants.

It must be noted that proving this uniqueness of strands independently of 3D space remains an open challenge.

2. The quantization of action and of angular momentum are consequences of discreteness. Now, only extended objects can model discreteness, because quantization and discreteness are inextricably linked to extension. Extension, in turn, implies the importance of topology. That is the reason that in the strand tangle model, at the Planck scale, the discreteness of action and of angular momentum are due to three-dimensional topology, even though the discreteness of space and time is unobservable. Dirac’s belt trick for fermion tangles realizes this connection. A literature search for other possible models yields a simple but clear result: No other model apart from the strand tangle model explains the quantization of angular momentum and the quantization of action.

3. In the strand tangle model, wave functions are oriented crossing densities of fluctuating tangles. In the literature, no other model explains and visualizes complex wave functions. Likewise, no other type of constituent deduces wave-particle duality and the Schrödinger equation ab initio. Generally speaking, no alternative specific model for the emergence of quantum mechanics is available in the literature, despite numerous investigations. Combining countable particles with an (almost) infinite-dimensional Hilbert space requires an (almost) infinite number of configurations or shapes; this is only possible using strands. For example, fluctuating membranes do not reproduce quantum interference, and neither do fluctuating constituents that are Planck-sized or of finite size in all three dimensions. In simple words, apart from oriented crossing densities, no other model for wave functions combines particle discreteness with continuous probability densities.

4. Strands describe quantum jumps as almost discrete processes. Several publications showed that the strand description of photons and their coupling to tangles reproduces quantum electrodynamics, without any measurable deviation, but including the g-factor of the electron. The literature provides no other description of electrodynamics or of quantum jumps. In simple words, at the Planck scale, no alternative description for quantum jumps is currently known.

5. Strands explain and visualize the non-commutativity of quantum observables. No model that uses continuous fields can yield non-commutativity, countable particles and discrete action values. In particular, the non-commutativity of rotations and of particle exchange can only be modeled with a topological model based on strands. In contrast, deformations of constituents with ends, with finite size, with closed strands, or with membranes are commutative. In simple terms, no model other than strands without ends reproduces non-commutativity.

6. The maximum speed, the maximum force, and the quantum of action, taken together, imply that there is a smallest measurable length and that space is fuzzy at Planck scales. (The surface dependence and the magnitude of black hole entropy imply that horizons consist of countable fluctuating strands of Planck radius without ends, without branches, and without knots. Because black holes can form by collapsing particles, particles must also consist of fluctuating strands.) The strand shape fluctuations lead to the fuzziness of quantum mechanics. In simple words, only fluctuating strands explain Heisenberg’s indeterminacy relation and the appearance of probabilities in the quantum domain.

7. Visualizing quantum entanglement with topological models is an old dream. Attempts so far do not reproduce all the observed properties of multi-particle entanglement. Apart from strands, no past approach (bit threads, Borromean rings, etc.) provided a complete description of entanglement, of ħ, or of the relation between the two. Only strands have a topology that allows multi-particle entanglement in complete agreement with observations. In simple words, no model other than strands explains and visualizes quantum entanglement.

8. Strands model photons. A similar model for photons, the unpublished “schmoton” model, is found on the internet. That model, however, lacks the identification of ħ with a crossing switch. Despite numerous attempts, no other model accurately describes electromagnetic waves as composed of discrete photons. In simple terms, no model other than strands describes the countability of photons, their spin ħ, their boson behaviour, their wave behaviour, and all their other properties.

9. Modeling the probabilistic outcomes of quantum measurements with a partially discrete and partially continuous substructure that differs from strands but reproduces particles has not been successful so far. Only observable crossing switches of unobservable strands reproduce Born’s rule, the collapse of the wave function, and decoherence. In simple terms, only observable crossing switches of unobservable strands eliminate hidden non-contextual variables and reproduce wave functions.

10. Deducing the Dirac equation from deeper principles is challenging because of the difficulty of encoding spinor behavior and antiparticles. Apart from Battey-Pratt and Racey’s approach (presented in the preprint at the end), several attempts to derive the Dirac equation from an underlying discreteness have been published. (They are listed by Simulik in his book.) None starts at the Planck scale. In simple terms, only rational 3D tangles yield spinors, yield continuous transitions from particles to antiparticles, and deduce the Dirac equation ab initio.

