3.1 The notion of incidence geometry
3.3 Examples of incidence geometries
3.3.1 Some basic examples of incidence geometries
3.3.2 The main incidence geometries
3.3.3 Generalizing the real cartesian plane
3.5 Behavior of parallelism in our examples
4.1 Betweenness structures, segments, triangles, and convexity
4.2 Separation of the plane by a line
4.3 Separation of a line by one of its points
4.6 Betweenness structure for the real cartesian plane
4.7 Betweenness structure for the hyperbolic plane
5.1 Congruence of segments structure and segment comparison
5.2 The usual congruence of segments structure for the real cartesian plane
5.3 The usual congruence of segments structure for the hyperbolic plane
6.1 Congruence of angles structure and angle comparison
6.2 Angle congruence in our main examples
6.2.1 Congruence of angles in the real cartesian plane
6.2.2 Congruence of angles in the hyperbolic plane
1.1 A Short History of Geometry
It is safe to say that the first geometric facts recorded in human history are found within the Egyptian and the Babylonian civilizations. There is strong evidence suggesting that even the Pythagorean Theorem was well known to these civilizations. However, these discoveries were only empirical facts, geometrical regularities that seemed to occur in every case considered. From this evidence, they would come to believe that these were universally true statements, although it seems that nobody bothered to find out why these phenomena took place, or how to “prove” that they were indeed valid in any case. I was not until the Greeks that mathematicians discovered a trustworthy method to know for sure the validity or falsity of any given geometric statement. The method, known today as the axiomatic method, consists in first taking certain geometrical facts, called axioms, or postulates, or principles, as self-evident, and then, based only on them, and by means of pure reasoning, to derive any other geometrical truth. This is one of the most important inventions of humankind. It initiates mathematics as we understand it today, and provides the paradigm for half the scientific method, which is nothing else but the addition of experimentation to the axiomatic methodology.
Within the Greek world, the peak of maturity of the axiomatic method was attained with the “publication” of Euclid’s Elements. Euclid lived approximately between the middle of the fourth century B.C. and the middle of the third century B.C., mainly in Alexandria, in the Hellenistic part of Egypt. The Elements is a collection of thirteen books, containing an axiomatic development of plane and space geometry, elementary number theory and incommensurable lines. Until the beginning of the twentieth century, the Elements was the main textbook for teaching mathematics, especially geometry.
After its publication, various authors detected two weak points in Euclid’s work: The feeling that the fifth postulate was not as self-evident as the previous four, and that it should be derived from them; and, secondly, the occasional departure from modern standards of