The spinning electron   Spinning electron.       The spinning electron   Evolving photon.

Beyond textbook physics – searching for the origin of colours

The present, grey-coloured pages propose a unified description of fundamental physics – including quantum gravity – called the strand tangle model or the strand conjecture, and compare its consequences and predictions with experiments. Strands describe all of nature with a single principle:

General relativity and the standard model are due to fluctuating strands of Planck radius, for which each change from an overpass to an underpass yields Planck's quantum of action ℏ.
Strands yield a number of previously unavailable results: strands explain the origin of each gauge interaction, its gauge groups and other properties, the origin of each elementary particle and its quantum numbers, and the origin and uniqueness of each fundamental constant: particle masses, the fine structure constant, the nuclear coupling constants, and the mixing angles. However, their exact values still need more precise calculations. Strands explain the origin of wave functions and allow deducing, without alternative, both the Lagrangian of general relativity and that of the standard model.

The strand tangle model is daring in its ideas, wide in its scope, and restrictive in its experimental predictions.

A simple introduction to strands in 12 one-minute steps is found on this page. Talk pdf
A compact introduction for physicists and physics students is this text. Talk pdf
A pedagogical introduction for physics students is here. Talk pdf
Talk slides with an introduction to the strand conjecture are here. Talk pdf They provide a complete description of motion, with animations and experimental predictions, explaining the origin of the fine structure constant 1/137.03, the electron mass, and thus all colours.
A motivating introduction for scientists and philosophers is this essay. Talk pdf
For the comparison of the strand conjecture with experiments,
read here. Talk pdf
For commented scientific publications and preprints about strands, read further. Talk pdf

●  Summary: what happens beyond the standard model?
●  Step 1: Maximum force– implies general relativity
●  Step 2: From maximum force to unification– requires strands
●  Step 3: From strands to black holes, general relativity, and quantum gravity
●  Step 4: The strand conjecture about wave functions and field theory
●  Step 5a: The strand tangle model yields the standard model of particle physics
●  Step 5b: Quantum electrodynamics deduced from strands
●  Step 5c: Quantum chromodynamics deduced from strands
●  Step 6: Cosmology deduced from strands– about dark matter and dark energy
●  Outreach
●  Acknowledgements
●  Volume VI of Motion Mountain
●  Strands in other languages

Summary: what happens beyond the standard model?

1. All observations and all equations of fundamental physics follow from crossing switches of fluctuating strands with Planck radius that reach the cosmological horizon.

2. All fundamental constants – the number of dimensions, the coupling constants, the particle masses and the mixing angles – are unique and follow from the strand tangle model. They can be calculated.

3. Strands predict – like textbook physics does and like Bronshtein's physics cube does – that there is no physics beyond the standard model with massive Dirac neutrinos and no physics beyond general relativity. All experimental predictions deduced from strands so far – listed here – agree with all experiments. No observation remains unexplained in fundamental physics, including the principle of least action. Finally, strands predict that no other, inequivalent model yields these results.

4. Particles are rational tangles of strands, wave functions are blurred strand crossing densities, black hole horizons are blurred weaves of strands, space is made of blurred crisscrossing strands, curvature and gravity are due to inhomogeneous strands, and the three gauge interactions are due to the three Reidemeister moves of strands.

5. Properties: One principle. Based on Lorentz invariance. Implies 3 dimensions. Deduces the 3 gauge interactions. Derives the 3 generations and the known elementary particles. Reproduces spinors and perturbative quantum field theory. Reproduces curvature and general relativity. Reproduces black hole entropy. Unifies general relativity and quantum theory. Makes testable predictions: no new physics, no supersymmetry, no science fiction.

All derives from the fundamental principle:

The fundamental principle of the strand
tangle model
The fundamental principle animated by Ben Kilgore (just click):

The fundamental principle states that each strand crossing switch produces a quantum of action ℏ. The principle allows deducing the complete standard model of particle physics and full general relativity. This includes deducing spinor wave functions, antiparticles, Dirac's equation, the gauge groups U(1), SU(3) and broken SU(2), electromagnetic fields, Maxwell's equations, QED, the nuclear interactions, QCD, the particle spectrum with its three generations, massive Dirac neutrinos, PMNS mixing, the Higgs mechanism, weak CP violation but strong CP conservation, and the quark model, but also curvature, the metric, Einstein's field equations, the Hilbert Lagrangian, and cosmology.


