Skip to content
Foundation-0
fractals
Public

About

on fractals

foundation-0.github.io/fractals/

Topics

foundation-0fractalsmathematics

Resources

Readme
Apache-2.0 license
Activity
Custom properties

Stars

0 stars

Watchers

0 watching

Forks

1 fork
Report repository
BranchesTags

Repository files navigation

fractals

Does reality repeat itself across scales? Atoms look a little like solar systems, solar systems a little like galaxies, galaxies cluster into clusters. This repository follows that intuition with first principles and checks it against what is measured: where the repetition is real, where it breaks, and why.

Status: exploration, not peer reviewed, no new result claimed. Started 16 October 2026 by Aska Digital with Claude (Anthropic), for Foundation0.

Where the intuition holds, and where it breaks

  • Atoms are not little solar systems. In a stationary atomic state, an electron has a spread of possible positions rather than a planet's orbit. Repeated measurements on identically prepared atoms reveal that spread, even with better instruments. Each reading gives one position. Planets obey the same quantum rules, but records left in their surroundings keep their motion close to classical orbits.
  • Except in one case, where gravity builds a real atom. Very light particles around a spinning black hole settle into a cloud with the same shapes and the same pattern of energy levels as a hydrogen atom, with gravity in place of electric attraction (the "gravitational atom"). It is still a wave cloud, not planets on orbits. The same equation, a wave held by a pull that weakens with distance, gives the same solution at any scale. The nesting lives in the equations, not the objects.
  • Quantum paths are fractal, between two lengths. A free particle's path, recorded by position measurements finer than ħ/p, has fractal dimension 2, against 1 for a classical trajectory (Abbott and Wise, 1981); below ħ/mc, in a single-particle model, the speed of light makes it smooth again. Its surroundings draw that roughness only if they kick it harder than its own momentum: a cold atom in a room-temperature vacuum, yes; nothing as warm as, and heavier than, the gas around it (quantum-paths).
  • Cosmic structure is fractal only up to a point. Galaxies cluster in a fractal-like way on small scales, but beyond about 80 to 300 million parsecs (h⁻¹ Mpc, 80 to 300) they are spread almost evenly (Scrimgeour et al., 2012). The nesting stops, the way a coastline's roughness stops at the size of molecules.
  • Turbulence repeats, but not exactly. Between the size where a flow is stirred and the size where viscosity takes over, the same energy flux passes through every scale. In a cascade model it holds to 0.2% over the first nine of 21 shells (2.4 decades), and a third-order law that follows from it to 0.1%. The bursts do not scale exactly: the distribution's tails fatten from scale to scale (intermittency). Where the cascade ends depends on the medium: about a millimetre in air and water, give or take a factor of 10; the ion scales (about 100 km) in the solar wind, where collisions are too rare for viscosity and a steeper kinetic cascade takes over. In molecular clouds the supersonic range ends at the sonic scale (about 0.1 pc), and a shallower subsonic cascade continues. The fractal edges of the Taurus clouds (dimension 1.36) agree in number with laboratory turbulent surfaces, but the two measure different things, and across clouds the dimension runs from about 1.2 to 1.6 (turbulence).

Some repetition across scales has to be tuned, and some sets its own range. At a critical point, such as water's liquid-vapour critical point or the Curie temperature of a magnet whose spins point only up or down, fluctuations of every size appear, but only while the system is held at that point. Turbulent cascades and gravitational clustering are scale-free over a range the system sets itself (third study).

First study: the Witness Map

Why do electrons act as waves and planets do not? Ask any object one question: can it be in two places at once, a gap Δx apart?

  • Wave clock: how long its wave needs to spread across the gap, m·Δx²/ħ. Heavier things spread slower.
  • Witness clock: how long until something around it (a scattered photon, a gas atom, its own heat glow, a black hole horizon) carries off a record of which place it is in.
  • Witness number W = wave clock ÷ witness clock. Below 1 the wave wins and the object can act quantum. Above 1 the witnesses win and it acts classical.