11. The fundamental principle implies that the principle of least action is the principle of the fewest crossing switches. Because of the coupling of crossing switches to photons, strands also explain the principle. A literature search shows that no alternative model for the principle is known, mainly because no alternative visualization, model, or explanation exists for photons and for quantum measurements. In simple terms, no model other than strands explains the principle of least action.

In summary, the strand tangle model appears to be the only model that reproduces wave-particle duality. It appears that no alternative model to crossing switches of strands explains, visualizes, or describes Planck’s quantum of action ħ or quantum mechanics. Therefore, quantum mechanics requires that elementary particles have tethers. Finding an alternative model or a definite counter-argument to any of the above points would immediately falsify the strand model.

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If you found or suspect a loophole in the above arguments, let me know. As usual, for each good candidate loophole, the first sender will get a dinner invitation.

An introduction to strands in quantum mechanics and in quantum theory is here: https://www.researchgate.net/publication/361866270.

For completeness, the strand model is the description of nature with fluctuating strands of Planck radius, for which crossing switches define Planck’s quantum of action ħ. It is a candidate for quantum gravity. Details are found at www.motionmountain.net/tiny.html and www.motionmountain.net/research.html. The pages explain how strands imply general relativity and the complete standard model of particle physics, including the particle spectrum, the gauge interactions, and the fundamental constants.

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Observation O1: All change and all motion in nature is observed to consist of evolving localized particles, evolving three-dimensional empty space, and evolving two-dimensional horizons.

Observation O2: All change in physical systems is observed to be limited by a minimum action ħ, a maximum energy speed c, a maximum force c⁴/4G, and a minimum system entropy k ln 2.

No hint of a deviation from these two observations has ever been found. This allows us to start a chain of arguments consisting of several steps.

(Kevin Todora, via wikimedia)

Step 1: Observations O1 and O2 distinguish elementary fermion rotations by one turn from those by two turns, and likewise, simple exchanges of fermions from double exchanges.

Step 2: Observation O2 includes that fermions of different masses are observed to have the same spin value ħ/2, independently of their mass and rotation speed. Therefore, in contrast to the rotation of a ball or a planet, the spin of fermions cannot be a geometric effect but must be a topological effect.

Step 3: Because of observation O2, elementary fermions are observed to be smaller than their Compton wavelength. Because of observation O2, in contrast to the rotation of a ball or a planet, the rotations of elementary fermions cannot be detected by following the motion of their geometric structure, shape or features. In addition, because of observation O2, partial rotations of elementary fermions are not observable.

Dirac’s trick shows that two rotations of a tethered object are equivalent to none,
in contrast to a single rotation. The trick also shows that tethered
objects can rotate continuously.

Step 4: Because of observation O2, the only remaining way to distinguish elementary fermion rotations - for observations, for measurements, and for nature itself - is by counting full rotations with respect to a distant environment using the topological effects of Dirac’s trick. More precisely, whatever distant environment is used in practice, the effects of full rotations of elementary fermions are always topological and always relative to the cosmological horizon, i.e., to the black night sky. Fermions are thus related and connected to the cosmological horizon. (Clarification: Without connections, a rotation of an elementary particle relative to the cosmological horizon is not detectable.)

Step 5: Observations O1 and O2, together with the properties of horizons, imply that black holes have an entropy of the order of the Bekenstein-Hawking expression. The existence of black hole entropy implies that black holes are made of fluctuating constituents. The surface dependence of black hole entropy implies that the constituents are filiform, open, without ends, and without a finite length.

Step 6: The order of magnitude of black hole entropy implies that the constituents of black holes and horizons have a projected cross-section given by the minimum area, four times the Planck area. The minimum area itself arises by combining the limits of Observation O2.

Step 7: Black holes can be described both as compressed fermions and as curved space. Therefore, black holes, fermions, and space consist of common constituents. Observations O1 and O2 imply that these constituents of nature occupy subsets of three-dimensional space. Therefore, black holes, horizons, fermions, and space consist of common, open, filiform constituents with minimum diameter, i.e., with Planck radius, reaching the cosmological horizon.

