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Sample Models/Computer Science/Cellular Automata/CA 1D Simple Examples

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CA 1D Rule 250

[screen shot]

If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web


This program models one particular one-dimensional cellular automaton -- the one known as "rule 250".

A cellular automaton (aka CA) is a computational machine that performs actions based on certain rules. It can be thought of as a "board" which is divided into cells (such as the square cells of a checkerboard). Each cell can be either on or off. This is called the "state" of the cell. The board is initialized with some cells on and some off. A clock is then started and at each "tick" of the clock the rules are "fired" and this results in some cells turning "on" and some turning "off".

There are many kinds of cellular automata. In this model, we explore a one-dimensional CA -- the simplest type of CA. In this case of one-dimensional cellular automata, each cell checks the state of itself and its neighbors to the left and right, and then sets the cell below itself to either "on" or "off", depending upon the rule. This is done in parallel and continues until the bottom of the board.

This model is one of a collection of 1D CA models. It is meant for the beginning user. If you have experience with CAs, we suggest you check out a more sophisticated model such as CA 1D Elementary.

In his book, "A New Kind of Science", Stephen Wolfram argues that simple computational devices such as CAs lie at the heart of nature's patterns and that CAs are a better tool than mathematical equations for the purpose of scientifically describing the world.


As the CA computes, each patch checks the color of itself and the patches directly to the left and right of it, and then paints the patch below it according to Rule 250:

```text B B B B B Y B Y B B Y Y B Y B Y

Y B B Y B Y Y Y B Y Y Y Y Y Y Y ```

For example, if we have a Rule 250 CA, and the current cell is yellow and its left neighbor is black and its right neighbor is black, the cell below it is painted black.


Initialization & Running: - SETUP initializes the model with a single cell on in the center. - SETUP-CONTINUE copies the last row of the previous run to the top so that you can continue running the model in "wrapped" mode when you click GO. - GO begins running the model with the currently set rule. It continues until the end of the view.


Although the rules are very simple, interesting patterns emerge in Rule 250. These patterns are highly regular.

Note that the pictures generated by this model do not exactly match the pictures in Wolfram's book, "A New Kind of Science". That's because Wolfram's book computes the CA as an infinite grid while the NetLogo model "wraps" around the horizontal boundaries. To get pictures closer to the ones in the book, you may need to increase the size of the world. You can increase the size of the world up to the available memory on your computer. However, the larger the world, the longer time it will take NetLogo to compute and display the results.


What happens to the regularity when SETUP-CONTINUE is used a number of times? Why do you suppose that is? (Note that in this model, the CA wraps around the sides.)

Is there any consistent pattern to the way this CA evolves?

Can you predict what the color of the nth cell on the vertical midline will be?


What if you wanted to observe the behavior of a CA over many iterations without having to click continue every time the CA reaches the bottom of the view? Simply replace the stop with setup-continue in the go procedure:

  if (row = min-pycor)
    [ stop ]


  if (row = min-pycor)
    [ setup-continue ]

What if a cell's neighborhood was five -- two to the left, itself, and two to the right?

Classical CAs use an "infinite board". The CA shown here "wraps" around the edges of the world (sometimes known as a periodic CA or CA with periodic boundary condition). How would you implement a CA in NetLogo that comes closer to the infinite board?

Try making a two-dimensional cellular automaton. The neighborhood could be the eight cells around it, or just the cardinal cells (the cells to the right, left, above, and below).


Life - an example of a two-dimensional cellular automaton CA 1D Rule 30 - the basic rule 30 model CA 1D Rule 30 Turtle - the basic rule 30 model implemented using turtles CA 1D Rule 90 - the basic rule 90 model CA 1D Rule 110 - the basic rule 110 model CA 1D Elementary - a model that shows all 256 possible simple 1D cellular automata CA 1D Totalistic - a model that shows all 2,187 possible 1D 3-color totalistic cellular automata.


Thanks to Eytan Bakshy for his help with this model.

The first cellular automaton was conceived by John Von Neumann in the late 1940's for his analysis of machine reproduction under the suggestion of Stanislaw M. Ulam. It was later completed and documented by Arthur W. Burks in the 1960's. Other two-dimensional cellular automata, and particularly the game of "Life," were explored by John Conway in the 1970's. Many others have since researched CA's. In the late 1970's and 1980's Chris Langton, Tom Toffoli and Stephen Wolfram did some notable research. Wolfram classified all 256 one-dimensional two-state single-neighbor cellular automata. In his recent book, "A New Kind of Science," Wolfram presents many examples of cellular automata and argues for their fundamental importance in doing science.

See also:

Von Neumann, J. and Burks, A. W., Eds, 1966. Theory of Self-Reproducing Automata. University of Illinois Press, Champaign, IL.

Toffoli, T. 1977. Computation and construction universality of reversible cellular automata. J. Comput. Syst. Sci. 15, 213-231.

Langton, C. 1984. Self-reproduction in cellular automata. Physica D 10, 134-144

Wolfram, S. 1986. Theory and Applications of Cellular Automata: Including Selected Papers 1983-1986. World Scientific Publishing Co., Inc., River Edge, NJ.

Bar-Yam, Y. 1997. Dynamics of Complex Systems. Perseus Press. Reading, Ma.

Wolfram, S. 2002. A New Kind of Science. Wolfram Media Inc. Champaign, IL. See chapters 2 and 3 for more information on 1 Dimensional CA See pages 25, 224, 694, 869-870, 952,1089 for information specifically on Rule 250.


If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

Please cite the NetLogo software as:


Copyright 2002 Uri Wilensky.


This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at

This model was created as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227.

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