Non-Euclidean Geometry in School

Margarita Spirova and Yulian Tsankov
Faculty of Mathematics and Informatics
University of Sofia, Bulgaria

Non-Euclidean geometries are a common subject of serious studies at
universities.  But by recently developed computer algebra systems this topic
can be made better understandable for students, and even suitable class-room
presentations in school can be easily prepared.  For instance, in [1] a method
for studying Euclidean and hyperbolic geometry in the spirit of Cayley-Klein is
discussed, using the computer system Cinderella.

In our talk we present some computer programs created by Mathematica.  These
programs might support the study, even in school, of the simplest of all
Cayley-Klein geometries, namely the isotropic geomety.  We think that this
geometry is very useful for a general understanding of non-Euclidean
geometries.  In order to introduce the basic notions of the isotropic plane
(cf. [2]),  one needs only knowledge about coordinate systems, whereas in the
hyperbolic plane it is necessary to introduce notations like complex numbers,
infinite elements, homogeneous coordinates, the cross ratio and so on.  The
presented programs have two aims:

- to visualize the basic objects in the isotropic plane (segments, angles,
  circles,...) which are different to their analogues from usual Euclidean
  geometry,

- to "prove" in basic relations concerning a triangle in the isotropic plane in
  an experimental way (the law of sines, etc.).

In this way the (non-Euclidean) isotropic geometry would be not only
understandable for students with special interests in mathematics, but also for
a wider circle of persons (e.g., for pupils in school).

[1] Kortenkamp, U., Richter-Gebert, J.: Euclidische und nicht-Euklidische
    Geometrie in Cinderella, J. Math.-Didaktik (2002), no. 3/4, p.301-322.

[2] Sachs, H.: Ebene isotrope Geometrie, Vieweg, Braunschweig-Wiesbaden, 1987.