Step 8: Observation O2 also implies minimum length measurement errors. The minimum measurement errors imply that, in every measurement attempt, filiform constituents with Planck radius are unobservable. Such constituents are intrinsically unobservable because nature prevents reaching any Planck-scale limit that is defined with all three quantities c, ħ, and G. This contrasts with Planck limits containing just one or two of these constants, which can be reached. Therefore, horizons, black holes, space, and particles consist of fluctuating strands with Planck radius that reach the cosmological horizon and are unobservable.

Together, steps 1 to 8 yield three conclusions (edited for clarity):

Only tangled strands with Planck radius describe and reproduce localized fermions of spin ħ/2 - with their wave functions, chirality, countability and different mass values - and thus the basis of quantum field theory.

Only fluctuating strands with Planck radius also reproduce two-dimensional horizons and black hole thermodynamics - and thus general relativity.

Only fluctuating strands with Planck radius also reproduce the three dimensions of space, because three is the only case for which strands can form tangles.

As a consequence, observations O1 and O2 lead to the strand model, without alternative. Vice versa, only the fundamental principle of strands explains observations O1 and O2, without alternative.

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If you suspect a loophole in the above arguments, let me know. For each good candidate loophole, the first sender will get an invitation to dinner.

For completeness: the strand model is the description of nature with fluctuating strands of Planck radius, for which crossing switches define Planck’s quantum of action ħ. It is a candidate for quantum gravity. Details are found at www.motionmountain.net/tiny.html and www.motionmountain.net/research.html . The pages explain how strands imply general relativity and the complete standard model of particle physics, including the particle spectrum, the gauge interactions, and the fundamental constants.

* * *

After the talk, a listener asked a question: if fluctuating strands of Planck radius form particles and empty space, as the model maintains, how dense are the strands in the vacuum?

For flat vacuum, the strand model appears to yield a surprising statement. Flat vacuum has no entropy, no temperature, and no energy. Flat vacuum has no observables. Now, in the strand model, any aggregate of untangled strands has no entropy, no temperature, and no energy. This is valid independently of the density of strands. This means that in a flat vacuum, the density of strands is undefined.

This result is unexpected and raises several questions.

Do different strand densities affect particles in different ways and thus correspond to different vacuum states? It might seem that the chiral motion of fermions - so similar to a maple seed moving through air - could be influenced. However, a second look shows that no such effect is expected. All densities have the same effect. All densities put chiral particles into motion. Lower densities might induce lower rotation rates at first sight, but they will also provide less hindrance for the Dirac trick.

Do different strand densities correspond to different potential values? This is a legitimate question. First of all, the strand model for gauge interactions implies that this is not the case for gauge potentials. They are much more complex that simple strand densities. But then, might different strand densities correspond to different gravitational potentials? The answer is not straightforward. At present, it appears more likely that the gravitational potential is related to the gradient of the strand density.

If the absolute density plays no role, might the strand density in flat space still be determined by cosmological aspects, for example, by the cosmological horizon? This leads into the fascinating topic of strand cosmology. It will be treated later on. The strand model has so many topics still to be explored… A whole research department could be kept busy with similar questions.

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The full slide presentation is https://www.researchgate.net/publication/401508100. A short preprint about it is researchgate.net/publication/397264142, a long preprint is researchgate.net/publication/361866270. Enjoy.

* * *

Worse, today we know that the idea of point particles moving through space points is wrong: there is a minimum length in nature, given by the Planck length. So we know that there are no points. And we also know that nature has wave functions. So, what is the correct description of the motion of a single particle through space?

(Photo by Navicore, via Wikimedia)

The nature of particle motion requires a description of what happens at the boundary between a particle and the surrounding space. Today, particles are described by quantum theory and space by general relativity. Thus, the answer to the question about particle motion requires quantum gravity! And, as we all know, that is still a topic of research.

Let us start from the beginning. What can move in nature? Only three systems move: particles, space, and horizons. Because black hole horizons have entropy, they are made of constituents. Because black holes can be said to be both curved space and many compressed particles, all three moving systems must consist of the same constituents.

Because black hole entropy is proportional to their area, they must be made of filiform constituents without ends. Because black hole entropy contains the Planck area, the constituents must have Planck cross-section. Because the constituents have Planck cross-section, they are invisible. Because the invisible constituents - called strands - also form space and particles, space must consist of untangled strands and particles of tangled strands. The tangled regions are observable and form the wave function. In each particle tangle, the strands connect the particle core to the cosmological horizon. The connections of each particle to the cosmological horizon determine its spin and its statistical behaviour.

