Muddy ants

I was watching my real ants today. Strangely, it is very obvious in a complex solution of topological paths between goal and source, that it is a successive approximation. It is very interesting to watch over time as the paths approach optimal. What looks like randomness combines to perform solution to a function, which is shortest path. What I would like to determine is which method they use. Is it: least squares, Taylor series, Fourier or more likely a mathematical method designed by 4 billion years of evolution. I would guess that the 3 dimensional optimum path function would be the most determinant factor in the relative success of predator-prey or foraging.

How does it derive that number and how do I derive that derivation. The sum of their apparent random wandering eventually resolves to a line that is the minimal path without having access to a calculator (AFAIK). I suppose I will apply all the known path approximation methods individually and in combination to see if there is a match. If there is no match then that makes it all the more interesting and so I can analyze to see if there is a new technique in the process they represent and this would ultimately be expressed in some neural arrangement that itself was a path that is compressed to a molecular sequence. It certainly is complex, however I have an advantage as I can simulate and I do have a calculator and a video recorder.

The odd thing is, that it almost looks like a sum of a series of complex prime frequencies. That actually would make some sense.

ADDED: Something odd is that I think this solves something I had been unable to solve before. It is not a waggle dance like bees, but is apparently vector math communication in chemicals. I can lay down chemicals in density so that they point and if you just follow a chemical by intensity it can cause a vector product reaction. Actually it could be seen in the same way as arrow signs on a highway that say "This way >>> to the restaurant at the end of the universe". I could even imagine that they theoretically could lay down trails with hooks and other features to indicate "Danger Ahead!". A kind of universal sign language for ants.