STUDY #30 · 2026 · IN OBSERVATION

Abelian Sandpile — Self-Organized Criticality

A model-driven visual study of a pile that finds its own breaking point.

MOVING IMAGE — GRAIN BY GRAIN 1080² lattice · 1.7M grains · feed ∝ frame, R ∝ √N · photoelastic

WHAT IS THIS

Bak, Tang and Wiesenfeld asked in 1987 why so much of nature produces events of every size: earthquakes, solar flares, avalanches, the crackle of 1/f noise. Their answer was a pile of sand. Add grains slowly and the pile steepens until it reaches the angle where the next grain can start a slide of any size — and then it stays there, without anyone tuning it. They called this self-organized criticality: a system that walks to its own critical point.

The lattice version is the entire model. Each cell holds an integer 0–3. Add a grain; if a cell reaches 4 it topples, keeping nothing and giving one grain to each of its four neighbours; grains that leave the edge are lost. Toppling can set off neighbours, and the chain runs until every cell is stable again. That chain is the avalanche.

Two things make this study possible. First, Dhar proved in 1990 that the result of a collapse — both the final heights and the number of topplings at every cell — is completely independent of the order in which cells are toppled. Second, that theorem is exactly the licence a GPU needs: topple every unstable cell at once is one of the legal orders, so the shader can be checked against the reference simulation bitwise, with no tolerance at all.

the whole plate — 1.7 million grains on a single point, stabilized N 1.7M · 1080² lattice · open boundary · photoelastic · zoom 1

Motif abelian sandpile / Bak-Tang-Wiesenfeld / self-organized criticality / avalanche / odometer / sandpile group

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation Criticality is not a setting here. The rule has no parameters at all — a cell holding four grains gives one to each neighbour — and yet the pile walks by itself to the state where a single grain can start a collapse of any size. Because the model is abelian, the order of the collapses cannot change the result: dropping a million grains one at a time and dropping them all at once produce the identical mosaic, which is also why the GPU may topple every unstable cell simultaneously and still match the reference simulation integer for integer.

Reference Per Bak, Chao Tang & Kurt Wiesenfeld, "Self-organized criticality: An explanation of the 1/f noise," Physical Review Letters, vol.59, no.4, 381-384 (1987); Deepak Dhar, "Self-organized critical state of sandpile automaton models," Physical Review Letters, vol.64, no.14, 1613-1616 (1990).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

AFTER THE THRESHOLD, THE EDGE THAT SETS ITSELF

Percolation is tuned to its critical point. The sandpile walks there.

Study #26 — Percolation Study #30 — Abelian Sandpile

The critical point reached by tuning — openness must be set to 0.5927 reached by itself — the pile arrives, and stays

The dial one knob, p, and one question asked of it no dial at all — the rule has not a single parameter

Chance the lattice is a coin toss; the flood reads it nothing is random — integers only, the same picture every time

What happens at the line connectivity switches — somewhere becomes everywhere, once avalanches of every size — the system lives on the line for ever

PARAMETERS EXPLORED

param meaning effect on the image

N grains dropped on the source radius of the plate: R ≈ 0.376·√N (measured, stable to 4 digits from N 10⁴ to 10⁵); each ×4 in N adds one generation to the hierarchy of triangles

lattice 1080² stills and hero · 540² critical state · 192² instrument how fine the mosaic is — the pattern itself is scale-invariant, so this is magnification, not shape

mode one point / rain / group identity the three faces: a deterministic mandala, the critical noise it settles into when grains fall everywhere, and the neutral element of the sandpile group

feed grains added per frame speed of the growth front only — by abelian-ness the finished image is the same for any feed rate

trail sweeps a cell keeps glowing after it topples the live edge of the collapse — the growth front in the hero, the avalanche footprint in the critical state

relief / region odometer shading / local-mean tone how strongly the record of the collapse lights the mosaic, and how visible the sectors are

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

One integer rule, no parameters. A cell that reaches four grains gives one to each neighbour; the theorems below are why the pictures can be checked exactly, with no tolerance at all.

z_x \ge 4: \qquad z_x \mapsto z_x - 4, \qquad z_y \mapsto z_y + 1 \quad (y \sim x)

The entire model. Heights are integers; grains that leave the open boundary are lost, which is why every configuration stabilizes in finitely many topplings.

a_x\, a_y = a_y\, a_x

Dhar's theorem (1990): the stabilization operators commute — the final heights and the per-cell toppling count (the odometer) are independent of the order of collapse. Toppling every unstable cell at once is a legal order, so the GPU engine matches the serial reference bitwise: 0 differing cells of 116,281, over 16,293,237 topplings.

\langle u(y) \rangle = G(x, y), \qquad \langle s \rangle = \textstyle\sum_y G(x, y)

The mean avalanche is the shadow of a random walk: adding a grain at x topples cell y on average G(x,y) times — the Dirichlet Green function, the expected visits of a simple random walk before it exits the lattice. Measured ⟨s⟩ 176.620 vs 176.702 exact (0.05%); the whole spatial profile matches to 0.02%.

\langle h \rangle = \tfrac{25}{8}, \qquad P(1) = \tfrac{2}{\pi^2} - \tfrac{4}{\pi^3}

The critical state is exactly known: mean height 25/8 (measured 3.13172, bulk window at L = 160) and the height-one probability 2/π² − 4/π³ (measured 0.07391 vs 0.07364). The number of recurrent configurations equals det(4I − A) — the number of spanning trees: 100,352 on a 3×3 lattice, matched by full enumeration.

A visual interpretation of the model, not a claim of scientific precision. The checks are integer-exact where the theory is (order independence, recurrent counts, conservation) and statistical where it is not (stationary heights, avalanche exponents on finite lattices).