The invisible strands constantly fluctuate, being pushed by neighboring strands. These fluctuations yield the entropy of black holes (and of curved space) and the wave function of particles. Tangles also yield spin 1/2, as Dirac showed in 1929.

As a consequence, particle motion is a sequence of Dirac’s belt tricks, also called string tricks, in a sea of untangled vacuum strands. This description supersedes the old, conventional idea that particles are points moving through a sea of vacuum points. When moving, the core of the tangle (see image) moves like a maple seed or a pinwheel moving through the air. Particle cores move like little propellers, and the strands connecting them to the cosmological horizon continuously perform Dirac tricks. In particular, this description, when contracted to a point and when recalling that the connections are unobservable, yields Feynman’s description of a particle as moving, rotating arrow.

Can this description be tested? Yes. Describing particle motion as a sequence of Dirac’s tricks in a sea of untangled vacuum of Planck radius strands also models the minimum length, explains the existence and the properties of wave functions, and explains the three-dimensionality of space. And strands allow even more tests: strands explain the known forces and the known elementary particles. In fact (see the references below), tangled strands imply and explain the Lagrangians of general relativity and of particle physics, including massive neutrinos, in all details. No other approach yields these explanations. In addition, as required above, strands also explain the quantum numbers of all particles, the particle masses, the gauge coupling constants, and the mixing angles. And these are the main open questions in modern physics.

In short, quantum particles move like tethered propellers continuously performing Dirac tricks. At present, strands provide the only known way to describe particle motion in accordance with all experiments, to describe quantum gravity in accordance with all experiments, and to deduce all of physics in accordance with all experiments.

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A short preprint about how the strand tangle model deduces all of physics is researchgate.net/publication/397264142, a long prprint is researchgate.net/publication/361866270. Enjoy.

* * *

(Photo by John Samuel, via Wikimedia)

Nature has a smallest length which arises when combining the maximum speed c, the maximum force coming from black hole limits, and the smallest action ħ:

However, in complete contrast, every equation in physics you ever learned assumes that points can be as close as desired. Every equation in physics either assumes that G vanishes or that ħ vanishes.

If we assume that G vanishes and assume ħ differs from zero, we call the description ‘quantum theory’ or ‘particle physics’ and get its equations.

If we assume that ħ vanishes and G differs from zero, we call the description ‘general relativity’ and get its equations.

If, however, we assume that neither vanishes, we get the minimum length, and equations are not possible.

Well, there is one way out. We could assume that the speed of light c is infinite. In other words, believing in a single equation is like believing in speeds faster than light.

This is a dream. Alas, nature does not allow it. Only fantasy does.

In technical terms, there are no trans-Planckian effects in nature. In simple words: believing in a single equation describing all of nature is science fiction.

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References:

The maximum force: motionmountain.net/maximumforce.html

The smallest action: motionmountain.net/actionquantum.html

The minimum length: motionmountain.net/minimumlength.html

(Image of an earthquake damage by Martin Luff, via Wikimedia)

Nature also has a limit to jerk. In American English, it has become common to (humorously) use the term jerk, which is usually abbreviated as j, for the time derivative of acceleration:

Like all quantities, it has a Planck-scale limit. The limit is

Like all higher time derivatives of acceleration, jerk has no practical use in the description of nature. Except one.

The internet gives a platform to many strange opinions. This includes the mistaken opinion that physical systems - or nature - can exceed the Planck limits. This is wrong.

Sometimes, the Planck mass is cited as an argument. But the Planck mass only limits elementary particle mass. And indeed, all elementary particles have smaller mass values than the limit.

In reality, there is no experiment whatsoever that allows trans-Planckian values to be achieved, to be measured or even to play a role in nature. Trans-Planckian quantities simply do not exist in nature. They are as real as pink flying elephants.

In fact, all measured jerk values are much smaller than the limit. Earthquakes can reach 300 m/s^3 - and such then lead to effects as those shown in the photograph above. Macroscopic jerk values are rarely larger than 10^5 m/s^3. Such values are found in crash tests and mechanical failures. For particles in accelerators, values in excess of 10^20 m/s^3 are achieved. But all these values are much smaller than the theoretical limit given above.

In other words, whenever you encounter a jerk, recall that every jerk is limited. This realization will greatly increase your quality of life.

