0101 A system of axioms in geometry as introduced in the geometry class
The most commonly used axioms in geometry are classified as follows:
- incidence axioms,
- order axioms,
- congruence axioms,
- continuity (measuring) axioms,
- axiom of parallels.
The first four groups constitute the system of absolute geometry — as it was designated first by János Bolyai — that does not include any form of the axiom of parallels. Depending on which further axiom we choose as the axiom of parallels we can develop absolute geometry into Euclidean geometry or hyperbolic geometry. The latter was discovered by János Bolyai (1802-1860) and Nikolai Lobachevsky (1792-1856). Carl Friedrich Gauss (1777-1855) also made fundamental discoveries in this field, but did not publish his results. Consider three important statements that refer to three different kinds of geometry respectively (line means straight line in the given system):
- Lines perpendicular to a given line of the plane are not intersecting, thus there are non-intersecting lines, i.e. parallel lines in the plane.
- Given a line and a point outside it, there is at most one line through the given point which lies in the plane of the given line and point so that the two lines are parallel.
- Given a line and a point outside it, there are at least two lines through the given point which lie in the plane of the given line and point so that the two lines are parallel.
The first statement is a theorem that can be proven without the axiom of parallels. The second and third statements are the two forms of the axiom of parallels in Euclidean geometry and hyperbolic geometry respectively. They are negations of each other. In other words, it cannot be deduced from the axioms of absolute geometry how many lines can be drawn through a given point which are parallel to a given line in the plane. This question can only be settled by introducing a new axiom, but this step results in a split within the—previously unique—geometric system.
Actually, Bolyai elaborated two geometric systems: absolute geometry that did not use any form of the axiom of parallels, and hyperbolic geometry which admitted another form of the axiom. Lobachevsky also built up a system of hyperbolic geometry, independently of Bolyai’s work. Moreover, Bolyai showed that Euclidean geometry is a limiting case of hyperbolic geometry.
Most of our geometry teaching in secondary school is built on Euclidean geometry. Visualizing hyperbolic geometry—as we have done with Euclidean geometry—helps enlighten our students on “well-known” Euclidean notions and statements which do not require the axiom of parallels. The contrast between Euclidean and hyperbolic geometry leads to a different approach, a new perspective in the study of geometric concepts. This experience may have a strong impact on the perception of students and their level of abstraction.
In this book we give examples of this idea by using dynamic geometry software GeoGebra. All our illustrations are available as online interactive applets as well. The GeoGebra book illustrates all the topics in the following sections by introducing the Poincaré disk as an electronically visualized object, and by making available to create points, hyperbolic lines, segments, rays and circles in the hyperbolic plane, and to create additional objects by using various other tools consecutively. 