What destroys the wave is the record, not the push of the detector: in 1998 Dürr, Nonn and Rempe marked which path atoms took with a classical kick far too small to matter, and the interference still vanished. The idea that scattered light carries records of an object's position is also studied by Riedel and Zurek (2010).

At a 250 nm gap, each object in its usual surroundings:

Object W Main witness
Electron (lab vacuum) ~10⁻¹³ leftover gas
Rubidium atom ~10⁻⁷ leftover gas
Buckyball, C₆₀ ~10⁻⁴ its own heat glow
Giant molecule, ~2,000 atoms ~0.03 its own heat glow
Interstellar dust grain ~10⁷ starlight
Virus (air) ~10¹⁶ air molecules
Cat ~10⁴⁸ air molecules
Earth ~10⁸¹ sunlight
Milky Way ~10¹¹⁹ its own starlight
Dark matter particle 0 nothing

What it shows:

  1. The four objects seen acting as waves in experiments all land on the wave side.
  2. Mass enters through the wave clock. Silence the witnesses and big things can cross: a virus in a 4 K vacuum scores W ≈ 0.04.
  3. A lone dark matter particle has no witnesses at any gap or particle mass. That is a statement about one particle. The galaxy-sized wave of wave dark matter is a crowd, and its lumps are watched by every star they pass (see The crowd wave below).
  4. A black hole horizon is a real witness (Danielson, Satishchandran and Wald, 2022) but a feeble one: ten horizon radii from the Milky Way's black hole, a wave dark matter particle split by 1 µm waits about 10¹⁷⁰ years to be recorded in the formula. This is a formal extrapolation below the particle's reduced Compton wavelength, ħ/(mc), about 0.06 pc for 10⁻²² eV. It is not a validated prediction for a particle confined to such a small region.

Interactive version: https://foundation-0.github.io/fractals/ (index.html at the root of this repository, served by GitHub Pages; it loads witness-map/model.js directly).

The crowd wave: where the map stops

The map asks about one particle at a time. Wave dark matter in a galaxy is never one particle. If its particle weighs 10⁻²² eV, each lump of the wave near the Sun is shared by about 10⁹³ particles. So many particles in one wave act together as an ordinary (classical) wave, the way countless photons make one radio wave (Hui, Ostriker, Tremaine and Witten 2017). The crowd keeps its own clocks.

The lumps. The wave interferes with itself into lumps (granules), seen in simulated halos (Schive, Chiueh and Broadhurst 2014; Schive et al. 2014). For particle mass m and speed spread σ, the density is correlated over ƛ = ħ/(mσ) (Bar-Or, Fouvry and Tremaine 2019, eq. 62), and a lump is about half a de Broglie wavelength, πħ/(mσ), across (Hui et al. 2017). Each lump pulls like one body of mass π^(3/2)·ρ·ƛ³, where ρ is the local density (Bar-Or et al. 2019, eq. 59).

Three clocks.

  1. Lump lifetime, ħ/(mσ²). The lumps fade and re-form as parts of the wave drift past each other. The density at a point forgets itself on this time (Bar-Or et al. 2019, eq. 62), and simulations measure the same (Veltmaat, Niemeyer and Schwabe 2018). It is exactly the map's wave clock at a gap of ƛ: m·ƛ²/ħ = ħ/(mσ²).
  2. The core's beat. At a halo's centre the wave settles into one standing wave, the core or soliton (Schive, Chiueh and Broadhurst 2014). Its wave turns over every 2πħ/|E|, where E is its energy level: about 120 million years for a core of 0.1 Suns per cubic parsec (Schive, Chiueh and Broadhurst 2020). It also breathes: the central density swings by more than a factor of two at f = 10.94 (ρ_c / 10⁹ Suns per kpc³)^(1/2) per billion years, and the swing hardly fades over the halo's life (Veltmaat et al. 2018, eq. 16). Stars inside a core feel it: the beat would heat the old star cluster at the centre of Eridanus II (Marsh and Niemeyer 2019), and the core's wandering can tear that cluster apart within about a billion years (Schive, Chiueh and Broadhurst 2020).
  3. Stirring time. Passing lumps tug on stars and on each other. The stirring (relaxation) time is how long the tugs take to change a star's speed by as much as the halo's own spread of speeds: 3m³σ⁶ / (16π²G²ρ²ħ³ ln Λ), with ln Λ ≈ ln(2r/ƛ) at distance r from the centre (Bar-Or et al. 2019, eqs. 61 and 85). Hui et al. (2017, eq. 32) give the same form; Bar-Or et al. find the true rates 2 to 11 times slower. The dark matter's own relaxation, which grows the core, runs on a time of the same form (Levkov, Panin and Tkachev 2018).

