Vizualizacija i povijest hiperboličke geometrije

U ovom diplomskom radu opisan je povijesni razvoj geometrije, posebno hiperbolične geometrije, te vizualizacija hiperbolične geometrije pomoću fizičkih modela, s naglaskom na kukičanim modelima. U poglavlju o povijesnom razvoju geometrije, počinjemo od samog nastanka geometrije motiviranog potrebama iz svakodnevnog života tadašnjeg čovjeka. Nastavljamo s razvojem euklidske geometrije i problemom petog Euklidovog postulata. Pokušaji dokazivanja petog Euklidovog postulata pomoću prva četiri često su bili na pragu otkrivanja hiperbolične geometrije. Navodimo i ukratko opisujemo povijest neeuklidskih geometrija općenito. Pravi preokret u razumijevanju petog postulata dogodio se u 19.stoljeću, kada su, neovisno jedan o drugome, Bolyai i Lobačevski razvili geometriju u kojoj peti postulat ne vrijedi, kasnije poznatu kao hiperbolična geometrija. Kasnije Riemann hiperboličnu geometriju povezuje s negativnom zakrivljenošću. Javlja se problem vizualizacije hiperbolične geometrije o kojem više u drugom poglavlju. Drugo poglavlje započinjemo vizualizacijom hiperbolične ravnine modelima u euklidskoj ravnini. Takvi su modeli zapravo mapa (karta) i ne čuvaju sve karakteristike hiperbolične ravnine. Zato je potrebna vizualizacija u euklidskom prostoru. Najprije navodimo modele od papira i način njihove izrade. Takvi modeli, zbog materijala od kojeg su izrađeni, imaju kratak vijek trajanja i teško je na njima vizualizirati određene karakteristike hiperbolične ravnine. Nastavljamo s modelima koji nemaju te nedostatke, a to su kukičani modeli. Nakon opisivanja načina izrade kukičanih modela hiperbolične geometrije, praktičnom aktivnošću presavijanja otkrivamo svojstva tih modela i povezujemo ih s karakteristikama hiperbolične geometrije. Na kukičanim modelima hiperbolične ravnine promatramo: paralelne i okomite pravce, kutove trokuta i mnogokute. Osim kukičanja hiperbolične ravnine u radu je opisano i kukičanje simetrične hiperbolične ravnine i kukičanje pseudosfere.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