The study of space 1............ with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that 2................ relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries, and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which 3................ structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics, and 4................ the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been 5................... by a human.

 

Example of a four-colored map

A four-coloring of an actual map of the states of the United States (ignoring water and other countries).

 

1.      Attitude of mapmakers

2.      Which number of colors is enough?

3.      Second proof using computers

4.      Theorem proven, doubts remain

5.      What is it about?

A. In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. Two regions are called adjacent only if they share a border segment, not just a point.

B. Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century, however, proving four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.

C. Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May (Wilson 2002, 2), “Maps utilizing only four colours are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property.”

D. The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proven using a computer. Appel and Haken's approach started by showing there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a special-purpose computer program to check each of these maps had this property. Additionally, any map (regardless of whether it is a counterexample or not) must have a portion that looks like one of these 1,936 maps. To show this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples existed because any must contain, yet not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and the theorem is true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand.

E. To dispel remaining doubt about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally in 2005, the theorem was proven by Georges Gonthier with general purpose theorem proving software.

a)      5

b)      1976

c)      1,936

d)      1852

e)      2005

f)        1997

 

1.      What are adjacent regions?

2.      What kinds of maps need four colors?

3.      When was the five color theorem proven?

4.      Is the four color theorem used by mapmakers?

5.      How was the four color theorem proven?

6.      Why was the proof unacceptable?

7.      Which methods of proving the theorem vere used in 2005?

 

In the opinion of Kenneth May, ..........................

Some mathematicians doubted..............................

Under the conclusion of Appel and Hanken.............................................

The theorem is not of interest ................................................................

Robertson, Sanders, Seymour, and Thomas dispeled the doubts......................................