Elementary Cellular Automaton Simulator & Rule Table

1. Prepare the initial state

  A row of cells, each holding 0 or 1.

  Gen 0: [0] [0] [1] [0] [0]

2. Look at each cell's neighborhood

  For each cell, look at left, center, and right as a set.
  The cells at each end wrap around to the opposite side, forming a ring (periodic boundary conditions).

       left center right
          ↓    ↓    ↓
  [0]  [0]  [1]  [0]  [0]

3. Look up the next state in the rule table

  Match the 3-cell combination against the rule table to get the next value.

  e.g. Rule 110:
  (0,1,0) → rule table lookup → 1

4. Update all cells simultaneously

  Apply the rule to every cell at once to produce the next generation.

  e.g. Rule 110:
  Gen 0: [0] [0] [1] [0] [0]
                ↓ all cells update at once
  Gen 1: [0] [1] [1] [0] [0]

5. Repeat

  Repeat 2→3→4 to advance through generations. Stacking them vertically creates a 2D pattern.

Elementary Cellular Automata (ECA) were systematically studied by Stephen Wolfram in the 1980s and represent the simplest form of cellular automaton. Despite having only 256 possible rules, their behavior is remarkably diverse. Wolfram classified this behavior into four classes.

Class 1 — Uniform

Regardless of initial conditions, all cells converge to the same state within a few generations. All information from the initial state is completely lost — the simplest possible behavior.

Class 2 — Periodic

Settles into stable patterns such as stripes or fixed points and repeats them indefinitely. Some influence from the initial conditions is preserved locally, but overall behavior remains predictably orderly.

Class 3 — Chaotic

Generates aperiodic, random-looking patterns indefinitely. Tiny differences in initial conditions lead to entirely different outcomes, making long-term prediction impossible. Yet because these patterns are generated by deterministic rules, they can be used as pseudorandom number generators.

Class 4 — Complex

Sits at the boundary between order (Classes 1–2) and chaos (Class 3). Information neither freezes as in Classes 1–2 nor disperses as in Class 3. In this delicate balance, "gliders" — structures that maintain their shape while moving — emerge spontaneously. Gliders can carry information, and their collisions produce new patterns. This ability to preserve, transmit, and transform information is what makes computation possible.

Among the 256 rules, several stand out: Rule 30 is used for chaotic random number generation, Rule 110 has been proven Turing complete, Rule 90 generates the Sierpinski triangle, and Rule 184 models traffic flow. Below, we explore these and other notable rules by class.

Learn more about the history and applications of cellular automata →

In Class 1, rules like Rule 0 (all cells converge to 0) and Rule 255 (all cells converge to 1) cause every cell to reach the same state within a few generations. Class 2, on the other hand, contains many interesting rules — including Sierpinski-family rules that generate fractal patterns and rules that simulate traffic flow.

One of the most famous patterns in ECA is the Sierpinski triangle (Sierpinski gasket). Starting from a single ON cell, a self-similar fractal triangle emerges.

Exact Sierpinski triangle

Rules 18, 90, and 146 produce symmetric Sierpinski triangles. Rule 60 produces a right-leaning triangle, and Rule 102 produces a left-leaning triangle (these two are mirror equivalents).

Sierpinski from single cell only

Rules 26, 154, 210, and 218 produce a Sierpinski triangle from a single-cell initial condition, but exhibit different behavior with random initial conditions.

Sierpinski-like fractals

Rules 22, 122, 126, and 150 produce fractal patterns that resemble the Sierpinski triangle but are not strictly identical to it.

Rule 184 is a particle-conserving rule used for traffic flow simulation. Treating black cells as cars and white cells as empty space, a car advances one cell if the space ahead is empty, and stops if blocked.

Produces aperiodic, random-looking patterns. Rule 30 is famously used in Mathematica's random number generator. Rules 45 and 106 also exhibit chaotic behavior.

The "gliders" described above can be observed in action by running Class 4 rules in the simulator. In Rules 110 and 54, structures that maintain their shape emerge within the periodic background pattern, moving at different speeds. When gliders collide, they produce complex interactions — annihilation, merging, and splitting.

Turing completeness means the ability to perform any computation given the right initial conditions. Class 4 (complex) is neither too orderly nor too chaotic, which makes this possible. In orderly rules, information becomes fixed; in chaotic rules, information disperses and is lost. In Class 4, gliders can preserve and transmit information, enabling computation.

Rule 110

Proven Turing complete by Matthew Cook in 2004, demonstrating that this simple rule can simulate any computation.

Rule 54

Conjectured to be Turing complete, but not yet proven. Like Rule 110, it produces gliders. From a single-cell initial condition, it produces left-right symmetric output.

Among the 256 rules, there are "equivalent rules" that generate patterns with the same structure. Equivalent rules are related by the following three transformations:

Left-right symmetry

Swapping the left and right cells in each neighborhood pattern (swap) produces a left-right symmetric pattern. When a rule produces symmetric output from a single cell (e.g., Rules 26 and 82, Rules 30 and 86), the transformation looks identical, so use random initial conditions to see the difference.

Black-white symmetry

Reversing the bit order and then flipping each bit (reverse+flip) produces a black-white symmetric pattern. This is different from a simple bit-flip (complement). When the initial condition is also color-inverted, the output becomes the color-inverted version of the original pattern. However, some pairs such as Rule 110 and Rule 137 produce color-inverted patterns even from the same single-cell initial condition.

Left-right + Black-white symmetry

Combines the two transformations above. First swaps left and right cells in each neighborhood pattern, then reverses the bit order and flips each bit.

Special Case of Black-White Symmetry — When the Rule Is a Palindrome

When a rule's bit string is a palindrome (reads the same forwards and backwards), flipping the input bits alone causes the output bits to flip as well. Normally, black-white symmetry requires reverse+flip (reversing the order then flipping), but for palindrome rules the bit string is unchanged when reversed, so flip alone is sufficient. For example, Rule 90 (01011010) is a palindrome, so 255 - 90 = Rule 165 (10100101) is its equivalent.

Note The swap in left-right symmetry swaps the left and right cells of each neighborhood pattern (3 bits). For example, neighborhood pattern 110 (position 6) becomes 011 (position 3) when left and right are swapped, so bit positions 6 and 3 are exchanged in the rule's bit string (8 bits). Similarly, 100 (position 4) and 001 (position 1) are exchanged. Symmetric patterns (111, 101, 010, 000) remain unchanged. For example, Rule 110 (01101110): bit positions 6 and 3 are both 1 (unchanged), while bit position 4 (0) and position 1 (1) are swapped, giving 01111100 = Rule 124.

Applying the three equivalence transformations above reduces the 256 ECA rules to 88 independent representatives. Every other rule is equivalent to one of these.