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A literature reference on the jerk limit is Physical Review D 104 (2021) 124079.

The Sower - by Vincent van Gogh

The theory of (relativistic) quantum gravity describes, as the name says, the quantum effects of gravity. Quantum effects are all those effects that contain Planck’s quantum of action ħ. In the case of gravity, this would include quantization of length, area or time, which might arise at the Planck scale. However, the smallest possible measurement errors in nature are of the same Planck size. In other words, there is no way to detect the quantization of space and time.

For example, many papers have discussed that gravitons cannot be detected one by one. There is no way to detect single gravitons, unlike photons. The fundamental reason: graviton effects are indistinguishable from the effects of quantum theory inside detectors.

As another example, quantum theory produces superpositions, such as particles partly in one position and partly in another. Will they yield new effects for gravity? Of course not: the world around us is already full of superpositions, and nothing surprising was found.

One can also ask the opposite question: are there any observations in contrast with general relativity or with particle physics? No. But wait, the mass media say otherwise. They mention “dark matter”, but it has never been observed to differ from normal matter and black holes. The mass media also tell about dark energy. This topic is in flux, at present. The only thing that is sure, today, in 2026, is that the Nobel Prize for the “acceleration of the universe” was premature.

The exploration can be continued. A relativistic quantum gravity effect is one containing all three constants ħ, c and G. No such effect has ever been measured. Only effects containing one or two of these constants have been measured. And it is a safe bet that no effect with all three constants is detectable.

So, how can a theory of quantum gravity prove that it is correct? In general, it has to derive general relativity (which is easy) and the standard model of particle physics (which is hard). But in particular, a theory of quantum gravity must explain the masses of the elementary particles, the gauge coupling constants, and the mixing constants. This is the decisive test.

The standard model of particle physics does not derive the values of the masses, the gauge coupling constants, and the mixing constants. The values are assumed from the start. However, these values are the only detectable effects of quantum gravity in nature.

In other words, to understand quantum gravity, we physicists have first to measure the constants as precisely as possible. This includes the neutrino masses and neutrino mixing parameters. We should invest as much effort as possible in this task.

Secondly, we physicists have to produce as many theories of relativistic quantum gravity as possible, calculate these numbers, and then compare theory and observations. And yes, this implies that any theory that does not allow calculating these numbers is of no help.

Go for a try! (But do not use Ai. It cannot help.)

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In one of the next posts, I will present a quantum gravity prediction that has been tested with a precision of 15 digits.

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In the literature, only two approaches are candidates for quantum gravity, thus deduce general relativity and particle physics and the fundamental constants. (The mass media tell otherwise.) The first approach is the octonion model by Tejinder Singh: see https://arxiv.org/abs/2209.03205; the other is the strand tangle model. A short preprint about the strand tangle model is researchgate.net/publication/397264142 and a long one is researchgate.net/publication/361866270.

Imagine a rope of constant radius that is infinitely flexible. Tie a tight knot into it. By how much did the ends of the rope come closer?

In other terms, find a formula to calculate the length of a tight knot in a rope.

Mathematicians call this the ropelength problem. It is unsolved for every non-trivial knot! The problem is unsolved for open knots (as in the figure) or for closed knots (as mathematicians like to define them). Only computer approximations to the ropelength exist. A formula does not. Clearly, different knots have different ropelengths:

In simple words, nobody in the world has a clue for the solution of the ropelength problem, despite the efforts of several mathematicians who have spent their lives on the problem.

The ropelength problem is fun

(1) A famous knot theorist humorously called the lack of solution a “shame” for the whole field of knot theory, and even a shame for mathematics as a whole.

(2) A few days ago, in early February 2026, I posted the problem on social media as a challenge for Ai systems. One big and proud mathematics Ai project - which I do not want to name here - wrote me back and stated that the problem was “ill-defined” and asked me whether I could provide a “reference”. Now, if an Ai system is even unable to find a wikipedia article, will it be able to solve the mathematical problem described in that article? I doubt it.

(3) Other Ai projects might succeed. One of them even liked my social media post. It will be fun to see what happens.

Mathematics has many beautiful and simple open problems. The ropelength problem might well come out at the top.

The ropelength problem is interesting

Apart from testing the abilities of mathematical Ai systems, the solution to the ropelength problem would be of interest because solving similar geometrical problems would solve the main open issues in fundamental physics.

