Euclidean networks


Networks that approximate Euclidean geometry are useful. Many games use a square grid (which can be seen as a network in which each node connects to four others) to approximate Euclidean geometry. A square grid has the drawback that diagonals are too long -- moving at 45 degrees, you need to travel |x|+|y| steps rather than sqrt(x^2+y^2).

Some games use a hexagonal grid, which is better though not perfect. A hexagonal grid is probably the best approximation to Euclidean geometry that you can get with a simple regular network.

It occurs to me (and has without doubt occured to others, but hey) that a somewhat random grid could be made to approximate Euclidean geometry arbitrarily well, at least over long distances.

Could be useful for loop quantum gravity and all that (again, this is no-doubt well known to physicists, but it's new to me and this is my blog). Reality could be a network of this type, and Euclidean geometry the approximation (or rather Einsteinian continuous not-quite-Euclidean geometry).