Artificial Wasteland · Instruments · hearable mathematics

Percolation, as Texture

Open each square of a grid at random, with probability p. Below a sharp threshold the open squares are scattered islands; above it, one cluster suddenly spans the whole grid. Drag p across that knife-edge and you hear it — the sound thickens, and a deep tone snaps in the instant the giant appears.

Each square is open with probability p (drag below). Open squares that touch join into clusters; the spanning cluster, when one exists, reaches from the top edge to the bottom. Drag p up from zero: nothing, nothing — then, at the threshold, a single cluster floods across the grid. The marker on the dial sits at that lattice's pc. Press Sweep to ride through it.

p = 0.30 · P∞ = 0.00
no spanning cluster
Lattice
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The order parameter, heard. The bass tone is P∞, the fraction of the grid in the largest cluster. Below pc there is no giant and the low partial is silent; above it, the giant dominates and the bass swells. The moment it enters is the phase transition.

The curve is P∞ — the size of the largest cluster, as a fraction of the whole — plotted against p for the exact grid you are playing. It clings to zero, then turns sharply upward at pc (the dashed line): the signature of a continuous phase transition. The dot is where you are standing.

P∞(p)

Honest about the edge. On an infinite lattice the turn at pc is a true kink; on this finite grid it is steep but rounded (finite-size). And the curve is one random realisation — press New grid for another. The threshold itself is exactly ½ for bond-□ and site-△ (provable), but for site-□ it is only the measured 0.59274605… — no closed form is known.

This is what you are hearing. Every cluster is sorted by size into a bin, and each bin drives one tone — big clusters low, small clusters high — at a loudness equal to its share of the grid. So the sound's spectrum is literally the cluster-size distribution: a thin high hiss of islands below pc, a broad critical roar at the edge, and a single deep tone once the giant takes over.

cluster sizes → tones

What you are playing

Percolation is the simplest model of connection. Take a grid; open each square independently with probability p; ask whether the open squares link up into a path that crosses the whole thing. Bake a porous stone, light a forest, gossip through a crowd — it is the same question, and it has the same startling answer: connection does not arrive gradually. There is a precise critical probability pc below which every cluster is small and above which an infinite one exists. Add one more open square at the edge and an entire continent of connection blinks into being.

You are dragging p through that edge. Below it the grid is gravel — scattered islands, each a small high tone. As you climb toward pc the clusters of every size appear at once (the distribution goes scale-free), and the sound fills out into a broad critical roar. Cross over, and the islands fuse: one giant cluster swallows most of the grid, and its size — the order parameter P∞ — drives a deep tone that was silent a moment before.

Connection is not a dimmer. It is a switch, thrown at pc, and you can hear the throw.

Three lattices share the dial. For bond percolation on the square grid the threshold is provably pc = ½ exactly (Harris 1960; Kesten 1980), and for site percolation on the triangular grid it is ½ too, by a self-matching symmetry. But for site percolation on the square grid — the default here, sitting right between them — no one has ever found a formula. The best anyone has is the measured 0.59274605079210… (Jacobsen 2015), and it may have no closed form at all. The same machine plays all three; mathematics can prove the threshold of only two.

Why the sound is the object

Nothing here is decoration. The grid is a single random realisation — every cell carries a fixed random number, and "open at p" means that number is below p, so dragging the dial floods the same landscape coherently. The clusters are found live by union-find, the standard algorithm; P∞, whether a cluster spans, and the full list of cluster sizes are read straight off that. The spectrum you hear is those sizes, binned and sounded — when the bass enters, it is because a giant cluster has genuinely formed in the grid in front of you.

What is fact, and what is a choice

Fact: which cells are open, how they cluster, the size of every cluster, whether one spans top-to-bottom, the order parameter P∞, and the sharp rise at pc — all recomputed live, and checked offline against the cited thresholds in research/percolation-played (the spanning probability crosses ½ at each lattice's pc; P∞ jumps across the transition; the binning conserves mass).

Choice: mapping cluster size to pitch (big → low) on a pentatonic ladder, the root pitch, and the drone timbre. These set what the transition sounds like; they cannot move pc or change whether a cluster spans. The pc marker for site-□ is drawn at the measured value and labelled as a measurement, never claimed exact.

Sources

An exhibit in the Wasteland's instrument room — hearable mathematics, where every sound is the object. It is the playable companion to the stratum When the Stone Lets Water Through, which proves two of these three thresholds exactly and leaves the third — and Cardy's formula — loudly open.