The traveling salesman problem

 Came up in Quanta, a fun one.

Let me say we have a euclidean flat map wit a finite number of cities, known in location to sufficient precision to solve the shortest route problem. Now, I construct a radial resolution based on the two closest cities.  I call that minimal resolution. I can increase that radius and find the count of cities within that lower resolution, counting the previous two as one. city so there is no double count. And repeat until the finite cities are marked.

The count in each resolution band becomes a probability, or proportion, really. If this were a finite element summation of these as an arbitrary set of dots, then I wold be adding up areas with fast rate of change more often, in smaller amounts. I am trying to get a Huffman map of the cities that at least, breaks up the routes by sets having a common visitation counts. I want minimum redundancy in the number of time I add stuff, so breaking the cities into set of finite size all within a quanta of resolution is maximum entropy.

Thus, my Huffman tree can take any city, encoded into its resolution, and find the number of visits it is going to have, not necessarily in the right order. But it can certainly limit the number of possibilities.

I think it works out to a solution on the Markov 3 tuples, up the 1,y,z axis. The z are the steps forward, the y the return steps. But your city count does not make a Markov triple. So what, you end up with a small trail, a bit of the old round about, kinetic energy.

This was the deal years ago, all about finding an appropriate ruler but saying nothing about order or sequence. The traveling salesman problem as a ruler, find that first, then flatten that hyperbolic salad bowl onto a flat Lie paper, and try different rotations.  The ruler you made become little dials about the Lie center. You have many combinations, still, but a great deal less then you started, and you have a regular sequence you can use to find solutions.

Part two of the salesman problem

Work it backwards. This time we do not have lie paper, we have the real euclidean map. But way have found out Markov 3 tuple, and constructed a tiny salad bowl to use as a magnifying glass.  I take my little lens and move it about, and when centered on a city, the lens will magnify the map along my two residual resolutions, and I can read off shortest path directions from one city to another. It will maintain the proper resolution axis, you look more vertical to the salad bowl and you see out to a blurry distance. From any starting point you can pick the most likely shortest path.