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<html><head>
<title>Fractal Geometry Summer Workshop</title></head>
<body bgcolor="WHITE">
<h1 align="center">Two-Dimensional CA Examples</h1>
<p align="justify">Here are a few examples of patterns grown by two-dimensional
CA. Click the small picture for a larger version.
</p><p align="justify">First, the gasket can be grown from an initial condition
consisting of a single live cell. Can you find rules to grow gaskets in
different orientations?
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx1R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx1.gif"><img src="2DCAExamples_files/2DCAEx1S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">From a single live cell, this rule grow a pattern of concentric
diamonds that fill to a maximum density, then empty and refill. The far right
picture is a time record: the middle cross-section of the two-dimensional
picture (horizontally) vs time (vertically, generations increase downward). Note
a familiar pattern.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx2R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx2.gif"><img src="2DCAExamples_files/2DCAEx2S.gif" width="65" height="65"></a>
</td><td><img src="2DCAExamples_files/2DCAEx2CS.gif" width="144" height="93">
</td></tr></tbody></table>
</p><p align="justify">Another example, grown from a single live cell.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx3R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx3.gif"><img src="2DCAExamples_files/2DCAEx3S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">Now three examples, grown from a single live cell, for Moore nbhd CA.
In the first, the central region freezes and the growth occurs only along the periphery.
The others continue to change throughout.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx4R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx4.gif"><img src="2DCAExamples_files/2DCAEx4S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx5R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx5.gif"><img src="2DCAExamples_files/2DCAEx5S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx6R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx6.gif"><img src="2DCAExamples_files/2DCAEx6S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">Now we give just a hint of sensitivity in two-dimensional CA.
First, here is the pattern evolving from a single live cell for this rule.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx7R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx7.gif"><img src="2DCAExamples_files/2DCAEx7S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">Delete the part of the rule under 0 and a homogeneous
pattern grows from a single live cell. From an initial condition of two
live cells separated by about 1/4 the state space size, the pattern on the
far right grows.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx8R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx8.gif"><img src="2DCAExamples_files/2DCAEx8S.gif" width="65" height="65"></a>
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx8D.gif"><img src="2DCAExamples_files/2DCAEx8DS.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">
</p><p align="justify">Now we consider several patterns grown from random initial
distributions of live and dead cells. First, the "majority rules" von Neumann
and Moore CA. (Do you see why these CA are called majority rules?)
The Moore CA pattern freezes completely; the von Neumann pattern
oscillates at the "checkerboard" regions.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx9R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx9.gif"><img src="2DCAExamples_files/2DCAEx9S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx10R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx10.gif"><img src="2DCAExamples_files/2DCAEx10S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">Here is another CA that evolves from random initial distributions
of live and dead cells to an organized pattern. This is an example of "self-organization"
in CA.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx11R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx11.gif"><img src="2DCAExamples_files/2DCAEx11S.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">This is another self-organizing CA. The second picture
is generated using the same rule, but with a different initial distribution of
live and dead cells. The pictures differ in fine detail, but not in
general form.
<table>
<tbody><tr><td><img src="2DCAExamples_files/2DCAEx12R.gif" width="290" height="65">
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx12f.gif"><img src="2DCAExamples_files/2DCAEx12fS.gif" width="65" height="65"></a>
</td><td><a href="http://classes.yale.edu/99-00/math190a/2DCAEx12s.gif"><img src="2DCAExamples_files/2DCAEx12sS.gif" width="65" height="65"></a>
</td></tr></tbody></table>
</p><p align="justify">Return to <a href="http://classes.yale.edu/99-00/math190a/CAPatterns.html">
4. C. Examples of Cellular Automaton Patterns</a>
</p><p align="justify">
</p></body></html>