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...)
The four conic sections
arise when a plane
cuts through a double cone
in different ways. If the plane cuts through parallel to the side of the cone (case 1), a parabola
results (to be specific, the parabola is the shape of the planar graph
that is formed by the set of points of intersection of the plane and the cone). If the plane is perpendicular to the cone's axis of symmetry
(case 2, lower plane), a circle
results. If the plane cuts through at some angle between these two cases (case 2, upper plane) — that is, if the angle between the plane and the axis of symmetry is larger than that between the side of the cone and the axis, but smaller than a right angle
— an ellipse
results. If the plane is parallel to the axis of symmetry (case 3), or makes a smaller positive angle with the axis than the side of the cone does (not shown), a hyperbola
results. In all of these cases, if the plane passes through the point at which the two cones meet (the vertex), a degenerate conic
results. First studied by the ancient Greeks
in the 4th century BCE
, conic sections were still considered advanced mathematics by the time Euclid
300 BCE) created his Elements
, and so do not appear in that famous work. Euclid did write a work on conics, but it was lost after Apollonius of Perga
c. 190 BCE) collected the same information and added many new results in his Conics
. Other important results on conics were discovered by the medieval Persian mathematician Omar Khayyám
(d. 1131 CE
), who used conic sections to solve algebraic equations