Is it artificial or absent intelligence?

The ropelength problem is a typical research problem that cannot be solved by reading the literature. Some new insight is necessary. Just ask your favorite Ai system: “Can you provide formulae for the ropelength of the open and the closed overhand knot?”

Is Ai absent innovation?

This specific question is highlighted by the ropelength problem. It is not required that mathematical Ai systems be like Ramanujan. (Though that would be fascinating, of course.) But can Ai systems produce any innovation at all?

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References

The image is from P. Pieranski, S. Przybyl, A Stasiak, Tight open knots, The European Physical Journal E 6 (2001) 123-128, doi 10.1007/s101890170012, and is also found on ResearchGate.

The wikipedia article on ropelength is here: https://en.wikipedia.org/wiki/Ropelength. Some papers follow.

On the Minimum Ropelength of Knots and Links: https://arxiv.org/abs/math/0103224.

Tight open knots: https://www.researchgate.net/publication/225672330_Tight_open_knots.

The shapes of physical trefoil knots: https://www.sciencedirect.com/science/article/pii/S2352431621000043.

More questions that check the abilities of Ai systems are here: https://www.motionmountain.net/tiny.html#Ai. At present, they all disappoint - deeply.

The above animation by ronwnor is an example and a slight extension of Dirac’s trick. Dirac used the trick in his lectures, but never published anything about it. Imagine the strands to have Planck radius. Such strands thus are just thin enough to be unobservable in practice but thick enough to still be physical.

To be precise, Dirac’s trick shows a ħ/2 particle (or spin 1/2 particle for short), where ħ is Planck’s quantum of action. The quantum of action ħ is due to a crossing switch of strands, as this the only difference between the unrotated structure and one rotated by one turn. (Such structures are called tangles.) This was deduced by Kauffman in 1987.

The animation uses Dirac’s trick to realize and visualise spin 1/2. The rotating structure is a spinning fermion. Why? First, the structure comes back to the original state every two rotations. This is the definition of spin 1/2. Then, if you take two such structures and exchange their positions, the whole sets comes back to the original state only after two such exchanges (try it at home with two paper stripes!). This is the definition of a fermion.

This video was made by Antonio Martos. It is also found at at motionmountain.net/videos.html#fermion. Instead of a strip, you can substitute each edge by a strand, so that each strip is changed into two strands.

The animations are also interesting for a second reason: this is the only way to visualize a spin 1/2 particle. There is no other way - across the scientific literature. There are various attempts, but none succeeds in visualizing locality, spin 1/2, fermion behaviour, and particle countability. Of course, must always recall that the tethers are invisible, and that the fluctuating central structure is observable as a cloud, i.e., as a wave function.

In other words, spin 1/2 implies that particles are tethered - even though we do not see such tethers in everyday life. Spin is rotation, particles are tangles.

But the wonders go on. In Dirac’s trick, the little central triangle is chiral. So the structure represents a chiral fermion. Now imagine that the spinning particle is surrounded by a heap of untangled strands. (They represent the vacuum.) The chiral core will act like a screw or a maple seed, and advance. This visualizes a chiral fermion moving through vacuum.

If we imagine a little arrow attached to the central triangle, while advancing, the arrow will trace out a helix. The wavelength of that helix is the wavelength of a quantum particle. Dirac’s trick thus implies the wave-particle duality.

A particle with high momentum has many rotations per length. Every rotation of the arrow corresponds to a Planck quantum of action ħ. Dirac’s trick thus yields de Broglie’s wavelength! The energy E of such a particle then is the angular frequency 𝜔 times the Planck quantum of action ħ. This is Planck’s relation!

Now let us take Dirac’s trick and not forget the random shape fluctuations. We then only get and see a fuzzy cloud. The oriented crossing density of that cloud is what is usually called the wave function. If instead, we imagine the strands to be infinitely thin and the chiral rotating region to be as small as a point, recalling that the strands are unobservable, we get Feynman’s path integral description: we get an advancing point with a rotating arrow.

In both cases, combining the wave properties with the relations for energy and momentum E=p²/2m yields Schrödinger’s equation. But even more follows from Dirac’s trick.

