## Sunday, April 08, 2007

### Code testing completed.

I needed to adjust the iterations down to 500 for the interactive plots for the 4 dimensional random walk but now all ten programs have tested as working at least once. I now realize I do not need four 1 dimensional random walks and in 1 dimension there is no diagonal v square walking. So for 1 dimension I just need an interactive and non interactive walk meaning I need only 14 programs.

I am tempted to make a 3 dimensional random walk that walks to all points in an origin centered cube. In other words the next step can be any combination of $(-1,0,1)$ for any of the three axis. Thus giving 6 choices with walking towards a face, 8 choices of walking to a corner, and 12 choices of walking to an edge. This could be $p=\frac{1}{26}$ or in my alternative version we would weight the probability based on the vector length of the walk. The probability would be inversely related to the vector length of the step. Thus setting the probability of walking to a face as unity with probability $p_{f}=\frac{1}{u}$ and then set the others to the inversion of their vector lengths. Thus walking along a diagonal would be walking to an edge and would have a probability of $p_{e} = \frac{1}{u \sqrt{2}}$ and walking to a corner would have probability $p_{c} =\frac{1}{u \sqrt{3}}$.

Then a basic law of probability would be used to create this equation $1= 6 \frac{1}{u} + 12 \frac{1}{u \sqrt{2}} + 8 \frac{1}{u \sqrt{3}}$ which would solve for

$u=6+12 \frac{1}{\sqrt{2}} + 8 \frac{1}{\sqrt{3}} = 19.10408353$ .