A grid of cells running one local rule can draw a fractal, build a highway or carry a glider. The rules here run from one-bit cells to continuous Lenia.
A cellular automaton is a grid of cells and one rule that every cell applies to its immediate neighborhood, over and over. The systems here get progressively less discrete: Wolfram's one-dimensional rules, Langton's ant, and Lenia, where state, space and time are continuous.
Cells in a row hold 0 or 1, and each cell's next state depends on itself and its two neighbors: a rule is a table of eight outputs, giving 28 = 256 rules. Wolfram's rule number is those outputs read as binary, from neighborhood 111 down to 000, as in the table under Figure 1.
From one live cell, Rule 90 makes each cell the exclusive-or of its neighbors and draws Pascal's triangle mod 2. Rule 30's center column looked random enough for Wolfram to use it as Mathematica's random number generator. Rule 184 is the traffic essay's automaton with a speed limit of one and no randomness: jams drift backward while cars pass through them. Rule 110 carries particles over a periodic background, and Matthew Cook proved in 2004 that their collisions can emulate any Turing machine. Conway's Life is their two-dimensional cousin (Life and symmetry attractors).
Wolfram sorted the rules by eye into four classes by their behavior from random starts: uniform, fixed or repeating, random-looking, and long-lived particles like 110's. One measurement does much of that sorting: run a rule twice from the same random row, one cell flipped in the second run, and mark where the runs disagree.
Nothing travels faster than a cell per step, so the disagreement stays inside a cone. Rule 90's reaches both edges in every run; Rule 30's moves right one cell per step and left about 0.2; Rule 110's spreads 0.40 cells per step; Rule 184's stays one to three cells wide in 15 of 16 starts. Mirror images and 0/1 swaps behave identically, leaving 88 distinct rules for Figure 3.
Eight rules in the corner reach a uniform row from every start: class I. Along the bottom, 61 keep a flip's influence under a fifth of a cell per step: class II, frozen or repeating, though a snapshot cannot tell them from chaos (Rule 204 copies every cell and leaves a row as disordered as Rule 30's). Eleven, among them 30 and 90, spread a cell per step or more: class III. Rule 110 sits between with seven others, with no wide gap on either side. Langton's λ, proposed as a dial from order to chaos, gathers the spreading rules at 1/2 (28 of 70, and none below 1/4), but most rules at 1/2 do not spread.
In 1986 Chris Langton moved the rule onto a single cursor. An ant on a white grid turns right on a white cell and left on a black one, flips the cell and steps forward. From a blank grid it draws small, nearly symmetric figures for a few hundred steps, an irregular blob for thousands, and then a straight line, the highway.
From move 9,977 on, the turns repeat with period 104, each period carrying the ant 2 rows and 2 columns along a diagonal. Leonid Bunimovich and Serge Troubetzkoy proved in 1992 that the ant's path is unbounded from any finite start; whether every finite start ends on a highway is an open conjecture. Figure 5 tests it on random squares of black cells around the ant.
All 3,200 runs at black fraction 0.5 reached the highway, as did all 28,800 across the nine fractions; the slowest took 394,020 moves. At side 40 the median is 8,533 and 222 of 400 runs beat the blank grid; the median grows from 3,420 at side 10 to 41,796 at side 160. That supports the conjecture, though no sample can prove it.
Bert Chan's Lenia (2019) makes each ingredient of Life continuous: states between 0 and 1, a ring-shaped kernel of radius cells, a smooth growth function , and steps of length :
where the brackets clip to [0, 1]. The potential averages the states around each cell, with weight at distance
zero at the cell and at , peaking at and normalized to sum 1, so is the filled fraction of a fuzzy ring. Growth is
positive only within of . The orbium, Lenia's best-known glider, has and : a cell grows only while its ring is 13.2% to 16.8% full and fades otherwise, the shape of Life's table with soft edges.
In the Growth view the front sits in the green band and the rear in red: the orbium adds material ahead, dissolves it behind and slides forward at nearly constant mass. starves it within a few time units, freezes the field into speckle within about twenty, kills it, and 0.020 leaves it slightly heavier.
If these creatures are continuous objects, refining the grid and time step should only sharpen them as the update approaches . Figure 7 runs the catalogued orbium, resampled to kernel radius , at steps per unit time for 30 time units.
At the catalogued they do not. For the orbium survives at and 10 but dies at 20 and 50, and Davis and Bongard (2022) found Lenia gliders failing at step sizes both too small and too large. The parameters are not to blame: an orbium relaxed at with or 50, then brought to 0.015, glides on at about 0.91 of its starting mass. The catalogued shape, recorded at , fails to settle into the lighter form finer steps favor (Chan reports the trend). At every run with and survives, as the continuum argument predicts. Coarse grids still kill it: at the kernel peaks three cells out, and at or 3 one step can move a cell half or a third of its range.
Rounding the states is the last test. A step at , changes a cell by at most 0.1, less than half of a 4-level step (0.125), so every change rounds away and the orbium freezes; at 6 levels it dies, 8 to 16 are erratic, and from about 24 it glides as with continuous states.
The agents of the next essay move through open space and steer by their neighbors, and whether a crowd of them comes to share one heading depends on that motion.