# Hyperbolic Geometry using Cabri

by Tim Lister, t.c.lister@open.ac.uk Last updated:  24/09/01

 A tessellation of the hyperbolic plane H2 (the Poincaré unit disc model) by (2, i, i) triangles, that is, with angles (90, 0, 0). Every triangle has two vertices on the disc boundary (at infinity). The diagram was built up from an initial triangle with a vertex at the centre, by reflection (inversion in a disc line) about one of its sides. A line in this model is the arc of circle orthogonal to the disc boundary. Parts of the tessellation are shown in varying degrees of layers in each of the quarters.

 During the summer of 97 I had great fun playing with some marvelous software, Cabri Geometry  , and devising constructions for use in teaching the basic ideas of a geometry course put on by the Open University. These started with some figures to demonstrate the transformations of Inversive Geometry, and progressed to figures for the Arbelos, the inversors of Peucellier and Hart, Coaxial Circles and so on, much of which was driven by the discovery of a Dover edition of a small pearl of a book "Advanced Euclidean Geometry (Modern Geometry) An elementary Treatise on the Geometry of the triangle and the Circle" (to give its full title) written by Roger A. Johnson and first published in 1929. It had languished on my bookshelves, having been bought years ago for 20 cents (South African) in some sale or other. I can recommend it as a fascinating read, or just for taking in the breathtaking complexity of the many hand crafted diagrams to be found on its pages.  Compared to these mechanic and static drawings the beauty of a Cabri figure is that you can "grab and drag" various objects in it and this movement of the figure often gives a clear or most certainly new insight into the underlying result or concept. The Cabri icon shown above, with its "grabbing hand", perfectly captures the major strength of this program. It was with this in mind that I started to construct a series of Cabri macros and an ever growing menu for constructions for hyperbolic (or non-Euclidean) geometry in the Poincaré disc model. The menu commands can be used to draw figures that illustrate some of the fascinating results and figures to be found in the hyperbolic plane.

 Download the hyperbolic geometry menu, plus notes for its use. The latest innovation is Cabri Java. Fully interactive Cabri figures displayed on the page via a Java applet. d-lines in the Poincare disc, parallels and ultra parallels Tiling the disc with regular triangles non Euclidean distance Talk given at CabriWorld 2001, Montreal
The Cabri Java Handbook