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. "Color by Number" worksheets and exercises, which combine learning art and math for people of young ages, are a good example of the four color theorem.
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 disputed by some 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!" (Full article...)
An animation showing how an obliquely cut torus
reveals a pair of intersecting circles known as Villarceau circles
, named after the French astronomer and mathematician Yvon Villarceau
. These are two of the four circles that can be drawn through any given point on the torus. (The other two are oriented horizontally and vertically, and are the analogs of lines of latitude
drawn through the given point.) The circles have no known practical application and seem to be merely a curious characteristic of the torus. However, Villarceau circles appear as the fibers in the Hopf fibration
of the 3-sphere
over the ordinary 2-sphere
, and the Hopf fibration itself has interesting connections to fluid dynamics
, particle physics
, and quantum theory