node witness-map/table.mjs prints these for three places:

Milky Way, Sun's orbit, 10⁻²² eV Same place, 3×10⁻¹⁹ eV Core of a dwarf like Eridanus II, 8×10⁻²³ eV
Size lumps 370 pc across lumps 0.12 pc across core 0.7 kpc in radius
Mass 10⁵ Suns per lump 1.2 Earths per lump 5×10⁸ Suns
Particles sharing it 10⁹³ 10⁷⁹ 10⁹⁷
Fastest clock lumps live 0.7 million years lumps live 240 years wave turns over every 110 million years, breathes every 260 million
Stirring time 5×10¹² years, 340 times the age of the universe 5×10²² years not modelled
Kick from one lump on a star 44 m/s 5 µm/s not modelled

Inputs. The Sun is 8.18 kpc from the centre (GRAVITY Collaboration 2019). The circular speed there is 229 km/s (Eilers et al. 2019), and σ = 229/√2 = 162 km/s, as in an isothermal halo (Bar-Or et al. 2019, eq. 63). The local dark matter density is 0.4 GeV/cm³ (de Salas and Widmark 2021 review: 0.3 to 0.6). The age of the universe is 13.8 billion years (Planck 2018). The dwarf is the simulated Eridanus II-like halo of Schive, Chiueh and Broadhurst (2020), with core radius 0.7 kpc. The 3×10⁻¹⁹ eV particle is the lowest mass allowed (at 99% confidence) by the stars of two ultrafaint dwarf galaxies, which the lumps would stir (Dalal and Kravtsov 2022); the Lyman-α forest requires more than 2×10⁻²⁰ eV (Rogers and Peiris 2021). Both rule out the 10⁻²² eV benchmark used by the map and by most of the papers above, if wave dark matter is all of the dark matter. Heavier particles make smaller, lighter, shorter-lived lumps: lump mass falls as m⁻³ and stirring time grows as m³.

One particle against the crowd.

  • One particle has no witnesses. The crowd has many. Nothing in the map records where one particle is (W = 0). A lump is about 10⁹³ particles pulling together, so its pull is 10⁹³ times one particle's. As a lump passes a star near the Sun, it changes the star's speed by about 2G·m_lump/(ƛσ) ≈ 44 m/s (impulse estimate), and the star meets a fresh lump about every 0.7 million years. A change of tens of metres per second in a star's motion is enormous by quantum standards, so the star holds a record of where the lump was. Summed over near and far lumps, these kicks give Bar-Or et al.'s stirring time to within a factor of 3 (checked in npm test). Allali and Hertzberg (2020, 2021) compute gravitational records of this kind, with the galaxy's gas as the recorder, for dark matter put in two places at once. In the Milky Way, a superposition of where the dark matter is does not survive, while a superposition of the wave's phase does, since the phase shows up in gravity only as faint, fast ripples.
  • The crowd's wave survives being watched. It is a classical wave. Recording the shape of a classical wave does not erase it, any more than measuring a sea wave flattens the sea. Witnesses erase superpositions of different shapes, and the lumps are interference inside one shape. So the records keep the halo in one definite pattern, and that pattern keeps its lumps (Allali and Hertzberg 2020: dark matter condensates decohere fast and are well described by classical field theory). Bar-Or et al. (2019) make the same point from the other side: their calculation is classical field theory, with quantum mechanics entering only through the way the wave spreads.
  • What W = 0 does say. Nothing knocks single particles out of the crowd. A particle's whole momentum in the wave is about mσ = ħ/ƛ, tiny. A hit from any photon or gas atom, whose wavelength is far shorter than ƛ, would carry much more (ħk ≫ ħ/ƛ) and throw the particle out of the wave. Where W is not zero, the per-particle witness rate is the rate at which particles leave the crowd.

