exhibition · 256 works · catalogue raisonné

Atlas of the 256

Every elementary cellular automaton: one row of cells, two states, three neighbours, one byte of rule. Hung on one wall and measured by this page.

Ptractable

There are exactly 2⁸ = 256 ways to answer eight yes-or-no questions, and each answer set is a universe. Swap left and right, or black and white, and you get the same universe seen differently; that folds the 256 into88 families. Stephen Wolfram sorted their behaviour into four classes in 1984. This page redoes the sorting itself, from random starting rows, and prints the numbers on every label.

class I · homogeneous: 8 families
class II · periodic: 65 families
class III · chaotic: 13 families
class IV · complex: 2 families
additive over GF(2): 16 rules
number-conserving: 170, 184, 204, 226, 240
reversible: 15, 51, 85, 170, 204, 240
with Gardens of Eden: 250 of 256

Method. Classes I–III are assigned by this page: four random rows on a ring of 149 cells run for 600 steps; if every run ends uniform the rule is class I; if every run ends in a (possibly sliding) cycle of period ≤ 160 it is class II; otherwise class III. Class IV (54, 110 and their families) cannot be certified by measurement and is taken from the literature. Damage spreading is averaged over three seeds at non-power-of-two horizons. Reversibility and Gardens of Eden are checked exhaustively on every ring of 3–12 cells. Compressibility is deflate on a 256 × 256 diagram. Rule 73 and Rule 106 land in class III here; some authors file them elsewhere.