World

What a world is

A world is a small universe whose rules are written exactly, with no approximation. Today, every Slean world is a one-dimensional cellular automaton.

• A row of cells forms a ring: the last cell touches the first.
• Each cell has a state, a small number: 0 or 1 in two-state worlds, 0, 1 or 2 in three-state worlds.
• At each time step, every cell changes at once. A cell’s new state depends only on itself and its two neighbours.

Reading a rule

The rule is a table. For each possible neighbourhood (left neighbour, cell, right neighbour) it gives the cell’s new state. With two states there are 8 neighbourhoods, so 8 entries. Here is the table of rule 184, the traffic model:

That is where the number 184 comes from. Write the 8 outputs in the order of the neighbourhoods 111, 110, …, 000: you get 10111000, and that binary number is 184. This is Wolfram’s numbering, used everywhere for the 256 elementary automata.

A three-state world has 27 neighbourhoods, so a table of 27 entries. A four-state world has 64.

The world in Lean

In Lean, an elementary automaton is written eca 184, and an automaton given by its full table is written tableCA 3 2 [ … ]: 3 states, a cell and its 2 following neighbours, then the table. The Lean definition is the world. A simulator that gives a different result is wrong, so a phenomenon cannot come from a simulator defect.

One difference in reading to know about

The figures on this site read the table in the centred way: new cell i looks at cells i−1, i and i+1. The Lean world reads the same table with a one-cell offset: new cell i looks at i, i+1 and i+2. The two pictures are identical up to a translation of one cell per step. Totals are exactly the same, so a conservation law true in one reading is true in the other.

For structures that move, the offset changes how many cells the structure appears to travel. The page A glider does the arithmetic.