Step 1: Maximum force

The story started with the discovery, in the years 2000 to 2003, of the maximum force value c4/4G in nature, by Gibbons and, independently, by the present author. In the past decades, a few papers tried to refute the result; over twenty papers by various groups across the world have confirmed it and corrected the apparent refutations. The author is also the discoverer of the principle of maximum force, i.e., of the result that general relativity can be completely deduced from nature’s limit c4/4G. The last papers by the author on the topic are

A. Kenath, C. Schiller and C. Sivaram, From maximum force to the field equations of general relativity - and implications, International Journal of Modern Physics D 31 (2022) 2242019, 10.1142/S0218271822420196. This paper won an honourable mention in the 2022 Competition of the Gravity Research Foundation. Download the pdf here. pdf

C. Schiller, Tests for maximum force and maximum power, Physical Review D 104 (2021) 124079. Preprint here. pdf

C. Schiller, Comment on "Maximum force and cosmic censorship", Physical Review D 104 (2021) 068501 10.1103/PhysRevD.104.068501. Free preprint here. pdf

C. Schiller, From maximum force via the hoop conjecture to inverse square gravity, Gravitation and Cosmology 28 (2022) 305–307, 10.1134/S0202289322030082. Download the pdf here. pdf

All deduced results on maximum force c4/4G – or on the equivalent maximum power c5/4G, maximum mass flow rate c3/4G or maximum mass to length ratio c2/4G – agree with all observations so far, including those of the LIGO and Virgo collaborations, and of all neutron star and pulsar measurements. All these limits are equivalent; each one defines general relativity. For details, see the dedicated web page on maximum force and maximum power.


Step 2: From maximum force to unification with strands

The necessity to use strands to achieve unification is deduced in simple terms in C. Schiller, From maximum force to physics in 9 lines and towards relativistic quantum gravity,   pdf published in Zeitschrift für Naturforschung A (2022)

A deeper dive towards unification with strands, still accessible to every physicist, is C. Schiller, From the Bronshtein cube of limits to the degrees of freedom of relativistic quantum gravity.   pdf

A popular account has been published online in Essentia.

These texts make clear predictions on how to pursue unification, and deduce the lack of possible progress in most other directions. In particular, they summarize all experiments ever made and show that unification takes place in three spatial dimensions, that unification must take into consideration the smallest length in nature, the double Planck length, and that the fundamental, common constituents of space and particles must be filiform, fluctuating and of Planck radius. In short, the texts show that the fundamental constituents of nature must be strands.


Step 3: From strands to black holes, general relativity, and quantum gravity

The description of nature with strands reproduces the field equations and quantum gravity. This is shown in the following publications.

C. Schiller, Testing a conjecture on the origin of space, gravity and mass, Indian Journal of Physics 96 (2022) 3047–3064. Read the published paper online for free at Dowload the preprint here.   pdf

A dedicated discussion of the quantum gravity aspects of black holes appeared as a chapter in the book by A. Kenath ed., „A Guide to Black Holes“, Nova Science Publishers in January 2023: C. Schiller, Testing a microscopic model for black holes deduced from maximum force. pdf

The first publication on gravitation from strands was C. Schiller, A conjecture on deducing general relativity and the standard model with its fundamental constants from rational tangles of strands, Physics of Particles and Nuclei 50 (2019) 259–299. pdf

See also the dedicated page on quantum gravity.

50 756 solar masses per second. Strands provide a microscopic model for space and horizons. This allows deriving the field equations of general relativity and a model for quantum gravity. Numerous tests of the strand conjecture in the domain of gravitation and quantum gravity are deduced, starting from a single principle. All tests agree with observations so far.

For example, strands confirm that gravitation – like nature itself – has a power or luminosity limit c5/4G, a momentum flow or force limit c4/4G, a mass flow limit c3/4G, and a mass to length limit c2/4G. The limits are given by one quarter Planck mass per Planck time, or 50 756 solar masses per second (times c-1, times c, or times c2). No observation ever exceeded these limits.

Many predictions about gravitons and quantum gravity are deduced, including a direct derivation of black hole entropy from strands. Above all, strands also explain the existence of gravitational masses of elementary particles, solve the hierarchy problem, and provide upper and lower limits for the mass values. All predictions agree with the data.

Strands seem to be the simplest quantum gravity proposal in the literature. Strands agree with and predict all observations: strands provide a microscopic model of space, black hole horizons and gravitons, explain mass, particles and black hole radiation, imply general relativity without modifications, prevent singularities and wormholes, reproduce cosmology (see step 6), but predict the lack of elementary dark matter particles. Strands are also complete: no question of quantum gravity is unanswered.

Any complete description of nature has to be strange. To satisfy this requirement for gravitation, the following animation, made by Jason Hise, shows how black hole rotation is modelled in the strand conjecture. (The flattening of the horizon at the poles is not shown.) With a bit of imagination, you can determine the location of the ergosphere.



Step 4. The strand conjecture about wave functions and field theory

Pedagogical. For physicists, the best introduction to the strand tangle model is the preprint showing how wave functions, quantum theory, fermions and bosons, the three gauge interactions, the three elementary particle generations and unique elementary particle masses follow from strands. A longer introduction is found on this page.

pdf  C. Schiller, An emergent model for wave functions explaining gauge interactions and elementary particles, preprint on Researchgate at

Wave functions and interactions. A single principle is used to derive the Schrödinger equation, the Pauli equation and the Dirac equation. Spin 1/2, fermion behaviour, mass and, above all, the observed elementary particles and the observed gauge interactions are deduced - without any modification.

Complete. The strand tangle model explains why all measurements are electromagnetic, why only massive particles can have electric charge, why the spin-statistics theorem holds, and why the origin of gauge interactions settles the Yang-Mills millennium problem. (The latter topic is also explained here.) In short, it is shown that all "laws" of physics are uniquely defined, without any possible variation or alternative. There is only one possible universe.

From 9 lines to 1 line. In 2022, the strand tangle model, with its one line, explains about 8.2 lines of the 9 lines that describe all of physics and of nature. The strand tangle model also agrees with all experiments. The remaining constants of line 9 still have to be deduced. The task is not finished. But: no other theory in the literature has achieved this much. (In fact, two other approaches have similar results. The octonion model by Singh,, and Connes' non-commutative geometry,, both explain more than 8 lines, but both appear to predict additional unobserved particles. A more detailed evaluation is found here.)

Using strands, nature is summarized in just 1 line: General relativity and the standard model are due to fluctuating strands of Planck radius, for which each crossing switch yields a quantum of action.

Qubits. The strand tangle model also shows how to give concrete meaning to Zizzi’s expression “it from qubit”: qubits can be modelled with strands. So can entanglement and decoherence. And gravity.

Lepton tangles. Use your mouse to play with the 3d visualizations of the three simplest tangles (derived in the various papers) for the electron neutrino, the muon neutrino, and the tau neutrino:

and the simplest tangles for the electron, the muon and the tau:

All these beautiful 3d visualizations were realized with Blender by Aleksandr, by Lucas and by Mitchell.


Step 5a: The strand tangle model for the standard model of particle physics

Beautiful. When we look at the starry sky, we admire the vast space, the coloured twinkling stars, and the deep blackness. The strand conjecture proposes an explanation for their origin, their properties and their motion. The foundations of what we find around us – particles, space, horizons and colours of everything we see – are explained.

C. Schiller, A conjecture on deducing general relativity and the standard model with its fundamental constants from rational tangles of strands, Physics of Particles and Nuclei 50 (2019) 259–299. Download the published paper at Read the published paper online for free at Download the preprint here, with films.   pdf

Testable. The paper argues that modern physics arises, directly and inevitably, from the Planck scale. Below, the more pedagogical papers and preprints deduce additional experimental predictions and tests. A detailed list of experimental tests is found on the bet page, by clicking here.

Simple. The strand conjecture starts with deducing Dirac’s equation from Dirac’s trick for tangles. Then, tangle classification yields the particle spectrum. Tangle deformations yield, via the Reidemeister moves, the particle gauge interactions groups U(1), SU(3) and broken SU(2). Working out the details gives usual particle physics, with no additions, modifications, or omissions. More details are found on this page.

