WANDER.

Self-avoiding walk · 3D cubic lattice

Wander A walk that never crosses its own path

Let a random walker step through a lattice with one rule — never touch a cell it has already used — and pure chance stops looking like chaos. The strand coils, folds, corners itself, and knits into a random polymer. What emerges is a hidden law: end-to-end distance climbs as N3/5.

Steps laid · N 1214
Tangle span 28.4a
Scaling ν ≈0.60
Folds 7
R vs N — log·log guide: R ∝ N0.6

F refold · space pause

Reading the tangle

One rule turns a walk into a shape

01 — The one ruleAn ordinary random walk is free to double back and sit on cells it has visited. Forbid that — self-avoidance — and each step must land on a fresh neighbour. The walk can no longer forget where it has been; its own history becomes the wall it must steer around.

02 — Why it foldsBecause the strand pushes off itself, it swells outward rather than curling into a tight ball. This is exactly how a real polymer chain behaves in a good solvent — a molecule too long to sit still, propped open by the volume it already occupies.

03 — TrappedSometimes every neighbour is already taken and the tip is boxed in. There is no legal next step, so the fold ends and a new strand seeds itself in open space. Each trapped ending is counted below as a completed fold.

04 — The hidden lawMeasure the straight-line distance from start to tip and it does not grow like the number of steps. It rises as Nν with ν ≈ 3/5 — the Flory exponent. Chance, constrained, obeys an exponent it never agreed to.

Free random walk
R ~ N½
A walk allowed to cross itself spreads only as the square root of its length — a diffuse, ghostly cloud.
Self-avoiding walk
R ~ N³ᐟ⁵
Forbidding crossings props the chain open. It reaches farther for the same length — the Flory scaling of real polymers.
What you are watching
≈0.6
The live exponent fitted from this strand's own R-versus-N cloud, condensing out of thousands of blind steps.