Conditionally Reversible Elementary Cellular Automata

Of the 256 elementary cellular automata, 16 have the property that changing the left bit in the rule icon always changes the output bit. These rules are called left bijective or said to exhibit one-sided additivity, and they can be run uniquely backward in time from initial conditions with certain background colors. This Demonstration explores the 12 of these 16 rules that support unique histories from a white background. See what the history looks like from a single black cell or from a random initial condition.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Bijectivity in the left position causes each southeast diagonal to be periodic. By extending each diagonal up the page, you obtain the unique infinite history.
Left bijective rules exhibit a "local" nested structure on the right side. This means that the regions of the automata seen in this Demonstration occur infinitely many times in the evolution.
For more information, see E. S. Rowland, "Local Nested Structure in Rule 30," Complex Systems, 16(3), 2006 pp. 239–258.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+