A compact introduction for physicists and physics students is pdf  C. Schiller, On the relation between the three Reidemeister moves and the three gauge groups, preprint on Researchgate at

C. Schiller, Testing a conjecture on the origin of the standard model, European Physical Journal Plus 136 (2021) 79. Download it at Read the published paper online for free at Download the preprint here.   pdf

Elegant. It is regularly claimed that the standard model is complex, incomplete or even ugly. The strand conjecture argues the exact opposite: all of particle physics is due to tangled strands fluctuating at the Planck scale. A single fundamental process appears to explain the principle of least action, the Dirac equation, the observed interaction spectrum, the observed gauge symmetry groups, the observed elementary particle spectrum, and the fundamental constants (masses, mixing angles, and coupling constants) describing them. The Lagrangian of the standard model arises, without modifications. Over 100 additional tests and predictions about particle physics beyond the standard model are deduced. They agree with all experiments. So far, no other approach in the research literature appears to make (almost) any of these predictions. Indeed, it appears that the explanation of the standard model using tangled strands is consistent, complete, correct, hard to vary, and unique. Above all, it is beautifully simple.

A list of experimental predictions is also found here. Parity violation is illustrated in the videos on the topic.


Step 5b: Quantum electrodynamics deduced from strands

C. Schiller, Testing a conjecture on quantum electrodynamics, Journal of Geometry and Physics 178 (2022) 104551. Download the preprint here.   pdf

Colours and beauty. The strand conjecture shows how the tangle model leads to quantum electrodynamics, including electricity, magnetism and optics. Over 40 tests for the conjecture are given. So far, they are all positive. In particular, the strand conjecture appears to allow approaching two old challenges: how to calculate the fine structure constant and how to calculate the lepton masses – both from first principles. The preprint uses the tangle model of particles to deduce estimates. The fine structure constant with its measured value 1/137.036(1) and the lepton masses, in particular the electron mass, are the ingredients that determine all colours, tastes, smells, sounds and most shapes around us. In other words: it is argued that tangles of strands generate all beauty in nature.

Presently, tangles lead to a crude estimate of the fine structure constant that is correct within 30%. This is not good; but so far, it is one of just two attempts worldwide to explain the value ab initio, using a unified description of particle physics and general relativity.

The spin of leptons. Leptons consist of three strands. The animation by Jason Hise gives an impression about how they spin:
The central cube contains the tangle core of the specific lepton.  

The spinning electron tangle. Fabrice Neyret, inspired by Jason Hise, produced two animations showing two options for the spinning electron. Use your mouse to change point of view:

The radius of the strands is the Planck length. The green bar is only added for better visualization; it shows the orientation of the electron. The tangle tethers reproduce spin 1/2 and fermion behaviour under particle exchange. The wave function arises from the blurring of the tangle crossings. The tangle details determine electric charge (every chiral crossing produces an electric charge e/3), parities (behaviour under mirror reflection and rotation reversal), lepton number (results from the 3 strands), mass (not visible directly, via the average rotation speed), electromagnetic coupling and the fine structure constant (through the statistics of tangle shapes), and the behaviour in particle reactions (due to the topology of the rational tangle). Positrons are mirror tangles rotating in the opposite sense. More details are found in the published paper on quantum electrodynamics linked a few paragraphs higher up, and also in the pdf found at step 4. No other model of the electron achieves all these explanations.

Here is the photon, showing its rotating phase:

Here is a topologically equivalent version, also showing the photon and its rotatig phase, animated by Mitchell Wieringa and seen from two different directions:



Step 5c: Quantum chromodynamics deduced from strands

pdf   C. Schiller, Testing a conjecture on quantum chromodynamics, Int. J. of Geometric Methods in Modern Physics 20 (2023) 2350095.

Quarks and nuclei. The strand conjecture shows how the tangle model leads to the strong interaction, the quark model, gluon flux tubes, confinement and asymptotic freedom. The existence (new in 2022) of glueballs is predicted. Many other tests for the tangle model are deduced, including the lack of new generations, the lack of CP violation and the lack of deviations from QCD. All consequences agree with the data. In particular, the strand conjecture allows estimating the strong coupling constant and the quark masses ab initio.

The spinning motion of the simplest tangle of the down quark. Jason Hise also produced the animation for this case:
Down quark tangle

Indeed, the tangle model is peculiar – to say the least.


Step 6: Cosmology deduced from strands

pdf   C. Schiller, Testing a conjecture on cosmology and dark energy (preprint).