Where the map stops applying.

  1. When particles share their wave. If one lump holds more than one particle, a lone particle's W does not describe what is seen. Wave dark matter always shares: 10⁹³ particles per lump at 10⁻²² eV, 10⁷⁹ at 3×10⁻¹⁹ eV. Near the Sun the crowd thins to one particle per lump only for particles heavier than about 20 eV (model; Bar-Or et al. 2019: "a few tens of eV"). The map's heavy dark matter particle, at 100 GeV, has 10⁻³⁹ particles per lump: for it, the per-particle picture is the right one.
  2. For the galaxy-sized wave. The pattern renews every lump lifetime whatever witnesses do, and its lumps are watched by every star. W = 0 is silent on both.
  3. For gravity's witnesses. The map counts a black hole horizon as gravity's only witness, and only for one particle. For the crowd, every star is a witness too. Gravity-only witnesses of a single particle are issue #3.

Does the crowd belong on the map? Yes, but not on the W axis. W asks whether one object can be in two places a gap apart. The crowd's wave is classical, so that is not its live question, and a crowd W would put two different things on one scale. The interactive page has a separate panel below the map, marked "not a witness number", with the crowd's clocks set against the age of the universe. In code: crowd() and CROWDS in witness-map/model.js, a separate table in node witness-map/table.mjs, and checks in npm test.

What the crowd model leaves out. Each place is treated as a uniform patch of halo: no orbits, no tides. The stirring time needs the orbit to be much larger than a lump (Bar-Or et al. 2019), so it fails near the centre. The core is taken at the size found in one simulation. The breathing period is Veltmaat et al.'s fit; a rough first-principles estimate (the gap between the core's two lowest energy levels in its own frozen gravity) gives a frequency 36% higher.

Does a black hole shape wave dark matter?

A spinning black hole can grow a cloud of very light particles with hydrogen-like levels (superradiance). For wave dark matter at 10⁻²² eV around a galaxy's central black hole of 10⁸ to 10⁹ Suns, it cannot in practice: the cloud would take 10¹⁶ to 10²⁴ times the age of the universe to grow, and out where it would sit the galaxy, not the black hole, holds the particles. The black hole can still squeeze the galaxy's dense dark matter core by plain gravity, but only within about 100 pc of the centre, too far in for the lensed images to feel. The cloud is the gravitational atom of the second study; the note's gravAtom is a compact copy of its algebra, and npm test fails if the two disagree. Details, the lens systems with anomaly and black hole data, and a reshaped test: notes/bh-lensing.md (issue #1).

Run it

Needs Node 18 or later, nothing else. The checks run on every pull request, on Node 18, 20 and 22.

npm test                                 # checks the model against published numbers
node witness-map/table.mjs               # every object at a 250 nm gap, then the crowd
node witness-map/table.mjs 1e-6 cryo     # 1 µm gap, everything in a 4 K vacuum
node notes/bh-lensing.mjs                # every number in the black hole note
node critical/table.mjs                  # the critical-point study's numbers (about 65 s)

witness-map/model.js is the whole model, including gravAtom for the black hole note. It loads in Node (require) and in a browser (window.WM). npm test also checks gravitational-atom/atom.js (second study, below). The third study's checks, tests/critical.mjs, add about 65 s of seeded Monte Carlo runs, most of it percolation with nine seeds. CI runs every line above except the critical/table.mjs line: it repeats the tests' seeded runs and prints the results without checking them.

