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# examples of initial states in the Game of Life

Theoretically, the Game of Life is played on an infinite two-dimensional board. In practice, it is played on a finite board and only a small portion of the finite board is shown. Some implementations even use wrap-around (e.g., a spaceship configuration that moves horizontally to the right that hits $(+max,y)$ is moved to $(-max,y)$).

In most of the examples here, whatever positions are not shown are assumed to be empty. Living cells will be represented by the letter “C” until I think of a better symbol or better graphics.

Another theoretical element of the game is that after the user sets the initial state of the game, no further input is accepted; the game proceeds entirely as a consequence of Conway’s rules. In practice, some implementations allow the user to stop a game in progress, add or delete some cells and resume the game.

The smallest possible ‘living’ configuration possible is a 2 by 2 square.

C | C | ||

C | C | ||

Since each of the four cells has exactly three neighbors, they live forever (any larger, filled-in square or rectangular configuration is doomed to extinction in two iterations). This might not seem terribly useful, even for modelling zero population growth, but for game aficionados, these static blocks can be useful as decorative elements or as “roadblocks” to keep a motile configuration from running outside of the visible board.

This configuration is slightly more interesting than the 2 by 2 square, but it still stays put in one place.

C | C | C | ||

Initial state and even-numbered iterations (or odd-numbered iterations only)

C | ||||

C | ||||

C | ||||

Odd-numbered iterations (or initial state and even-numbered iterations)

Perhaps the most basic motile configuration is the glider. With the orientation shown below, it moves southwest indefinitely (unless it hits a block or a blinker that nudges it in a different direction or kills it altogether).

There are many implementations of the Game of Life available in C++, Pascal, Mathematica (version 5.0 of which even provides a built-in command for it), etc. On the Web, Life is usually implemented in Java. It is possible to implement it in JavaScript of VBScript, but these days such implementations are likely to crash the browser.

Of the Java implementations I’ve seen, the one I like the best is Mirek’s Java Cellebration v.1.50. Select ”Life” from the top drop-down menu and ”Conway’s Life” from the second drop-down menu. You can draw your own configurations, start and stop, step through, etc. By default, the board is finite and wraps around.

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## Comments

## another way of illustrating life states

You can use PSTricks to make graphics to illustrate states

in the game of life.

If you add the lines

\usepackage{pstricks}

\newcommand{\smallbox}{\psframe*(0,0)(0.9,0.9)}

to your preamble, all you need to do is make a \pspicture

of the appropriate size and \put copies of \smallbox

wherever you want.

For example, to illustrate the glider, you can do

\begin{figure}[hh]

\begin{center}

\begin{pspicture}(0,0)(5,5)

\put(1,1){\smallbox}

\put(2,1){\smallbox}

\put(3,1){\smallbox}

\put(1,2){\smallbox}

\put(2,3){\smallbox}

\end{pspicture}

\caption{The initial state of a glider.}

\end{center}

\end{figure}

## Re: another way of illustrating life states

It works. Thank you very much for your suggestion. I'll let the grammar soak up in my mind overnight. Tomorrow I will change the others and maybe add more examples.