Page 396 - Srednje skole - radovi

ZASNIVANJE NEEUKLIDSKIH GEOMETRIJA; GEOMETRIJA LOBAĈEVSKOG THE
FOUNDATION OF NON-EUCLIDEAN GEOMETRY; THE LOBACHEVSKY GEOMETRY

Autor: Stefanovid Milan prva godina Matematiĉke gimnazije

Mentor: Perovanovid Mirjana profesor matematike u Matematiĉkoj gimnaziji

Rezime

Neopipljiva... Neodoljiva... Zanimljiva... Jednom reĉju matematika je nešto bez ĉega bi mi školovanje bilo mnogo
dosadnije, kao i ţivot. Matematika je jedinstvena i neizostavna. Druge nauke ne bi mogle bez nje, dok je ona jedina
samostalna meĊu svim naukama.

"Neka ne ulazi onaj ko ne zna geometriju." Reĉe Platon. Ovaj rad posvetidu jednoj veoma apstraktnoj grani
geometrije.

"Nema ni jedne matematiĉke grane, ma koliko da je apstraktna, koja se jednom ne bi mogla primeniti na pojave
stvarnog sveta." N. I. Lobaĉevski

Ako bismo bar malo razmislili o smislu ove izreke shvatili bismo da je ona zaista istinita. To nameravam da dokaţem
ovim radom. U njemu obradidu jednu veoma zanimljivu i neopipljivu oblast iz dela matematike koji se zove Osnovi
geometrije. Slobodno bismo mogli da kaţemo da je Euklid udario temelje geometriji i prvi zapoĉeo njeno deduktivno
zasnivanje. Deo matematike koji du izuĉavati u radu nastao je u XIX veku, i naravno još uvek se razvija. Zove se
geometrija Lobaĉevskog ili Hiperboliĉka geometrija.

"Sve uzvišeno isto je tako teško, kao što je retko." rekao je Spinoza. To sam napomenuo zato što se mnogi od nas nisu
susreli sa ovim delom matematike, a velika vedina de ga odbaciti kao ne istinu. MeĊutim da li je geometrija
Lobaĉevskog kontradiktorna samoj sebi?

Kljuĉne reĉi: aksioma, teorema, euklidski prostor, hiperboliĉki prostor, Lobaĉevski
Summary

Impalpable…Irresistible…Interesting…In one word, mathematics is something without which my schooling would be
much more boring, just as life would be. Mathematics is unique and obligatory. The other sciences could not be
imagined without it, whereas mathematics is the only independent science among all the other sciences.

“Let no one ignorant of Geometry enter” said Plato. I will dedicate this work to the one very abstract branch of
geometry.

“There is any mathematical branch, no matter how much it is abstract, that could not be applied to the real world
phenomenon” said N.I.Lobachevsky. If we thought a little bit about this saying, we would realize that it is really true. I
intend to prove that by this work. In this work, I will deal with one very interesting and impalpable mathematical
domain called the Basics of Geometry. We could freely say that Euclid laid the foundations of geometry and was the
first one who began its deductive foundation. The part of the mathematics, which I will research in this work,
originated in the 19th century and it’s still developing. It is called Lobachevsky geometry or hyperbolic geometry.

“All that is sublime is as hard as it is rare” said Spinoza. I mentioned this because many of us have not encountered
this part of mathematics yet, and great majority will reject it as an untruth. However, is Lobachevsky geometry
contradictory to itself?

Key words: axiom, theorem, Euclidean space, hyperbolic space, Lobachevsky