University of Zagreb. Faculty of Science. Department of Mathematics.
Abstract
Važno mjesto u geometriji zauzimaju geometrijske konstrukcije pravilnih n-terokuta. Poznato je kako elementarno (samo uz pomoć ravnala i šestara) konstruirati jednakostraničan trokut, kvadrat, pravilni peterokut, šesterokut, osmerokut i deseterokut. Prvi koji nedostaje u ovom nizu je pravilni sedmerokut. Gauß je, proučavajući konstrukciju pravilnog sedmerokuta, dokazao koji se sve pravilni n-terokuti mogu elementarno konstruirati te da sedmerokut, nažalost, nije jedan od njih. Međutim, pravilni sedmerokut možemo konstruirati ako se poslužimo nekim sredstvima koje nisu dio elementarnih konstrukcija. Na primjer, možemo ga konstruirati umetanjem pravca ili presjekom krivulja drugog reda. U ovom radu opisujemo različite geometrijske konstrukcije pravilnog sedmerokuta, polazeći od Arhimedove analize i njegovih uvjeta za izvođenje konstrukcije. Arhimedov rad nastoje usavršiti srednjovjekovni arapski matematičari. U radu su opisane njihove konstrukcije Arhimedovih uvjeta ili Arhimedovog pravca, ali i neke originalne analize konstrukcije pravilnog sedmerokuta. Promatramo i neke algebarske konstrukcije. Rješenja ciklotomijske jednadžbe pridružene pravilnom sedmerokutu mogu se konstruirati uz pomoć krivulja drugog reda, najčešće presjekom parabole i hiperbole. U ovom radu analiziramo i nekoliko zanimljivih konstrukcija ovog tipa.The geometrical construction of regular polygons is an important part of geometry. It is known how to elementary (only by means of a ruler and a pair of compasses) construct an equilateral triangle, a square, a regular pentagon, a regular hexagon, a regular octagon, and a regular decagon. The first one missing in this sequence is a regular heptagon. While studying the construction of a regular heptagon, Gauß has determined the class of regular polygons that can be constructed elementary. Unfortunately, a heptagon is not one of them. However, a regular heptagon can be constructed if we use methods which are not parts of elementary constructions. For example, it can be constructed by inserting a line or using intersections of conics. We describe different constructions of a regular heptagon, starting from the analysis due to Archimedes and his conditions. Medieval Arab mathematicians tried to improve the work of Archimedes. We describe their constructions based on the work of Archimedes, as well as some of their original analyses concerning the construction of a regular heptagon. Some algebraic constructions have also been looked at. The solutions of an equation associated to a regular heptagon can be constructed using conics, mostly using intersections of a parabola and a hyperbola. We analyze a few interesting constructions of this type as well
