Rule garden / Experiment guide

Crossing an edge, meeting yourself

The picture has edges. The garden does not: column 800 neighbors column 0. Does crossing that seam immediately change how many cells disagree?

Predict, then cross

1. Open rule 90 at generation 400. The eight differences are at columns 0, 32, 256, 288, 512, 544, 768, and 800.
2. Predict the next count. Each differing cell sends a contribution to each neighbor; two contributions to the same cell cancel. Pay special attention to the two ends of the displayed row.
3. Inspect generation 401, then 402. The readout truncates lists longer than twelve positions, but always reports the full count. Compare the seam with the exact checks below.
4. Try generation 439, then change the scattered seed and press Grow. Does the count depend on the background?

• Generation 401: 16 differences. Near the seam they occupy columns 799, 800, 0, and 1. The contribution from column 0 wraps to 800; the contribution from 800 wraps to 0. These land at different cells, so they do not cancel.
• Generation 402: 16 differences. Near the seam the occupied columns are now 798, 799, 1, and 2; columns 0 and 800 cancel.
• Generation 439: 128 differences. Wrapping has changed the layout, yet the count still matches the infinite-line pattern.

Two different boundary events

The cancellation guide gives the infinite-line count: 2 raised to the number of 1s in the binary representation of the generation. It also warns that the displayed extremes stop following 400 − t and 400 + t after generation 400. These statements have different limits: the count formula remains valid on this ring through generation 800.

Imagine unfolding the ring into an endless row. At any one generation, rule 90's differences occupy only positions of one parity: all even or all odd. Two such positions are separated by an even number. For them to become the same cell when folded onto an 801-cell ring, their separation must also be a multiple of 801. The smallest possible separation is therefore 1602.

At generation t, the unfolded pattern spans 2t columns. Until generation 801, it cannot contain a pair 1602 columns apart. Its contributions wrap into distinct cells, preserving the count. This does not prevent the ordinary local XOR cancellations that also occur on the infinite line.

A prediction beyond the picture

The browser shows generations 0–439. A longer simulation first departs from the infinite-line count at generation 801: the two extreme contributions meet at column 400 and cancel. Binary 801 is 1100100001, with four 1s, so the line has 16 differences and this ring has 14. This later result was checked separately; it is outside the browser's inspection range.

The ring's size matters. An 800-cell ring permits an even separation of 800, so its first such collision occurs at generation 400. Do not transfer a boundary claim to a different width without checking it.

Try describing what changed using two observations: where differences appear, and how many remain. Crossing a drawing's seam and meeting an earlier branch of the pattern are different events.