20.500.12556/DKUM-68638 Hiperbolična geometrija in regularna tlakovanja Hyperbolic geometry and regular tilings Magistrska naloga obravnava hiperbolično geometrijo in regularna tlakovanja v njej. Hiperbolična geometrija je ena izmed treh možnih geometrij poleg evklidske in sferične. 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. 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. hiperbolična geometrija regularna tlakovanja hyperbolic geometry regular tiling true true false [B. Arcet] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2017-10-03 11:09:18 2017-11-08 11:00:57 2022-08-01 16:08:30 2019-07-11 16:59:44 2017 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 514(043.2) 23438344 URN:SI:UM:DK:ONLUVSBA MAG_Arcet_Barbara_2017.pdf MAG_Arcet_Barbara_2017.pdf 1 A933DD32646DB9A6EF3C7DEA1014797B b59380bdefd8253fbaf1e56e43e80c848108e969552631185c63990723268145 66a54934-7c0f-11eb-bb7a-00155d0001ca 20.500.12556/dkum/93a4edfc-1170-41a3-ad02-7d57fd8bd0b3 https://dk.um.si/Dokument.php?lang=slv&id=119287 Fakulteta za naravoslovje in matematiko 0 0 0