The relation for energy and momentum allow defining an effective inertial mass value of the chiral fermion. Mass describes how frequently and how rapidly Dirac’s trick takes place when a particle moves. When the strands fluctuate, the motion of the animation at the top is in fact quite unlikely. The rotations is thus slow. Therefore, the tangle structure has an effective mass that is much smaller than the Planck mass. Dirac’s trick thus solves the mass hierarchy problem, the question why elementary particles have such tiny mass values. But there is more.

The curved regions in Dirac’s trick are, as a deeper investigation shows, the virtual gravitons surrounding the central mass. Dirac’s trick thus describes both inertial mass and gravitational mass. Therefore, the two are equal. This is the equivalence principle that Galileo and Einstein made popular. It forms the basis of general relativity.

And yes, the fermion can also advance by rotating through curved space. And yes, it follows geodesics. (Empty space is itself made of strands as well, but in contrast to particles, the strands are untangled.)

All this follows from Dirac’s trick. In simple words, Dirac’s trick implies quantum theory and general relativity. For strands of Planck radius, the specific structure in the animation above, with the chiral central triangle, is well known. It is usually called an electron. And yes, every electron in our body is connected to the border of space with unobservable strands. This is the strand tangle model - the tiny theory of physics.

In simple words, if you take seriously the ideas of Dirac and Kauffman, Dirac’s trick contains the main wonders of nature and of physics: both quantum physics and general relativity follow - from topology. Nature is a topologist’s dream.

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A popular introduction to the mentioned dream, the tiny theory of physics, is available here.
A short technical preprint is researchgate.net/publication/397264142.
A long technical preprint is researchgate.net/publication/361866270.
The preprints show that Dirac’s trick also implies the Dirac equation, the electric charge of the electron, and black hole rotation. Mathematically speaking, Dirac’s trick visualises the double cover of SO(3) by SU(2) and the behaviour of the unit quaternions.

The sketch in the lower left is a flash image of a black hole for an observer at large distance. In reality, the strands fluctuate all the time: their shapes “tremble”. These fluctuations yield the entropy, the temperature and the mass-energy of black holes, as told in various papers and preprints (such as this last one). As mentioned, strands are unobservable, due to their thin radius. Therefore, black holes are spherical. The crossing switches of strands yield Planck’s quantum of action: crossing switches thus produce all physical observables. They are all on the surface of the black hole sphere, where the strands are interwoven.

Strand black holes are fascinating. Because strands yield the thermodynamics of black holes, they yield and imply general relativity, with Einstein’s field equations, as Jacobson showed in 1995. For the same reason, strands realize Verlinde’s entropic gravity. Strands also illustrate why the entropic description and the description with curved space are equivalent. (Rotating particles are show why gravity is also the exchange of gravitons, but we skip this here.)

Strand black holes can rotate, as the wonderful animation by Jason Hise shows:

This is a version of Dirac’s trick, for the case of large numbers of tethering strands. One recalls that strands are unobservable, only crossing switches are observable. Thus, only the black hole is observable, not its tethers. The animation also visualizes the frame dragging effect occurring around the black hole horizon.

Strands imply that the mass of a black hole is located on its horizon. This agrees with their moment of inertia. (And this disagrees with all the sycophants that repeat that black hole mass is located in a “singularity” at their “geometric centre”. It was known already before the strand model that this is not the case.) The moment of inertia I of a black hole is I=MR^2. Black holes thus behave similarly to rotating mass shells, which have the slightly smaller moment of inertia I = (2/3) MR^2, and de facto behave like a rotating massive ring, which has I=MR^2.

Strand black holes also can have an angular momentum. The angular momentum is given by the moment of inertia times the angular velocity. This is interesting, because the maximum angular velocity, for a horizon rotating with surface speed c, is c/R. As a result, using the moment of inertia of black holes, the maximum angular momentum J is cMR, or, in units of mass and using the fact that the radius of extremal rotating black holes is half the non-rotating value, J ≤ GM^2/c. This is the well known angular momentum limit for black holes.

Strand black holes further have a maximum electric charge. In the strand tangle model, electric charge is due to chiral crossings of the strands leaving the central structure. (Each such crossing yields a charge e/3.) Because the average strand length in the horizon is a fraction of the diameter, the number of strands in a black hole is proportional to its diameter. Therefore, the maximum electric charge is also proportional to the black hole diameter. This is a well-known result. Strands yield it automatically.

