Rule garden / Experiment guide

Fewer differences, a wider reach

Does a falling difference count mean an influence is retreating? Rule 90 gives a precise counterexample: most disagreements can cancel while the outermost disagreements move farther apart.

Predict and inspect

1. Open rule 90 at generation 127. Read the exact count and outermost columns below the chart. Predict what the next row will do.
2. Set Inspect generation to 128. It updates immediately. Compare the count and columns with your prediction. A low count is hard to see on the fixed chart scale, so use the numerical readout.
3. Try generations 255 and 256. Then change the seed to 12 and press Grow. Do the checkpoints depend on the background?

On a narrow screen, scroll the table sideways to see every column. Keyboard: focus the table, then use the arrow keys.

Why the middle disappears

Rule 90 takes the XOR of the left and right neighbors: one occupied neighbor gives 1, but two give 0. The difference between two rule-90 gardens obeys this same rule because their shared background cancels.

At generation 127, the differing cells occupy every second column from 273 to 527. On the next step, each interior candidate receives two contributions and cancels. Only the two outer candidates receive one contribution. The same mechanism applies from generation 255 to 256.

Before edge crossing, the difference count at generation t is 2 raised to the number of 1s in the binary representation of t. For example, 127 is 1111111 in binary (seven 1s, count 128), and 128 is 10000000 (one 1, count 2). The outermost columns remain 400 − t and 400 + t through generation 400.

Thus a falling Hamming distance does not by itself mean that the spatial reach shrank. It also does not tell you whether the two full gardens are becoming empty: switch to Original garden to check that separately.

Keep the boundary in mind

These checkpoints precede the first displayed edge crossing at generation 401. The garden is a ring, so do not extend the outermost-column formula past generation 400. After wrapping, the leftmost and rightmost displayed cells are not a measure of distance traveled around the ring.

Continue with crossing an edge, meeting yourself to test which predictions survive wrapping.

Next, compare a stationary and a traveling difference with identical counts. What extra observation would you record alongside a count history?