Checks

npm test compares the model with published numbers. A line marked DIFF is a disagreement recorded in CORRECTIONS.md; it still passes, but fails if the size of the disagreement changes.

Check Published Model
Closed-form localisation rates: small grain and large grain in light, grain in air, small glowing sphere formulas (Schlosshauer 2007; Romero-Isart 2011) within 2%
Joos and Zeh's table: three grain sizes in five surroundings powers of ten 11 of 15 within ×5
From it, a 10 µm grain: Big Bang afterglow / sunlight / air (m⁻² s⁻¹) 10¹⁰ / 10²⁵ / 10⁴⁰ 1.6×10¹⁰ / 4.4×10²⁵ / 8.6×10⁴⁰ (air: DIFF)
Schlosshauer's decoherence times: two sizes in four surroundings powers of ten 7 of 8 within ×5
Gravitational channel: Earth split by its radius beside a Sun-mass black hole (Danielson et al., v2, eq. 20) ~10 µs 22 µs
Electric channel: electron split by 1 m, same black hole (Danielson et al., v2, eq. 16) ~10⁴³ years 4×10⁴² years
Heated C₇₀ loses half its fringe visibility, 190 m/s beam (Hornberger et al. 2005, Fig. 9) about 2,500 K 2,475 K (the point the glow is fitted to)
Same, 100 m/s beam about 2,450 K 1,960 K (DIFF)
Photons given off while the visibility halves (Hackermüller et al. 2004) one to two 1.4
C₇₀ glow at 2,500 K: photons above 1.6 eV in a 4 ms transit about 3 0.09 (DIFF)
C₇₀ glow power at 3,000 K 1.1×10⁴ eV/s 770 eV/s (DIFF)
Crowd: lump mass, isothermal halo at 200 km/s, 1 kpc out (Bar-Or et al. 2019, eq. 64) 1.03×10⁷ Suns 1.03×10⁷
Crowd: stirring time × ln Λ, same place (eq. 104) 2.08 billion years 2.08
Crowd: stirring time over Hui et al.'s relaxation time (2017, eq. 32) 2 to 11 (Bar-Or et al.) 8.1
Core peak density, 10⁻²³ eV, 1 kpc core (Schive et al. 2014, eq. 4) 1.9 Suns per pc³ 1.94
Core mass inside the core radius (same paper, eq. 3) 5.5×10⁹ Suns 5.4×10⁹
Core wave period at 0.1 Suns per pc³ (Schive et al. 2020) ~120 million years 121 million
Core breathing frequency (Veltmaat et al. 2018, eq. 16) 0.16 per (Gρ_c)^(-1/2) 0.22 (frozen-gravity estimate)

The horizon rows use equation numbers from arXiv:2205.06279v2. Its electric formula (eq. 15) includes the SI vacuum permittivity ε₀ explicitly, as the model does.

The published tables give powers of ten only, so agreement within a few times is as close as the comparison can go. The tolerance is ×5: half a power of ten for rounding, and a little more for inputs the tables do not state. The crowd rows compare formulas and fits, so they agree more closely; the core is solved here from first principles (the ground state of the Schrödinger–Poisson equations), not taken from the fits. Through gravity alone, the electron of the electric row would wait 5×10¹²⁴ years. The black hole note's checks are listed in notes/bh-lensing.md, the gravitational atom's below, and the critical point's in critical/README.md.

What the re-check found (issue #5, details in CORRECTIONS.md): two coefficient errors, now fixed; a molecule's glow that is fitted, and gets the onset of decoherence right but the light wrong; and four places where the published tables are inconsistent, with each other, with their own labels, or with where their formulas apply. Schlosshauer's 2007 book and Joos and Zeh's 1985 paper were not available; the check uses open copies of their tables.

