GeoGebra Classroom

Outline

# A dynamic visualization of the hyperbolic geometry

“I have created a new, different world from nothing...” (János Bolyai) Most of the primary and secondary school curriculum is based on Euclidean geometry. Our students are somewhat familiar with the life and works of János Bolyai’s, but his geometry, what is more, non-Euclidean geometries in general are completely unknown even for most secondary or university students. Since it is impossible to study the topic precisely in an axiomatic way, teachers do not even exploit any opportunities for abstraction. In fact, if we can make non-Euclidean geometry visual, we can even enlighten the basic concepts of the “well-known” Euclidean geometry, including some simpler statements of it. We shall see that the well-known Euclidean geometry and Bolyai’s geometry, being visualized here, have common roots: all concepts before defining the axiom of parallels are valid in both geometries. In this way we can use the model presented here to provide a new approach to teach elementary geometry in secondary school as well. This GeoGebra Book uses only the basics from secondary school in order to define the main concepts of hyperbolic geometry. We do not want to explain general knowledge on non-Euclidean geometries. Instead, a visual picture book is presented to invite the reader to learn more on the topic. Let us see then now, how Bolyai’s “new, different world” looks like! Note: This book is the English translation of a Hungarian GeoGebra book on the same topic. The translation is an on-going work, so please expect several missing translations. However, all applets are already translated. In case the English text is missing, you may be interested in the original Hungarian text and try to translate it automatically in English, e.g. with Google Translator.