Elementary cellular automata: 256 small rules you can actually compare

The rule space is small enough to hold in your head

An elementary cellular automaton has one row of cells. Each cell is either 0 or 1. To compute the next row, the rule looks at three cells above it: the left neighbor, the cell itself, and the right neighbor. Three binary inputs give eight possible neighborhoods, from 111 down to 000.

That gives the familiar count: 2^8 possible output tables, or 256 rules. A Wolfram rule number is just a compact way to write one of those eight-bit tables. Rule 30 is 00011110 in binary. Rule 110 is 01101110.

The diagram is time, not a picture of a two-dimensional world

The canvas on this page draws each generation as a new row. The top line is the initial condition; every row under it is the next tick. That makes the diagram look two-dimensional, but the automaton itself is one-dimensional.

This distinction matters when comparing it with Conway Life. Life updates a two-dimensional grid in place. Elementary rules build a history stack. You are reading the trace of time from top to bottom.

A few named rules are enough for a first map

Rule 30 is the noisy reference point. Rule 90 is the triangular parity reference point. Rule 110 is the computation reference point because its localized structures can support universal computation under suitable initial conditions. Rule 184 is a good practical contrast because it is often read as a traffic-flow rule.

The class labels in Wolfram's writing are useful as a first reading tool: fixed behavior, periodic behavior, chaotic-looking behavior, and structured complexity. I would not treat those labels as a full mathematical classifier. They help a learner decide what to watch first.

The practical lesson is comparison

Do not try to understand cellular automata from one rule. Keep the starting row fixed and change the rule number. Then keep the rule fixed and change the initial row. The surprise is not that one rule looks strange. It is that the same eight-neighborhood format can produce order, apparent randomness, moving local structures, or traffic-like sorting.

That is why this module starts with an explorer instead of a static article. The article gives the vocabulary; the grid gives the evidence.

What to watch in the demo

• Start with Rule 30 and a single live cell. The left edge soon looks regular, while the center and right side look much less predictable.
• Switch to Rule 90 without changing the seed. The same one-cell start now draws a Sierpinski-like triangle instead of Rule 30 noise.
• Try Rule 110 with a random seed. Look for small moving structures against a repeated background, not for smooth randomness.
• Use Rule 184 with a random seed. It reads more like traffic flow than biological growth.