What the model leaves out

  • Objects sit still. Fast motion, internal structure and spin are ignored.
  • The wave clock assumes speeds far below light speed and is unreliable for gaps below ħ/(mc), the reduced Compton wavelength. The particle horizon example also assumes a source confined to a small region, which is not justified at those scales.
  • W is per particle. Wave dark matter's galaxy-wide wave is a crowd effect, enormous numbers of particles sharing one wave. Its clocks are sketched in The crowd wave; it gets no witness number.
  • Gravity's only witness is a black hole horizon. Nudges from passing stars are treated as zero.
  • Molecules glow through one fitted number and no electronic gap. The fit puts the loss of C₇₀ interference at the measured temperature, but real C₇₀ gives off no light below 1.6 eV and its glow rises as T¹¹, against T⁵ here (Hornberger et al. 2005). Below 2,000 K nothing checks the molecules' glow.
  • Light an object absorbs from its surroundings is not counted. For an object as warm as its surroundings it would add as much again as its own glow.
  • Every number is good to a factor of a few.

Second study: the gravitational atom

Hydrogen and a gravitational atom drawn side by side at the same scale: https://foundation-0.github.io/fractals/gravitational-atom/ (gravitational-atom/index.html, which loads gravitational-atom/atom.js, the whole model; window.GA in a browser, require in Node).

A particle of mass μ held by a pull that weakens as 1/r has levels E = μc²(1 − α²/2n²), a Bohr radius a₀ = ħ/(μcα), and the same wave shapes in units of a₀. Hydrogen: α = e²/(4πε₀ħc) ≈ 1/137. A black hole of mass M and a boson of mass μ: α = GMμ/(ħc), the hole's size over the particle's wavelength, so a₀ = r_g/α² (Baumann, Chia and Porto 2019, eqs. 2.5 to 2.9).

In their example, 5 million Suns and a boson of 10⁻¹⁷ eV:

  • α = r_g/λ_c = 7.4×10⁹ m ÷ 2.0×10¹⁰ m = 0.37, against 0.0073 for hydrogen.
  • a₀ = 0.35 AU, 10²¹ times hydrogen's 0.529 Å. The 211 cloud is most likely found 4a₀ = 1.4 AU out.
  • The ground state is bound by 7.0×10⁻¹⁹ eV, 5×10⁻²⁰ times hydrogen's 13.6 eV, but that is 7% of the particle's rest energy against 27 parts per million for hydrogen. So relativity matters more: the first correction to the 211 level is −11% (their eq. 2.10).
  • The horizon at the centre has radius 0.16a₀ at spin 0.99. A proton is 1.6×10⁻⁵a₀. The horizon is not a wall: a state whose pattern turns slower than the horizon grows by taking the hole's spin (superradiance), the rest drain in. Here both 211 and 322 can grow; at leading order 211 grows by a factor e every 45 years, 322 every 31,000 years.

Checks

Check Published Model
Hydrogen binding, fixed proton (CODATA 2022 Rydberg energy) 13.605693 eV 13.605693 eV
Bohr radius (CODATA 2022) 0.529177 Å 0.529177 Å
Hydrogen ionization energy, proton allowed to recoil (NIST ASD) 13.598435 eV 13.598287 eV; the gap is mostly fine structure
α for 3 Suns and 10⁻¹² eV (BCP 2019, eq. 2.6) ≈ 0.02 0.0224
α for 5×10⁶ Suns and 10⁻¹⁷ eV (BCP 2019, sec. 4.3) ≈ 0.37 0.374
Gravitational waves from a real-field cloud (BCP 2019, eq. 4.1 and sec. 4.3) 484 Hz per 10⁻¹² eV; ≈ 5 mHz 483.6 Hz; 4.84 mHz
Largest α at which 211 grows, spin 0.8 (BCP 2019, sec. 2.2, from Dolan 2007) 0.25 0.252
Most likely radius, n²r_g/α² (BCP 2019, eq. 2.9) 4a₀ and 9a₀ 4.000a₀ and 9.000a₀
211 e-folding time, 10 Suns, α = 0.3, spin 0.9 (Arvanitaki, Baryakhtar and Huang 2015, Table I) ~10⁻⁴ years 4×10⁻⁴ years
Fastest 211 growth, spin 0.99, α = 0.42 (Dolan 2007) 1.5×10⁻⁷ c/r_g 9.3×10⁻⁹ c/r_g, 16× too slow
Bosons in a 10-Sun cloud that took 0.1 of the spin (ABH 2015, eq. 8) ~10⁷⁶ 8×10⁷⁶

The two growth-rate rows show where the model stops: growth rates use the leading-order formula (Detweiler 1980, as corrected in BCP 2019, eq. 2.12), which is good at small α and several times too slow near the fastest growth. The page says so next to every growth time.

