Vizualizacija i povijest hiperbolične geometrije

Pugar, Ana (2014) Vizualizacija i povijest hiperbolične geometrije. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

In this diploma thesis the historical development of geometry and especially that of hyperbolic geometry is presented, as well as hyperbolic geometry visualization with the aid of physical models, and with emphasis on crochet models. In the chapter on the historical development of geometry, we start with the very beginnings of geometry motivated by the everyday needs of the past human life. We continue by giving an overview of the Euclidian geometry and the problem of Euclid’s fifth postulate. Attempts at proving Euclid’s fifth postulate by referring to the first four have often been on the verge of revealing hyperbolic geometry. We present and briefly describe the history of non-Euclidian geometries in general. The real shift in understanding the fifth postulate occurred in the 19th century when Bolyai and Lobachevsky independently developed a geometry in which the fifth postulate was invalid, which later became known as hyperbolic geometry. Later Riemann connected hyperbolic geometry with negative curvature. Thus the problem of hyperbolic geometry visualization appears, which will be dealt with in the second chapter. The second chapter starts with a visualization of a hyperbolic plane with models in the Euclidian plane. Such models are actually a map and do not represent all of the characteristics of the hyperbolic plane. That is why a visualization in the Euclidian space is needed. We first present the paper models and their assembly. Such models, because of the material from which they are made, do not last long and it is difficult to visualize certain characteristics of the hyperbolic plane on them. We continue by presenting models which do not have these shortcomings, i.e. crochet models. After a description of the fabrication process of hyperbolic geometry crochet models, through the practical activity of folding we reveal the properties of such models and connect them with the characteristics of hyperbolic geometry. On the crochet models of hyperbolic plane we observe: parallel and vertical lines, triangle and polygon angles. Apart from crocheting a hyperbolic plane, the thesis describes crocheting of symmetric hyperbolic planes and crocheting a pseudosphere

Item Type: Thesis (Diploma thesis)
Supervisor: Brückler, Franka Miriam
Date: 2014
Number of Pages: 46
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 19 Jun 2015 12:19
Last Modified: 19 Jun 2015 12:19
URI: http://digre.pmf.unizg.hr/id/eprint/4066

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