The four color theorem has been proved. It takes only four hues to color any 2 dimensional map of any kind. No matter how the area is subdivided, you can color the resultant map with four hues so that no two hues of the same type are touching.
I have noticed that in one dimension, such as my check ledger, it only takes three hues to color any one dimensional "map," or configuration of subdivisions. This is an anectdotal observation.
It seems that the number of hues required to color any type of subdivision of a space so that no two sides are touching with the same hue is (Number of dimensions + 2) For a two dimensional space it requires four hues. For a three dimensional space, it requires five hues. For a five dimensional space it requires seven hues to color any configuration of subdivisions. Note: the different dimensions should all be on orthogonal axes.
Number of hues = Number of dimensions + 2
Note: I have used the term "hue" to refer to the noun of "tint, shade, or color." I reserve the word "color" to be used as a verb describing an action taken. This is just to avoid literary confusion and make the ideas more clearly described.
Note: The check ledger example for one dimension allows discontinuous countries (i.e. one country can be surrounded by another country). If we disallow discontinuous countries, then the equation becomes Number of hues = Number of dimensions + 1. However, it seems that in the real world there are discontinuous countries, so the original equation is still good.
copyright (c) 2017
William Schaeffer
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