Popločavanje ravnine

master's thesis

• Author: Ana Bejić
• Mentor: Matija Kazalicki (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-16, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Još u antičko vrijeme, pitagorejci su znali da su jedini pravilni mnogokuti kojima se može popločiti ravnina, jednakostranični trokut, kvadrat i pravilni šesterokut, međutim sama matematička teorija popločavanja stara je tek nešto više od jednog stoljeća. U ovom diplomskom radu problem popločavanja zapisan je matematički i prikazana su moguća rješenja koja zadovoljavaju postavljene uvjete. Dan je uvid u tek nekoliko od mnogobrojnih primjera popločavanja Euklidske ravnine koji pokazuju ljepotu matematike i njenu sveprisutnost u svakodnevnom životu. Definirani su temeljni pojmovi i dana je klasifikacija popločavanja s obzirom na simetrije. Detaljnije su analizirana popločavanja ravnine pravilnim mnogokutima, dokazan je teorem da postoje točno tri pravilna popločavanja ravnine odnosno 11 različitih rub-na-rub popločavanja pravilnim mnogokutima kod kojih su čvorišta istog tipa.

Abstract (english)

Even in ancient times, the Pythagoreans knew that the only regular polygons capable of tessellation the plane are an equilateral triangle, squares and a regular hexagon, however, the mathematical theory of tessellation itself is just over a century old. In this master thesis the problem of tessellation is mathematically written and possible solutions that meet the set conditions are shown. Here is given an insight in just a few of the many examples of tessellation the Euclidean plane that show the beauty of mathematics and its omnipresence in everyday life. The basic concepts are defined and the classification of the tessellation is given in terms of symmetry. Tilings of the plane by regular polygons are analyzed more precise and demonstrated by the theorem that there are exactly three regular plane tessellations. Furthermore, there are 11 different edge-to-edge tessellation of the plane by regular polygons and with the same type of vertices.

• Keywords: popločavanje ravnine; popločavanje Euklidske ravnine
• Keywords (english): plane tessellation; plane tiling; Euclidean plane tessellation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:707640
• 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: 2019-09-10 11:45:24