Rule garden / Experiment guide
Can two different rule numbers describe the same experiment viewed from opposite sides? Start rules 30 and 86 from one cell in the middle of an 801-cell ring. Their pictures face opposite ways, but their difference charts agree at every generation.
One centered cell on an odd-width ring looks the same in a mirror. The empty perturbed row does too: flipping the sole occupied cell removes it. Both rules keep an empty row empty, so here the original and difference pictures coincide.
A scattered row usually does not equal its reflection. Using the same seed in both tabs gives the same row, not a reflected row. To make the general comparison, reflect the entire original starting row and the flipped-cell position as well as the rule. The current controls do not offer arbitrary reflected starting rows; the scattered links deliberately demonstrate the failed control, and the full reflected-seed check lives in the test script.
Keep the linked width at 801 for an ordinary left-to-right mirror around the flipped center cell. At any odd width N, column x maps to N − 1 − x and the center stays put. At even widths the two middle columns exchange places, so simply selecting One cell again does not place the impulse at its reflected position.
A neighborhood is written left, center, right. Reflection swaps its first and last digits. It leaves 111, 101, 010, and 000 unchanged, exchanges 110 with 011, and exchanges 100 with 001. Move the rule’s output decisions to those reflected neighborhoods.
| Neighborhood | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 |
|---|---|---|---|---|---|---|---|---|
| Rule 30 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
| Reflected rule | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 |
On narrow screens, scroll the table; keyboard users can focus it and use arrow keys.
The reflected outputs read 01010110: 64 + 16 + 4 + 2 = 86. This is a rearrangement of neighborhood decisions, not a reversal of all eight binary digits.
Every neighborhood in a reflected row encounters the swapped decision, so one rule-86 step after reflection equals reflection after one rule-30 step. Repeat that identity to obtain the same statement at any generation. Reflecting both gardens also reflects their XOR difference. Since a reflection only rearranges cells, its count stays the same.
This argument works for every starting row and every time, provided the starting row is reflected. The finite tests below check the code and examples; they are not the reason the claim holds forever.
Exchange left and right in rule 170’s “copy right neighbor” decision: which rule copies the left neighbor? The reflected rule is 240. Rules 90, 150, and 204 are their own mirrors. For those rules, a reflection-symmetric starting row remains symmetric. An asymmetric starting row need not become symmetric.
Record the rule pair, width, starting row, flip location, and reflection map. Equal counts alone do not establish a mirror relationship: the stronger test compares every cell at its reflected position. The traveling-difference guide shows another way counts can hide geometry.
Source: the Wolfram Atlas rule-30 properties lists 86 as its left-right equivalent. The table and whole-row argument above spell out that equivalence.
Checked 2026-09-09: all 256 rules on all bit rows of widths 1–8 (130,560 one-step reflection cases), plus 14,080 complete original/perturbed/difference row triples for rules 30/86 with reflected starting rows at widths 7, 31, 255, and 801 through generation 439. Exact checkpoints and the same-seed counterexample also pass.