## In the End – Just a Bunch of Pretty Pictures

I recently worked a problem that required counting the number of paths to a node in a lattice.  For something to do, I decided to try to produce a visualization of how the number of paths increases with increasing distance from the starting node – the origin.  Here is an example of a  path from the origin to the the point (5,5,5) created with Mathematica.

Random Path Through a 5x5x5 Lattice

The path randomly decides to increase one unit in either the x or the y or the z direction with equal probability.  The total number of paths to (5,5,5) is $\frac {(5+5+5)!}{5!5!5!}$.

I first thought to use gephi, a program that produces graph theory graphs but I couldn’t get the hang of it and managed this graph before I gave up.

Gephi Depiction of Lattice

This depiction uses  YifanHu’s Multilevel scheme.  This scheme shows the three dimensional structure but is not fully rotatable.

So I turned to Mathematica to produce my own drawings.  A first try produced this.

5x5x5 Lattice – Node Size equals Number of Paths

And then this.

5x5x5 Lattice – Node Size equals Number of Paths – Another Example

The node size corresponds to the number of paths to that node.
The pictures are not very informative obviously because the number of paths is a factorial expression.  So I decided to use the logarithm of the number of paths.  Thus this,

5x5x5 Lattice – Log Node Size equals Number of Paths

And this with color coding for number of paths added.

But now I have lost all sense of the size of the increase since I am using logarithms. I ended my exploration here.

Random Paths Through a 5x5x5 Lattice

I was thinking of creating an animation but to what end.  I have five classes to occupy me now. So just some pretty? pictures.