Zlatni rez i Fibonaccijevi brojevi

master's thesis

• Title (english): Golden ratio and Fibonacci numbers
• Author: Hana Strejček
• Mentor: Marija Galić (mentor)
• Committee member: Marija Galić (predsjednik povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Aleksandra Čižmešija (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Zlatni rez je simbol savršene skladnosti, ravnoteže te ga nalazimo već kod starih Egipćana. Zlatni rez često povezujemo s Fibonaccijevim brojevima te je ovaj diplomski rad posvećen istraživanju i opisivanju njihovih svojstava. Ovaj diplomski rad sadrži dva velika poglavlja. U prvom poglavlju nalazi se povijesni pregled zlatnog reza, opisana su njegova osnovna svojstva, što geometrijska, što numerička. Opisani su neki od geometrijskih likova koji nose epitet zlatni te gdje zlatnih rez možemo pronaći u našoj okolini. U drugom poglavlju opisano je kako su nastali Fibonaccijevi brojevi, a time i Lucasovi brojevi, dokazani su neki njihovi identiteti, opisana je kombinatorna interpretacija Fibonaccijevih brojeva i pokazano da omjer Fibonaccijevog broja i njegovog prethodnika teži zlatnom rezu.

Abstract (english)

The golden ratio is a symbol of perfect harmony and balance and has been in human history since the time of the ancient Egyptians. Fibonacci numbers are often associated with the golden ratio, and this thesis is dedicated to the golden ratio and Fibonacci numbers. This thesis contains two large chapters. In the first chapter, there is a historical overview of the golden section, its basic properties, both geometric and numerical, are described. Some of the geometric figures that bear the epithet golden are described, and where we can find the golden section in our surroundings. In the second chapter, it is described how Fibonacci numbers, and thus Lucas numbers, were created, some of their identities were proven, a combinatorial interpretation of Fibonacci numbers was described and it was shown that the ratio of the Fibonacci number and its predecessor approaches to the golden ratio.

• Keywords: simbol savršene skladnosti; ravnoteža; povijesni pregled zlatnog reza; Fibonaccijevi brojevi; Lucasovi brojevi
• Keywords (english): symbol of perfect harmony; balance; historical overview of the golden ratio; Fibonacci numbers; Lucas numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:094379
• 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: 2023-08-18 11:33:58