Hiperbolična geometrija in regularna tlakovanja

magistrsko delo

Barbara Arcet (Author), Matej Mencinger (Mentor), Simon Špacapan (Co-mentor)

Abstract

Magistrsko delo obravnava hiperbolično geometrijo in regularna tlakovanja v njej. Hiperbolična geometrija je ena izmed treh možnih geometrij ob evklidski in sferični. V tem magistrskem delu si podrobneje ogledamo regularna tlakovanja, t.j. pokritja ravnine s samimi skladnimi pravilnimi mnogokotniki. V tem pogledu smo v evklidski in sferični geometriji precej omejeni, saj v prvi obstajajo le trije primeri regularnih (platonskih) tlakovanj, v drugi pa pet. V hiperbolični geometriji jih obstaja neskončno. Magistrsko delo je organizirano v tri dele. Najprej spoznamo osnove hiperbolične geometrije, opišemo njen razvoj skozi zgodovino ter si ogledamo tri modele za njeno predstavitev. Drugi del je namenjen lastnostim hiperbolične geometrije ter njeni primerjavi z evklidsko in sferično geometrijo. Definiramo razdaljo med poljubnima točkama, ogledamo si ukrivljenost ploskve in kote trikotnikov v vseh treh geometrijah. Seznanimo se s Pitagorovim izrekom, zapisanim tudi v okviru sferične in hiperbolične geometrije. V zadnjem delu se osredotočimo na regularna tlakovanja v vseh treh omenjenih geometrijah ter predstavimo nekaj matematičnih umetnij neevklidske geometrije nizozemskega umetnika M. C. Escherja, katerih osnova so prav regularna tlakovanja v hiperbolični geometriji. Razmislimo tudi o razlogih in možnostih vpeljave hiperbolične geometrije v šolski pouk.

Keywords

magistrska dela;hiperbolična geometrija;regularna tlakovanja;

Data

Language:	Slovenian
Year of publishing:	2017
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[B. Arcet]
UDC:	514(043.2)
COBISS:	23438344
Views:	1096
Downloads:	201
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Hyperbolic geometry and regular tilings
Secondary abstract:	Master's thesis focuses on hyperbolic geometry and regular tilings in it. Hyperbolic geometry is one of the three classical geometries next to Euclidean and spherical. In this thesis we deal with regular tilings or tesselations of the plane with regular and pairwise congruent tiles. Only few regular tilings in Eucledean and spherical geometry exist: in the first one three, and in the second one five. In hyperbolic geometry there are infinitely many possibilities. The thesis is organized in three parts. First, we define hyperbolic geometry and basic concepts. We describe its development through the history and present three models of it. Then we write about the properties of hyperbolic geometry and compare them with Euclidean and spherical geometry. The distance between two arbitrary points is defined, we take a look at the curvature of the surface, inner angles of the triangles in all of the geometries. Finally we prove the Pythagorean theorem which is in hyperbolic and spherical geometry a little bit different than in Euclidean. In the last part we focus on regular tilings in all three geometries and present the work of the Dutch mathematical artist M. C. Escher, who was using exactly regular tilings in hyperbolic geometry. At the end we present some ideas about motivation and reasons for introduction of hyperbolic geometry at school.
Secondary keywords:	master theses;hyperbolic geometry;regular tilings;
URN:	URN:SI:UM:
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	IX, 68 f., [2] f. pril
ID:	10867489