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Example of a four color map

The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point.

It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.

The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.

The lack of mathematical elegance was another factor, and to paraphrase comments of the time, "a good mathematical proof is like a poem — this is a telephone directory!"

Picture of the month

In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. The fifth postulate or parallel postulate is equivalent to:

Given a line and a point not on that line, exactly one line can be drawn through that point which does not intersect the original line (see 1).

In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example elliptical geometry:

Given a line and a point not on that line, all lines drawn through that point will intersect the original line (see 2).

And hyperbolic geometry:

Given a line and a point not on that line, an infinite number of lines can be drawn through the point that do not intersect the original line (see 3).

These other forms of geometry, where the parallel postulate does not hold are called Non-Euclidean geometry.

Categories

Did you know...

• ...that every natural number can be written as the sum of four squares?
• ...that the largest known prime number is over 12 million digits long?
• ...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
• ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
• ...that it is unknown whether π and e are algebraically independent?
• ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• ...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.
• ... that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
• ...that the primality of a number can be determined using only a single division using Wilson's Theorem?

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