Good fences make good neighbors, and it would be nice to minimize the amount of fence you need to build and maintain. Assuming WORLD DOMINATION, partition a sphere into N equal area regions minimizing the perimeter of the regions. This is a problem that has been posed and studied by others. As N increases, most of the regions will probably be hexagons.
The problem generalizes: a farmer has a finite plot of land on a Euclidean plane to divide among N heirs in the following unequal ratio... Other manifolds, dimensions...
In general, an absolute minimum appears to be difficult to find or prove. What is an approximation algorithm that strictly enforces the area constraints but may not find the absolute minimum perimeter?
No comments :
Post a Comment