Cellular automaton

Imagine a grid of tiny squares.

Each square is called a cell. A cell can be in a few states. It might be black or white.

Each cell looks at its neighbors. These are the cells right next to it. Rules tell the cells how to change. The rules stay the same for every cell. This can make very cool patterns. Some patterns stay still. Others move like they are alive. It is like a tiny world of math.

Imagine a grid of tiny squares. Each square is a cell.

Each cell can be in different states. A cell might be on or off. It might be black or white. Every cell has neighbors. These are the cells right next to it.

Rules tell each cell how to change. The rules look at the cell and its neighbors. These rules stay the same for every cell. This creates a cellular automaton.

Stanislaw Ulam and John von Neumann first thought of this in the 1940s. They used these models to study how crystals grow. They also studied how liquids move. In the 1970s, a famous version called Conway's Game of Life became popular. In this game, cells live or die based on their neighbors. Some rules make patterns stay still. Other rules make patterns move or look messy. Some patterns become very complex. These systems can even act like computers. They can simulate things in biology or chemistry. You can even imagine the grid as a doughnut shape. This is called a torus. In a torus, cells on the edge wrap around to the other side.

Imagine a giant sheet of graph paper that goes on forever. Each small square is called a cell.

Every cell can be in a certain state, like being black or white. We call these states "on" or "off." These grids are used to study how things work in nature. Scientists use them to model physics and even biology. They are also called cellular spaces or iterative arrays.

To make the model work, we use rules. Each cell looks at its neighbors to decide what to do next. A neighbor is just a cell that is right next to it. One type is the von Neumann neighborhood, which uses the four cells touching the sides. Another is the Moore neighborhood, which includes the four diagonal cells too.

When the rules are applied, every cell changes at the same time. This creates a new generation of the grid. The rules stay the same for every cell throughout the whole process.

This idea began in the 1940s at Los Alamos National Laboratory.

Two scientists named Stanislaw Ulam and John von Neumann worked on it. Ulam wanted to model how crystals grow using a simple network. Von Neumann was interested in how a robot might build another robot. They eventually found a way to calculate how liquids move. They treated the liquid as many tiny, separate units. This was the birth of the cellular automaton.

In the 1970s, a famous version called Conway's Game of Life became popular.

It uses a two-dimensional grid with simple rules for living and dying. If a live cell has too few neighbors, it dies from underpopulation. If it has too many, it dies from overpopulation. A dead cell can become alive if it has exactly three neighbors. Stephen Wolfram later studied one-dimensional versions in the 1980s. He found that some rules create very complex patterns.

These models can connect to many things we see every day. They can simulate chemical reactions or how living things behave. Sometimes, scientists imagine the grid is shaped like a doughnut.

This shape is called a torus. On a torus, if a cell goes off the top, it reappears at the bottom. This helps the model act like it is infinite. Even simple rules can lead to amazing, complex worlds. These small cells can even act like a computer.

A cellular automaton, often abbreviated as CA, is a discrete model of computation. It is used to study complex systems in fields like physics, theoretical biology, and microstructure modeling. These systems are also known by many names, including cellular spaces, tessellation automata, and iterative arrays. At its core, a cellular automaton consists of a regular grid of cells. Each cell exists in one of a finite number of states, such as being "on" or "off." This grid can exist in any finite number of dimensions.

The mechanism of a cellular automaton relies on a specific process of updating states. First, an initial state is chosen at time t = 0 by assigning a state to every cell. This starting arrangement is called a configuration. To advance to the next generation, or time t + 1, a fixed rule is applied to the entire grid simultaneously. This rule is usually a mathematical function. It determines the new state of a cell based on its current state and the states of its neighbors. The neighborhood is a set of nearby cells defined relative to the cell being updated.

There are different ways to define these neighborhoods. The von Neumann neighborhood, named after a founding theorist, includes only the four cells that are orthogonally adjacent. The Moore neighborhood is larger, including the four orthogonal cells plus the four diagonally adjacent cells. For a cell with a Moore neighborhood, there are 512 possible patterns of neighbors. The total number of possible automata can be calculated using the formula k^s, where k is the number of possible states and s is the number of neighboring cells, including the cell itself.

History shows that these ideas emerged from a desire to model natural growth and complexity. In the 1940s, Stanislaw Ulam and John von Neumann worked together at Los Alamos National Laboratory. Ulam used a lattice network to model how crystals grow. Von Neumann was interested in self-replicating systems, specifically the idea of a robot building another robot. He eventually developed a tessellation model, known as the von Neumann universal constructor. This design used a 200,000 cell configuration to prove that a pattern could make endless copies of itself.

Stephen Wolfram later categorized cellular automata into four distinct classes based on their behavior. Class 1 consists of automata where patterns stabilize into homogeneity. Class 2 involves patterns that evolve into stable or oscillating structures. Class 3 contains automata where patterns evolve in a seemingly chaotic fashion. Class 4 is the most complex, where patterns become extremely intricate and may last a long time. These Class 4 automata are thought to be computationally universal, meaning they can simulate a Turing machine.

In the 1970s, interest in the subject expanded beyond academia due to Conway's Game of Life. This is a two-dimensional cellular automaton with simple rules for survival and reproduction. A live cell dies if it has fewer than two neighbors or more than three neighbors. A dead cell becomes alive if it has exactly three neighbors. This model demonstrated how simple local rules could create complex global behaviors.

To simulate these systems on a computer, researchers often use a finite grid. This creates a problem with the cells on the edges. One solution is to use periodic boundary conditions, which creates a toroidal arrangement. This shape is like a doughnut, where a cell moving off the top reappears at the bottom.

This method allows the finite grid to behave like an infinite plane. By using modular arithmetic, scientists can effectively simulate vast, complex universes through these small, discrete units.