Rule garden / Experiment guide

Erase everything, or return exactly?

Two rules, the same circular space, opposite fates. On a ring of 256 cells, rule 90 erases every starting row by generation 128. Rule 150 returns every starting row exactly at generation 128. A boundary can determine whether a system forgets everything or keeps enough information to recover its past.

Try the prediction

1. Open rule 90 on 256 cells, generation 127. There are 128 differing cells. Inspect 128: zero differences. Both gardens have merged.
2. Switch View to Original garden and press Grow. The inspector reports 0 lit cells at 128. That is stronger than agreement: both rows are empty. Later generations stay empty. Change the seed and repeat.
3. Open rule 150 on 256 cells, generation 128. Read “cells differ from its own generation 0”: 0. The original garden has returned exactly. Inspect 1 to see that this seed really changed along the way; inspect 256 or 384 to see further returns.
4. In Difference view, rule 150 has 1 differing cell at column 128 at generation 128. This compares the two gardens. It is a different comparison from the original garden returning to its own beginning.
5. Try rule 90 on 255 cells or 257 cells. Both have 2 differences at 128. A one-cell change in circumference breaks the erasure prediction at this checkpoint.

The image always shows 440 generations, stretched to the same display shape. Ring cells sets the number of columns; the inspector and exported PNG use that actual width. A shared URL includes the width. Earlier guides use the default 801 cells.

Why powers of two matter

Let S shift a whole row one place around the ring, and let I leave it unchanged. Addition below means XOR. Rule 90 is S + S⁻¹; rule 150 is I + S + S⁻¹. Applying a rule twice means multiplying its expression by itself.

When you square a sum of commuting shifts, every mixed term appears twice and cancels by XOR. Square again and the shift distances double again. After 1, 2, 4, 8, … steps, this gives:

Rule 90 after 2ᵏ steps: shift the starting row 2ᵏ places each way, then XOR the two copies.
Rule 150 after 2ᵏ steps: do the same, then XOR the unshifted starting row too.

On 256 cells, shifting 128 places left reaches exactly the same positions as shifting 128 places right. The two copies cancel. Rule 90 leaves nothing; rule 150 leaves the unshifted starting row. This reasoning applies to every initial row, not only the seeds offered by the controls.

More generally, for N = 2ᵐ cells, m ≥ 1, rule 90 erases by N/2 steps and rule 150 returns at N/2. Try 64 cells at 32 steps, or 512 at 256. Some rows erase or return sooner; N/2 is a universal bound/return time, not each row’s minimum time.

Memory you can recover

For rule 150 on these power-of-two rings, advancing another N/2 − 1 steps undoes one step. No two distinct starting rows can merge: if they did, their later exact returns would have to be equal too. This is stronger than the odd-parity survival promise, which applies to any circular width but only forbids merging for pairs with odd difference count.

For rule 90, knowing the rule, width, and an empty row at generation N/2 cannot tell you which starting row was used: every one leads there. The 255/257 experiment only disproves this particular erasure deadline at those widths; it does not establish a general classification of all ring sizes.

Keep the comparisons separate

Record rule, ring cells, first-row mode, seed, and generation. Then record three quantities: differences between the two gardens, lit cells in the original, and changes from the original’s own generation 0. They answer three different questions: did the gardens merge, did a row empty, and did a row return?

Background: the polynomial method for additive cellular automata. The erasure and return arguments above are derived here using commuting shifts and XOR cancellation.