{{Percolation Simulation}}
Let <math>c</math> be a 2D boolean square matrix of <math>n \times n</math> values of either <tt>1</tt> or <tt>0</tt> where the
probability of any value being <tt>1</tt> is <math>p</math>, (and of <tt>0</tt> is therefore <math>1-p</math>).
We define a ''cluster'' of <tt>1</tt>'s as being a group of <tt>1</tt>'s connected vertically or
horizontally (i.e., using the [[wp:Von Neumann neighborhood|Von Neumann neighborhood rule]]) and bounded by either <math>0</math> or by the limits of the matrix.
Let the number of such clusters in such a randomly constructed matrix be <math>C_n</math>.

Percolation theory states that <math>K(p)</math> (the mean cluster density) will satisfy <math>K(p) = C_n / n^2</math> as <math>n</math> tends to infinity. For <math>p = 0.5</math>, <math>K(p)</math> is found numerically to approximate <math>0.065770</math>...

;Task

Show the effect of varying <math>n</math> on the accuracy of simulated <math>K(p)</math> for <math>p = 0.5</math> and
for values of <math>n</math> up to at least <math>1000</math>.
Any calculation of <math>C_n</math> for finite <math>n</math> is subject to randomness, so an approximation should be
computed as the average of <math>t</math> runs, where <math>t</math> &ge; <math>5</math>.

For extra credit, graphically show clusters in a <math>15\times 15</math>, <math>p=0.5</math> grid.

Show your output here.

;See also
* [http://mathworld.wolfram.com/s-Cluster.html s-Cluster] on Wolfram mathworld.