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PARALELNOST I HIPERPARALELNOST U HIPERBOLIĈKA GEOMETRIJA. PARALLELS
AND HYPER-PARALLELS IN HYPERBOLIC GEOMETRY
Uĉenik: TAMARA ŠEKULARAC, III razred Matematiĉka gimnazija, Beograd
Mentor: VESNA RAJŠIĆ, profesor matematike Elektrotehniĉka škola „Nikola Tesla”, Beograd
REZIME
Razvoj nauke podsticao je razvoj tehnike i doveo do napretka sveukupnog ĉoveĉanstva. Veliki broj
problema koji se postavljao pred tadašnji uĉeni svet i nauĉnike zahtevao je rešavanje mnogobrojnih
problema u raznim prirodnim naukama, a meĊu njima i u matematici i geometriji. Izuĉavanja u
oblasti geometrije stara su hiljadama godina. Pokušaji razrešenja pitanja o petom euklidskom
postulatu doveli su do osnivanja potpuno nove geometrije koja je ne protivureĉna. Ova nova
hiperboliĉka geometrija je taĉna koliko i euklidska geometrija. Dokazi do kojih se izuĉavanjem ove
geometrije došlo doprineli su proširenju postojećih znanja i uvoĊenju potpuno novog pogleda na
geometriju. Ova nova saznanja uticala su na potpuno novo, drugaĉije i kompleksnije sagledavanje
odnosa taĉke, prave i ravni u prostoru.
Lobaĉevski je utvrdio da njegova neeuklidska geometrija ima neposrednu primenu kod
izraĉunavanja kosmiĉkih prostranstava. U okvirima obiĉnih zemaljskih dimenzija kod svih
proraĉunavanja ljudi koriste euklidsku geometriju kao geometriju koja je najjednostavnija i
najrealnije odraţava stvarnost. Dostignuća u ovoj oblasti govore o tome da se fiziĉki prostori
prevelikih dimenzija ponašaju kao neeuklidski prostori. Za njihovo izuĉavanje potrebne su
neeuklidske geometrije odnosno geometrija Lobaĉevskog.
Ideja o ovom istraţivanju o hiperboliĉkoj geometriji potekla je iz ţelje da se sazna nešto više o
mogućnostima matematiĉkog i geometrijskog rešavanja prostora, kao o neophodnom znanju bez
koga se ne mogu rešavati sloţeni prostorni problemi u raznim oblastima istraţivanja u domenu
geometrije, fizike i astronomije.
Kljuĉne reĉi: hiperboliĉka geometrija, aksioma Lobaĉevskog, paralelnost, hiperparalelnost, modeli
hiperboliĉke ravni, primena hiperboliĉke geometrije
Summary
The development of science encouraged the development of technology and led to the progress of
mankind. A great number of problems which scholars and scientists of that time were facing
demanded a solution of numerous problems in various natural sciences, among others, in
mathematics and geometry. The studies in the field of geometry have been thousands of years old.
Attempts of resolving questions about the fifth Euclidian postulate led to the creation of entirely new
geometry that is non-contradictory. This new hyperbolic geometry is as accurate as the Euclidian
geometry. The evidence obtained by studying this type of geometry led to the expansion of existing
knowledge and to the introduction of a completely new view of the geometry. The new findings
influenced a completely new, different and more complex understanding of the relations among
dots, lines and planes in space. Lobachevsky found that his new non-Euclidian geometry has direct
application in the calculation of user spaces. In terms of ordinary earthly dimensions in all
calculations people use Euclidian geometry as the simplest geometry which reflects reality in most
realistic way. Achievements in this area suggest that physical spaces of too large dimensions
behave as nonEuclidian spaces. Their study requires non-Euclidian geometries, i.e. Lobachevskian
geometry. The idea of this research on hyperbolic geometry came from a desire to learn more about
the possibilities of mathematical and geometrical solution of space, as the knowledge necessary to
solve complex spatial problems in various areas of research in the field of geometry, physics and
astronomy.
Key words: hyperbolic geometry, the axiom of Lobachevsky, parallelism, hyperparallelism, models
of the hyperbolic planes, the application of hyperbolic geometry