Cellular automaton

A cellular automaton (pl. cellular automata, abbrev. CA) is a simple model of computation studied in automata theory. It is made up of a grid of small units called cells. Each cell can be in one of a few simple states, like on and off. The grid can have many dimensions, and each cell looks at its nearby neighbors to decide what state to change to next.

The idea of cellular automata began in the 1940s, created by scientists Stanislaw Ulam and John von Neumann while they worked together at Los Alamos National Laboratory. Interest grew much larger in the 1970s thanks to a famous example called Conway's Game of Life. This special kind of cellular automaton shows how simple rules can create complex patterns that change over time.

These systems are useful in many areas, such as physics, theoretical biology, and modeling tiny structures in materials called microstructure. Scientists like Stephen Wolfram have studied these patterns carefully. Some cellular automata can become very complicated and might even be able to perform any calculation a computer can do, which researchers call being computationally universal or able to simulate a Turing machine.

Overview

A cellular automaton is like a grid of squares, where each square can be black or white. The color of each square changes based on the colors of its neighboring squares. The neighbors are usually the squares directly next to it, either just the side squares or also the corner squares. There are many ways to decide how the colors change, leading to many different patterns over time.

When we make these grids on a computer, we often use a smaller, finite grid instead of an infinite one. We have to decide what to do with the squares on the edges. One common way is to connect the edges, so the grid wraps around like a tube or a doughnut. This helps avoid special rules just for the edge squares and makes the patterns easier to calculate.

History

Stanisław Ulam studied crystal growth using simple patterns in the 1940s at the Los Alamos National Laboratory. His colleague, John von Neumann, worked on systems that could copy themselves. Together, they created the idea of cellular automata. This is a way to model how groups of small units change based on their neighbors.

Later, many scientists used this idea. One famous example is the Game of Life. It was created by John Conway and shared by Martin Gardner in Scientific American. With just a few simple rules, it shows surprising patterns and can even do calculations. Other researchers, like Stephen Wolfram, studied how simple rules can create complex behaviors in nature.

Classification

Wolfram made a way to sort cellular automata into four groups based on how they act. These groups help us see the different patterns and actions that can come from simple rules.

• Class 1: Patterns turn even and calm quickly. Any random parts in the start disappear.
• Class 2: Patterns become calm or moving back and forth. Some random parts may stay, but changes don’t spread far.
• Class 3: Patterns act in a wild, hard-to-predict way. Any calm parts that show up disappear fast.
• Class 4: Patterns grow complex, working together structures that last. These systems might be able to do any kind of math work, in theory.

These groups are ideas to help us understand. Some cellular automata might show parts of more than one group. Scientists have tried to make better groups, but some are still hard to explain clearly. The idea of sorting things into four groups started from studies in chemistry and thermodynamics.

Elementary cellular automata

Main article: Elementary cellular automaton

An elementary cellular automaton is a simple system where each cell can be on or off. Each cell checks its two neighbors to decide its next state. Because of the different ways cells and their neighbors can be, there are 256 possible rules for how these cells change.

Some rules, like rule 30, rule 90, rule 110, and rule 184, are very interesting. Rule 30 makes patterns that look random and messy. Rule 110 makes patterns that are not random and not repeating, letting complex shapes form and interact in surprising ways. This rule shows that even very simple systems can do complex things.

Rule space

A cellular automaton rule is like a set of instructions that tells each cell what to do next based on its neighbors. These rules can be thought of as points inside a shape called a hypercube. For simple rules, this shape has 8 sides, and for more complex ones, it has 32 sides.

We can measure how different two rules are by counting the steps needed to move from one point to another in this hypercube. This helps us see if rules that behave similarly are close together. Scientists have noticed that some rules that stay mostly the same are found in one area, while others that change a lot are found in another. This idea is linked to something called the "edge of chaos."

Applications

Further information: Patterns in nature

Cellular automata can show how many natural things work. For example, the pretty patterns on some seashells come from natural cellular automata. Each pigment cell on a seashell follows easy rules based on the cells next to it, making nice designs as the shell grows. Plants use a similar way to control tiny holes on their leaves that let gases in and out.

In chemistry, cellular automata can copy reactions that make swirling patterns, like those in some chemical mixes. In physics, these models help scientists learn how materials change in different situations, such as how magnets can lose their magnetism when they get hot.

Cellular automata are also used in computer science for jobs like making random numbers and solving some kinds of problems. They have been used to create music and build landscapes in video games.

Specific rules

Some well-known cellular automata rules include Conway's game of life, Langton's ant, and Rule 90. These rules help create interesting patterns in a grid of cells. Other examples are Brian's Brain, Langton's loops, and Wireworld. They show how simple rules can lead to complex behavior.