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<html><head>
<title>Fractal Geometry Summer Workshop</title></head>
<body bgcolor="WHITE">
<h1 align="center">Outer Totalistic Cellular Automata Rules</h1>

<p align="justify">For simplicity we consider only <a href="http://classes.yale.edu/99-00/math190a/StateNumber.html">binary</a> CA.
The extension to S &gt; 2 is straightforward.


</p><p align="justify"><b>Outer totalistic</b> CA rules depend on only the number of live cells
in a neighborhood, with the additional condition that we allow different outcomes if the
<a href="http://classes.yale.edu/99-00/math190a/Nbhd.html">central cell</a> is alive or dead.  For one-dimensional, N = 3 rules
the nbhd around the central cell can have 0, 1, or 2 cells alive.  Thus S = 2, N = 3
outer totalistic CA rules have six nbhd configurations:

</p><p align="justify">1. Central cell dead, 0 surrounding cells alive

</p><p align="justify">2. Central cell alive, 0 surrounding cells alive

</p><p align="justify">3. Central cell dead, 1 surrounding cell alive

</p><p align="justify">4. Central cell alive, 1 surrounding cell alive

</p><p align="justify">5. Central cell dead, 2 surrounding cells alive

</p><p align="justify">6. Central cell alive, 2 surrounding cells alive

</p><p align="justify">


<table>
<tbody><tr>
<td>Here is how we tabulate the nbhd configurations.  The number
corresponding to the configurations is given below each box.
</td><td><img src="OT_files/OTN3.gif" align="right" width="340" height="89">
</td></tr></tbody></table>

<table>
<tbody><tr>
<td>For example, selecting the left box under 1 means if the central cell is alive and
exactly one of the two surrounding cells in the nbhd is alive, then the central cell
remains alive in the next generation.  The corresponding nbhd configurations are
colored red.  Selecting the right box under 1 means if the central cell is dead and
exactly one of the two surrounding cells is alive, then the central cell becomes
alive in the next generation.  The corresponding nbhd configurations are colored blue.
</td><td><img src="OT_files/OTN3Ex.gif" align="right" width="340" height="98">
</td></tr></tbody></table>

</p><p align="justify">Outer totalistic rules can be formulated for one-dimensional N = 5,
two-dimensional von Neumann, and two-dimensional Moore CA.

<table>
<tbody><tr>
<td><img src="OT_files/OTN5.gif" align="right" width="340" height="60">
</td></tr></tbody></table>

<table>
<tbody><tr>
<td><img src="OT_files/OTvN.gif" align="right" width="290" height="65">
</td></tr></tbody></table>

<table>
<tbody><tr>
<td><img src="OT_files/OTM.gif" align="right" width="290" height="65">
</td></tr></tbody></table>

</p><p align="justify">Here is an example of an Outer Totalistic rule for a
von Neumann neighborhood.


<table>
<tbody><tr>
<td>Checking the left box under 1 means the central cell is alive and exactly one
of the four surrounding cells is alive.  The corresponding neighborhoods are shown
in blue.

<p align="justify">Checking the right box under 2 means the central cell is
dead and exactly two of the four sorrounding cells are alive.  The corresponding
neighborhoods are shown in red.

</p></td><td><img src="OT_files/OTvNEx.gif" align="right" width="384" height="189">
</td></tr></tbody></table>


</p><p align="justify">Continue to <a href="http://classes.yale.edu/99-00/math190a/CABasics.html">
4. B. Cellular Automata Basics</a>

</p><p align="justify">

</p></body></html>