Konstruktibilnost u planimetriji

master's thesis

• Title (english): Constructability in planimetry
• Author: Glorija Filipović
• Mentor: Igor Ciganović (mentor)
• Committee member: Vjekoslav Kovač (predsjednik povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Committee member: Aleksandra Čižmešija (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu raspravljali smo o konstruktibilnosti u planimetriji. U prvom poglavlju prikazali smo euklidske konstrukcije jednobridnim ravnalom i šestarom. U drugom poglavlju, bavili smo se algebarskim metodama konstrukcija i proširenjem polja racionalnih brojeva jer se konstrukcije ravnalom i šestarom mogu povezati s tipom proširenja polja racionalnih brojeva, koje sadrži korijene odgovarajućih polinoma. Kroz nekoliko teorema smo pokazali da se ne mogu svi geometrijski objekti dobiti konstrukcijama ravnalom i šestarom. Treće poglavlje bilo je o trima klasičnim problemima, kroz tri podpoglavlja smo opisali kako su oni nastali, naveli smo neke od pokušaja njihovog rješavanja i dokazali nemogu ćnost konstrukcije uporabom samo ravnala i šestara. Na kraju, u četvrtom poglavlju, kroz nekoliko teorema dokazali smo konstruktibilnost pravilnih mnogokuta.

Abstract (english)

In this master’s thesis we shall analyse the issue of constructibility pertaining to planimetry. In the first chapter we will display euclidean constructions using a straight edge ruler and a (regular) divider. In the second chapter we examined the algebraic methods of construction and widening of the field of rational numbers since constructions derived from the utilisation of a ruler and a divider can be associated with the type of expansion of the field of rational numbers that includes the roots of proper polynomials. The employed theorems demonstrate that not all geometric objects can be obtained by constructions using a straight edge ruler and a divider. The third chapter deals with the classical problems whose emergence is discussed in three subheadings, and some attempts at their solution are presented as well, and we demonstrated the impossibility of their construction by using solely a ruler and a divider. Finally, in the fourth chapter we employ certain theorems in order to prove the constructibility of regular polygons.

• Keywords: euklidske konstrukcije; algebarske metode konstrukcija; konstrukcije ravnalom i šestarom; konstruktibilnost pravilnih mnogokuta
• Keywords (english): Euclidean constructions; algebraic construction methods; constructions with a straight edge ruler and a divider; constructibility of regular polygons
• Language: croatian
• URN:NBN: urn:nbn:hr:217:293491
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-10-26 09:45:12