SELECTED STILLS — 7

the whole plate — the deterministic mandala of a single point source N 1.7M · 1080² · photoelastic · zoom 1

the lace band, where the triangles break into stipple N 1.7M · 1080² · photoelastic · zoom 3.6

the mosaic at cell scale — four heights, four fringes N 1.7M · 1080² · photoelastic · zoom 9

one grain, one avalanche — every site that toppled because of a single added grain critical state · 540² · one grain · brighter = toppled later

the identity element of the sandpile group — the neutral element has a face identity · 513² lattice · rendered 1080² · photoelastic

same pile, read with a different instrument N 1.7M · 1080² · rice (Oslo rice pile) · zoom 1

the flare reading, mid magnification N 1.7M · 1080² · solar (EUV false colour) · zoom 2.2

PROCESS — PARAMETER SWEEPS

The scale ladder — the same single point fed 10², 10³, 10⁴, 10⁵ grains (heights above, log odometer below). The measured radius keeps the ratio R/√N pinned at 0.3162 → 0.3606 → 0.37643 → 0.37609: the pattern is one fixed shape blown up, each ×4 in grains adding one generation to the hierarchy of triangles while total toppling work grows as N². The finished plate is this ladder continued to 1.7 million.

one rule, four scales N 10² → 10⁵ · R/√N → 0.376 · topplings ∝ N^1.99–2.00 · heights / log odometer

SIGNATURE — THE PILE MEASURES ITSELF

No scale is special — every collapse lands on the same straight line.

Rain single grains on the settled pile and record how many cells each one knocks over. Most do nothing; some topple a handful; rarely one takes a quarter of the plate. On log-log axes the sizes fall on a straight line across four decades (measured slope τ ≈ −1.085 at L = 64) — the power law that is the whole claim of self-organized criticality, reached here without tuning a single parameter. The state it hovers in is exactly known: the mean height converges on 25/8 (measured 3.13172) and the chance of the lowest height on 2/π² − 4/π³ (measured to 0.37%).

The strangest identity is the quietest: the average number of times each cell topples after one added grain equals the lattice Green function — the expected visits of a simple random walk before it leaves the grid (matched to 0.05%, its whole spatial profile to 0.02%). A rule with no randomness in it averages out to pure diffusion; the mean avalanche is the shadow of a random walk. And the two ways of building a pile do not build the same pile: grains stacked on one point pack the interior to density 2.25, while the rained-on critical state holds only 17/8 = 2.125 — the mandala is denser than the weather.

the avalanche law, and the exact height law of the critical state L 64 · 60,000 drops · τ −1.085 · ⟨s⟩ 153.2 · ⟨h⟩ 3.1172 vs 25/8 · P(1) 0.067 vs 2/π²−4/π³

COLOUR = PHOTOELASTIC FRINGES

Stress inside a real granular pile is not invisible: press photoelastic discs between crossed polarisers and the force chains light up as bright fringes on a dark ground. The fringes do not simply go from dark to light — they walk an order of colours. The reference palette, photoelastic, takes its four steps (near-black → deep blue → magenta → gold) from that fringe order and assigns them to the four heights a cell can hold: gold is a cell one grain away from collapse.

The variations are not moods; each is a place where the same power law was actually reported. rice — the Oslo rice pile, the laboratory experiment where avalanche statistics of this kind were measured directly. seismic — the Gutenberg-Richter law for earthquake magnitudes. solar — the energy distribution of solar flares in EUV false colour. neural — bursts in cortical tissue under calcium imaging. Same pile, different instrument.

the same mosaic read as the Oslo rice pile — where this avalanche law was measured in a laboratory N 1.7M · 1080² · rice palette · Frette et al. 1996

The colours are artistic approximations of those imaging conventions, not measured values.

REFERENCES

Per Bak, Chao Tang, Kurt Wiesenfeld. "Self-organized criticality: An explanation of the 1/f noise." Physical Review Letters, vol.59, no.4, 381-384 (1987).
Deepak Dhar. "Self-organized critical state of sandpile automaton models." Physical Review Letters, vol.64, no.14, 1613-1616 (1990).
Satya N. Majumdar, Deepak Dhar. "Height correlations in the Abelian sandpile model." Journal of Physics A: Mathematical and General, vol.24, no.7, L357-L362 (1991).
V. B. Priezzhev. "Structure of two-dimensional sandpile. I. Height probabilities." Journal of Statistical Physics, vol.74, no.5-6, 955-979 (1994).
Wesley Pegden, Charles K. Smart. "Convergence of the Abelian sandpile." Duke Mathematical Journal, vol.162, no.4, 627-642 (2013).
Lionel Levine, Wesley Pegden, Charles K. Smart. "Apollonian structure in the Abelian sandpile." Geometric and Functional Analysis, vol.26, no.1, 306-336 (2016).
Vidar Frette, Kim Christensen, Anders Malthe-Sørenssen, Jens Feder, Torstein Jøssang, Paul Meakin. "Avalanche dynamics in a pile of rice." Nature, vol.379, 49-52 (1996).

INTERACTIVE STUDY

Drop grains on a grid. A cell holding four grains topples and gives one to each neighbour; grains that fall off the rim are gone. There is no dial for “critical” anywhere in that rule — yet the pile walks to a state where one more grain can trigger a collapse of any size. The meter below the plate keeps the size of every collapse: no matter how you pour, the histogram is a straight line on log–log. Click the lattice to pour there. It is a deliberately simplified instrument, capped at a 192² lattice with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENT · 192² LATTICEABELIAN SANDPILE · PARALLEL TOPPLING

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.