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Tema ovog diplomskog rada Eulerova je formula. Otkriće Eulerove formule pripisuje se švicarskom matematičaru Leonhardu Euleru. Euler je u toj formuli povezao broj vrhova, broj bridova i broj strana svakog konveksnog poliedra. Otkrio je ako V predstavlja broj vrhova, B broj bridova, a S broj strana konveksnog poliedra, tada vrijedi formula: \[V-B+S=2\] Dokaza Eulerove formule ima mnogo, njih čak 17 različitih. Eulerova formula vrijedi za pravilne i polupravilne poliedre, te za neke konkavne poliedre kao što je veliki zvjezdasti dodekaedar. Pravilnih poliedara ima samo pet, to su tetraedar, kocka, oktaedar, dodekaedar i ikosaedar, dok polupravilnih ima mnogo više. Eulerovu formulu primjenjujemo u dokazu da postoje samo pet pravilnih poliedra, u dokazu Pickove formule, u dokazu Erdös-Gallai-Sylvesterovog teorema i mnogih drugih. Matematičari su se bavili i dali nekoliko generalizacija Eulerove formule, kako bi ona vrijedila za sve poliedre. Zanimljivo je to da je Eulerova formula usko vezana uz teoriju grafova. Naime, svakom se poliedru može pridružiti njegova ravninska mreža, takozvanim rastezanjem poliedra u ravninu, čime broj vrhova, bridova i stranica, koji se javljaju u formuli, ostaje isti. The theme of this graduate thesis is Euler’s formula. The discovery of the Euler’s formula is attributed to swiss mathematician Leonhard Euler. In this formula, Euler had associated with the number of vertices, the number of edges and the number of faces of each convex polyhedron. He found that if V is the number of vertices, the B number of edges, the S number of faces convex polyhedra, then the following formula is valid : \[V-B+S=2\] There are many proofs of the Euler formula, even 17 of them. Euler’s formula is valid for regular and semiregular polyhedron, and for some concave polyhedron such as a great star dodecahedron. There is only five regular convex polyhedron, that is tetrahedron, cube, octahedron, dodecahedron and icosahedron, while there is much more semiregular. Euler’s formula is used to prove that there are only five regular polyhedra, we also use it to prove the Pick’s formula, Erdös-Gallai-Sylvester theorem and many others. The mathematicians dealt with and gave a few generalizations of the Euler formula, so that it would apply to all polyhedra. It is interesting that Euler’s formula is closely related to the graph theory. Each of the polyedra has its plane graph, by streching them into the plane, thus the number of vertices, edges and faces, appearing in the formulas, remain the same. |
