Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Maths in a minute: Cellular automata

      16 May, 2017

      The name "cellular automaton" may sound a bit frightening, but the concept is actually quite simple. Think of a grid on the plane, for example a square grid or a honeycomb, in which each individual cell (each little square or hexagon) has one of two colours, say black or white. At each time step (say every second or every minute) the cells change colour in a way that depends on what colour the neighbouring cells are.

      For instance, in the honeycomb example above, which shows only three states separated by two time steps, a cell changes colour if at least four of its neighbours are the opposite colour. You could carry on evolving the grid indefinitely, changing the colours according to the rules at each time step.

      Cellular automaton

      More generally, cellular automata can be defined in any dimension (you could have just a one-dimensional row of cells, or a three-dimensional grid of cells, for example), they can involve more than one colour, and they can also involve an element of chance.

      Cellular automata are capable of amazingly complex behaviour. Even simple rules can give us patterns that evolve chaotically, leaving us no hope of ever predicting them accurately. But they can also produce stable patterns that change little over time, or patterns that look like they couldn't possibly be the result of mindless interactions between neighbours, but involve some grand overall design (technically, cellular automata can exhibit self-organisation and emergence).

      The video below shows the evolution of a cellular automaton in which the individual cells are so small, you can hardly see them (they're like the pixels on a computer screen). You can see how spiral patterns emerge over time. These actually resemble spiral patterns found in nature. Find out more in the article Spontaneous spirals by Wim Hordijk, who also made the movie.

      Cellular automata are used to simulate processes in nature (for example the pattern formation on animal skins). Theoretical computer scientists like them because they can represent a kind of universal computing machines. And some even wonder whether the whole Universe is a cellular automaton. You can find out more in these Plus articles:

      • Games, life and the game of life
      • Matrix: Simulating the world
      • Spontaneous spirals
      • Spotting lizards.
      • Log in or register to post comments

      Read more about...

      Maths in a minute
      cellular automaton
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms