A Single Difference.

Two copies of the same cellular automaton, started identically. At a chosen generation, exactly one cell is flipped in the second copy. Three panels show what happens next, on the same scale and the same timeline:

01 / Baseline

The original world, never touched.

02 / Perturbed

The same world with one cell flipped at the dashed line.

03 / Difference

The exact XOR of the two: orange where they disagree. Orange marks a difference, not a third state.

Below them, the measurements:

Different cells / Fraction different

How many cells disagree at the current generation.

Fronts from the flip

How far left and right of the flipped cell the outermost differences have reached.

Difference over time

The fraction of cells that differ, generation by generation: this run, and the mean of 128 runs with other seeds and flip positions.

What changed at this moment?

A plain-language reading of the current generation, with front speeds.

Vanish / Stay local / Spread / Mixed / Additive

Presets showing the five typical outcomes: Rule 232, 108, 30, 110 and 90.

Replay / Play / Step / Jump to flip

Move along the shared timeline. Rewinding replays the same computed experiment; nothing is rerun at random.

Speed

Steps per second during playback.

Inspect cell position

Read the baseline, perturbed and XOR values of one cell at the current generation.

Set up your own comparison

Choose the rule, boundary, width, length, starting state (seeded random or one central cell), the seed, and when and where to flip. Press Apply comparison.

Try this: flip the same cell under Rule 232 and Rule 30. In the first the difference dies out or freezes into a cell or two; in the second it fills the row. Then try Rule 90, where the difference grows into a perfect triangle whatever the background.

Damage spreading

Flipping one cell and tracking the disagreement is called a damage-spreading experiment. It is the cellular automaton version of the “butterfly effect” question: does a tiny change die out, stay put or take over? For continuous systems the answer is measured by a Lyapunov exponent, the rate at which nearby states separate. For cellular automata, the growth rate of the difference pattern plays the same role.

The speed limit

A cell only sees its immediate neighbors, so a change can travel at most one cell per generation in each direction: a light cone. Front speeds near 1 mean the rule passes changes on as fast as it possibly can; speeds near 0 mean the change stays local.

Additive rules

Rule 90 sets each cell to the XOR of its two neighbors. Rules like this are additive: the evolution of a difference doesn’t depend on the background at all, so the difference panel is exactly a single cell evolved under Rule 90. The page checks this on every Rule 90 run.

The five presets show the main outcomes, and they line up with the classes described in the Rule Observatory. In simple rules like Rule 232 and Rule 108 the difference vanishes or stays within a cell or two. In chaotic-looking rules like Rule 30 it spreads in every run and the fraction different settles near a half, as if the two worlds had become unrelated. In complex rules like Rule 110 it usually spreads too, but unevenly, carried by the moving structures, and settles lower (around a third of the row in a quick check of 64 runs).

The mean over 128 runs is the measurement; a single run is an anecdote. For example, the page reports what fraction of runs still carry a difference at the end and the mean fraction of cells that differ. Two rules that look similar can differ sharply on these numbers.

• The runs are finite and deterministic. Spreading damage is consistent with chaos but doesn’t prove it, or prove universality or irreducibility.
• The intervention is always a single flipped bit at one moment; other kinds of perturbation can behave differently.
• On a small ring with periodic boundaries, a spreading difference meets itself coming round the other side, so late generations reflect the ring’s size.

Background: damage spreading and Lyapunov exponents in cellular automata.