On 3/6/2014 7:37 PM, stumblin' in wrote: > I am working under the premise that the 4 color theorem works. > Working with the new unit pattern, > > a b > c d > > presents its own challenges and peculiarities. > For instance, > When the 2nd and 3rd countries unite, > > a b a b a b a b... we could do --> > > a b b a b a b a... swap the rest of the row or we could do --> > > b a a b a b a b... > > If other countries unite on the same row that could require > another swapping similar to above... > > If countries unite with countries on other rows... > It changes the whole landscape... > > Whoever said the 4 color theorem was just a tip of an iceberg was right... > There seems no end in sight... > Maybe there never will be... >
It is the four colors but mostly the three. It is the completions on the spherical that are five colors but those are not planar. The build in the four and the three, is in values that as a scale would work out easier to named ranges and classifications then as to interpolation.
Maybe if where the planar maps were also adjacent when they shared vertices besides when they shared borders, then more regions would be adjacent. For example, a pi chart could have any number of categories, all sharing a point. In systems and classifications where it is relevant to figure out how many colors of paint to mix, or rather indicators in an atomic alphabet then as to color for interpretation, planar adjacency on the map coloring, and that compared to a language L* of all the colors other regions of the map may well or not need that many other colors to have their own (for example a border around it, here in the planar of the graph). Then, components together and having so many colors, having a boundary around them all, adds the color of the boundary, where they are all adjacent in the middle the boundary is all of them and each has their own. Then, the four color problem would be a case of that, that with only edge connections as how the borders work out then that there are only adjacencies of edges and not vertices alone, that it is four colors in the planar. Basically the vertex is a clear region or colorless there.
Seems put more numbers at it on the graph characteristics simply, they'd simply build more numbers of their graph characteristic. Probably map to categories, too, probably already does.
Then the point of the academic is finding the right academic that (for example) has developed then what are for example the values of these, then for example of natural observations where there are chiralities, etcetera.
Then, building more regions into the map, that aren't colored, here could work to partitioning the map, that then in the planar, as those are closed in circuits, that all the subgraphs would be connected orthogonally from the borders. Then, any neighbor to the clear regions would have a two-color proof about them instead of a three-color proof.