Strands also imply that black holes have no other quantum numbers, because all other quantum numbers of elementary particles, such as strong charge, weak charge, etc., automatically vanish for black holes. This result is usually summarized by stating that black holes have no hair.

Strands also imply that black holes cannot be smaller than a Planck mass. Smaller strand structures are particles - they have no horizon.

Because black holes have a temperature, they evaporate. Strands imply that evaporating black holes leave no remnants, thus no leftovers. Only elementary particles are remain. This solves a further problem of classical black holes. (And obviously, the strand model of black holes and particles has no problem with information loss.)

In simple words, weaves of unobservable strands of Planck radius automatically yield black holes with all their properties. This implies - but do not tell it anybody else - that strands describe the quantum aspects of gravity.

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‪Bibliography:

A simple introduction into the strand description of gravity is this web page. It also contains many experimental predictions about gravity and quantum gravity.

For physicists, the strand tangle model is presented in this publication preprint. The strand tangle model deduces all of general relativity and all of particle physics from just one fundamental principle. A long publication preprint is here.

There is only one way out: one needs a model of space that is both continuous and discrete. This allows combining the advantages of continuity, namely the possibility to produce continuous fluctuations, with the advantages of discreteness, namely to model a minimum length.

As readers of this blog know, there is only one simple way to do so: thinking of space as consisting of fluctuating strands of Planck radius, with no end or branches.

1. Such strands have twice the Planck length, the minimum length in nature, as their diameter. Thus, they are barely observable (in principle) and thus “barely exist”. However, such strands cannot be observed in practice, because there is no way to reach those tiny lengths.

2. Such strands must fluctuate, to produce isotropic space.

3. Such strands cannot have branches or ends, because that would not yield black hole entropy or particle reactions as we know them. They also would not yield interactions or elementary particles.

These three points appear simple. However, it takes some time to get used to the idea that nature consists of strands. We are educated differently. And the approach produces many questions.

• Can the stands be cut? No.

• What are strands made of? They cannot be cut, so an answer is impossible.

• Can one prove their existence? Not directly, only indirectly: they produce quantum effects. They even describe superpositions and quantum measurements.

• Without strands, we do not understand spin 1/2 and fermion behaviour.

• Without strands, there is no Dirac trick! This trick is also called the belt trick or the string trick. Not onlydoes it explain spin1/2 ad fermion behaviour; it also explains how wave functions arise, its explains wave-particle duality, it explains the Dirac equation. The Dirac trick also explains fermion masses, inverse square gravity, the equivalence principle, and black hole rotation.

• Strands explain the principle of least action. Also this story is told in the text listed in the bibliography.

• Do strands make experimental predictions? Yes, many, listed here, and they all agree with observations. They deduce general relativity and the full standard model of particle physics with massive neutrinos.

• But do they predict something new? No! They predict that physics is already fully known. But that was said already many times. Indeed, but this time the predictions agrees with data! (Well, if you support this project a bit by clicking on the donate button found on this page, and I’ll tell you what can be predicted after all.)

• Strands are crazy! Our bodies are surrounded by strands? This is crazy! Yes, strands are crazy. But they answer all questions: For example, they tell why the elementary particles are limited, why there are only four interactions, where do the coupling constants come from, where particle masses come from, including neutrinos, why masses run with energy.

  

The last point, the craziness of strands, is the second reason that unification took so long. The first reason was the difficulty to accept minimum length. The second reason, the craziness, is related to the difficulty of finding a concrete description that incorporates minimum length. Strands do require a complete change in our ideas about the foundations of nature. Everything in nature is made of strands:

Strands require a change in thinking habits. It takes time to accept such a radical change of description. But strands have advantages: strands, for the first time, visualize wave functions. Thus, strands can be used in teaching quantum theory.

Above all, strands explain why we can scratch our nose and where the colours around us come from. Both the scratching and the colours are explained because strands, for the first time, explain the origin of the fine structure constant. In the research literature, there are only two approaches that explain the coupling constants and the interactions. Let’s find out who will win.

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‪Bibliography:

A popular introduction into the tiny theory is this web page.

For physicists, the tiny theory of physics, the strand tangle model, is presented in this publication preprint. The strand tangle model yields ab initio estimates for elementary particle masses, coupling constants and mixing angles. Strands also deduce the elementary particles, the gauge interactions and many experimental tests - all from just one fundamental principle.

A long publication preprint is here.