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!"
This is a graphical construction of the various trigonometric functions from a unit circle centered at the origin, O, and two points, A and D, on the circle separated by a central angle θ. The triangle AOC has side lengths cos θ (OC, the side adjacent to the angle θ) and sin θ (AC, the side opposite the angle), and a hypotenuse of length 1 (because the circle has unit radius). When the tangent line AE to the circle at point A is drawn to meet the extension of OD beyond the limits of the circle, the triangle formed, AOE, contains sides of length tan θ (AE) and sec θ (OE). When the tangent line is extended in the other direction to meet the line OF drawn perpendicular to OC, the triangle formed, AOF, has sides of length cot θ (AF) and csc θ (OF). In addition to these common trigonometric functions, the diagram also includes some functions that have fallen into disuse: the chord (AD), versine (CD), exsecant (DE), coversine and excosecant (under point F). First used in the early Middle Ages by Indian and Islamic mathematicians to solve simple geometrical problems (e.g., solving triangles), the trigonometric functions today are used in sophisticated two- and three-dimensional computer modeling (especially when rotating modeled objects), as well as in the study of sound and other mechanical waves, light (electromagnetic waves), and electrical networks.