Rule garden / Experiment guide

Does the background matter?

Every comparison starts with exactly the same disagreement: the center cell is flipped. Does that guarantee the same difference picture? Keep the rule fixed and change only the scattered seed to find out.

Predict, then compare

1. Open rule 30 with seed 7 at generation 1 and seed 12 at generation 1. Both inspectors show three differing cells at columns 399, 400, and 401. Predict whether the next rows will also match.
2. Set Inspect generation to 2 in each garden. Seed 7 has 3 differences, at columns 400, 401, and 402. Seed 12 has 5 differences, at columns 398, 399, 400, 401, and 402. The first match did not persist.
3. Now compare rule 90, seed 7 with rule 90, seed 12. Both have two differences at columns 398 and 402. Try other generations and seeds, pressing Grow after changing a seed.
4. Switch to Original garden and press Grow. The inspector still counts disagreements between the pair; it does not count occupied cells in the selected view. Check that different originals can produce identical difference pictures.

Why rule 30 can notice its background

Write the left, center, and right cells as L, C, and R. Rule 30 outputs L XOR (C OR R). OR outputs 1 when either input is 1; XOR outputs 1 when its two inputs differ.

Consider flipping C while keeping its neighbors fixed. If R is 0, changing C changes the output. If R is 1, C OR R stays 1 and the change is hidden. The surrounding state can therefore decide whether a disagreement passes through a particular update. In our two seeded gardens, their disagreement rows first differ at generation 2.

Why rule 90 cannot

Rule 90 outputs L XOR R. Taking the XOR of two updated gardens cancels their shared background: the new difference is simply the old left difference XOR the old right difference. Thus the same initial difference produces the same entire difference history, for any background. This remains true across the joined edges of the ring.

Two matching seeds would not prove that claim. The XOR identity does; experiments check that the garden implements it. Conversely, one unequal pair is enough to disprove background independence for rule 30. Neither result says that every pair of rule-30 seeds must diverge, or ranks the rules by “chaos.”

Keep a useful record

Record the rule, first-row mode, seed, generation, and differing columns. The URL saves these controls and the selected view. Compare one setting at a time, and distinguish a claim about two examples from a claim about every possible starting row.

Next: equal counts can hide different positions, and a smaller count can accompany a wider reach.