The universe. This and a subsequent preprint on cosmology complete the topic of gravitation. In the strand conjecture, the universe consists of a single closed strand that forms the cosmological horizon and also the particles and the space inside it. Over time, this strand gets more and more tangled. (As one reader said: the universe plays cat's cradle.) This description reproduces usual cosmology and leads to numerous tests and predictions: the universe expands; nothing – no matter, no radiation and no space – exists beyond the cosmological horizon; inflation did not occur; there are no cosmic strings and no higher dimensions; there is no non-trivial topology; there is no bouncing universe; there is just one universe; the luminosity of the universe is always limited by c5/4G; dark matter is not made of unknown elementary particles; if dark matter exists at all, it is made of known matter or black holes or both; dark energy, or vacuum energy, does exist and is a natural consequence of strands; the density of vacuum energy, the cosmological constant, is small; baryogenesis appears to be due to non-perturbative effects.

The strand description of cosmology is promising. However, calculating the vacuum energy density remains a challenge. Therefore, clarifying the relation of strands to modified gravity and to the baryonic Tully-Fisher relation remains a challenge as well.



Particle size. As the animation at the very top of this page shows, there is no way to define the size of an elementary particle. The wave function describes its average extension. The electric charge describes its interactions; but the charge is a consequence of Planck-sized crossings. Finally, every electron has tethers reaching the cosmological horizon. In short, an electron is at the same time wave-function sized, of Planck size, and of the size of the night sky.

The same is valid for all other elementary particles. Their size is always described by their wavelength, by the Planck length, and by the size of the universe. Everything in nature has three sizes. But not more. And not less.


Blog. The blog on research about fundamental physics and strand tangles tells more about general ideas, past mistakes, objections, encountered difficulties, and progress.


T-shirt. An important motivation for strand research has always been the support for the ailing physics T-shirt industry. For decades, it has been desperate for new designs. Now they are available.


History. The strand conjecture is a side result of the free Motion Mountain physics book series, in particular of Dirac's spin 1/2 demonstration, of the principle of maximum force, of the strand explanation of back hole entropy, and of the meditation time offered by the Munich subway. Strands reduce the 9 lines describing textbook physics to a single principle (that fits on a T-shirt), and make clear predictions for experiments and calculations. If you want to bet about the outcomes, to evaluate your chances, to comment, or if you want to help with animations similar to these, write to


Technicalities. The strand conjecture reproduces the Lagrangians of the standard model and general relativity, explains the number of generations and the particle spectrum, deduces all Feynman diagrams and propagators, explains the gauge groups U(1), SU(2) and SU(3), explains the fundamental constants ab initio, solves the hierarchy problem, explains neutrino masses without a see-saw mechanism, solves the strong CP problem, predicts the validity of the standard model and of general relativity up to the Planck scale without any intermediate energy scale, implies that the weak interaction violates parity maximally, explains the equality of proton and positron charge, has no problems with anomalies, predicts no issues with baryogenesis, has no grand unification, has no supersymmetry, has no additional spatial dimensions, has no inflation, no inflaton and no dilaton, solves black hole and singularity issues, implies gravitational waves, has no dark matter particles, has a naturally small cosmological constant, solves various problems with gauge theories, answers Hilbert's sixth problem, and explains the principle of least action.



Several of these articles were supported by grants from the Klaus Tschira Foundation: Eur Phys J Plus, Indian J Phys, J Geom Phys, IJGMMP, Z f Naturf, dark energy preprint, emergent quantum theory preprint, first Bronshtein cube preprint.


Volume VI of Motion Mountain

Older work. A more extensive, more passionate, but also older and less precise presentation is the original text on the strand model. It was written as a research volume that continues the adventure of the five textbook volumes.

pdfFree book:   The Strand Model – A Speculation On Unification
The quest for a complete, unified theory leads to a proposal with testable predictions, to ab-initio estimates of the W/Z and Higgs/Z boson mass ratios, and to a new ab-initio approximation for the fine structure constant and the other fundamental constants. 500 pages, 40 MB.


Strands in other languages

A `strand' is best translated in Dutch as draad, in French as fil, in German as Faden, in Italian as filo and in Spanish as hilo. The mathematical concept of `tangle' is best translated in Dutch as wirwar, in French as enchevêtrement, in German as Gewirr, in Italian as groviglio and in Spanish as enredo. A `tether' is best translated in Dutch as lijn, in French as lien, in German as Leine, in Italian as nesso and in Spanish, for example, as vínculo.