One bound in the sources is looser than the formula behind it. BCP 2019 give α < 0.5 as the upper limit for 211 growth, "assuming an initial BH spin of ã = 0.99" (eq. 2.14). Their own condition (eq. 2.1) gives about 0.44 at that spin; 0.5 is the limit for a maximally spinning hole (as in Arvanitaki, Baryakhtar and Huang 2015, eq. 6).

What the gravitational-atom model leaves out: everything past leading order in α; the cloud's own gravity and the hole's loss of mass; interactions between the particles; anything else near the hole.

Third study: the critical point

Why does anything repeat across scales at all? At a critical point, such as water at 647 K and 22 MPa or a magnet at its Curie temperature, fluctuations of every size appear, from the size of a molecule up to a correlation length ξ that grows without limit as the temperature approaches T_c. A 2D Ising lattice and site percolation, run with seeded random numbers in critical/model.js, come out at or close to the exact or high-precision answers: the Ising results within 1.5 standard deviations, the percolation threshold within one standard error once open edges are allowed for, and the spanning cluster's dimension 0.8% low (details in critical/README.md, issue #20):

  • T_c from where the Binder cumulants of two lattice sizes cross: 2.2704 ± 0.0017 J/k_B, against Onsager's 2/ln(1 + √2) = 2.2692. Yang's spontaneous magnetisation within 0.21%.
  • The exponents β/ν = 0.1245 ± 0.0026, γ/ν = 1.751 ± 0.004 and η = 0.250 ± 0.005, against 1/8, 7/4 and 1/4.
  • Percolation, averaged over nine seeds: p_c = 0.59292 ± 0.00023, against 0.592746 (Newman and Ziff 2000), 0.00017 high, of which open edges account for 0.00012 (Ziff and Newman 2002). A spanning cluster of fractal dimension 1.880 ± 0.002, 0.8% below 91/48 = 1.896: a finite-size shortfall, as the effective exponent rises with L over these sizes.
  • Where the nesting stops: at the lattice spacing below, and at ξ above. The measured ξ falls from 9.9 lattice spacings 6% above T_c to 1.1 at 76% above, within 3% of the exact value. At T_c the size of the box is the only limit.

Here the scale-free range has to be tuned: the temperature to T_c and the magnetic field to zero. Turbulent cascades, and gravitational clustering grown from a nearly scale-invariant start, are scale-free over a range that nobody tunes: the flow's own numbers, or the time gravity has had to act, set it (sources in critical/README.md; the clustering turns smooth on the largest scales, as the opening section describes). Real fluids have the same mechanism with three-dimensional exponents: ξ grows 4.27-fold for each tenfold approach to T_c (ν = 0.630; Kos, Poland, Simmons-Duffin and Vichi 2016), against tenfold for the 2D lattice. npm test checks the study against the exact or high-precision results and, for percolation at finite sizes, against Ziff and Newman's measurements (tests/critical.mjs).

Mistakes found along the way are in CORRECTIONS.md. Open questions are the issues.

Sources

The crowd wave:

The critical point (the full list is in critical/README.md):

Licence

Apache License 2.0.

About

on fractals

foundation-0.github.io/fractals/

Topics

foundation-0fractalsmathematics

Resources

Readme
Apache-2.0 license
Activity
Custom properties

Stars

0 stars

Watchers

0 watching

Forks

1 fork
Report repository

Releases

Packages

Contributors

Languages