Simetrije periodičkih popločavanja ravnine

master's thesis

• Author: Ana Kruhoberec
• Mentor: Franka Miriam Bruckler (mentor)
• Committee member: Franka Miriam Bruckler (predsjednik povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Simetrije svake ravninske figure čine grupu obzirom na kompoziciju. Ukoliko figura posjeduje translacijsku simetriju, govorimo o periodičnoj figuri, preciznije o frizu ili tapeti, ovisno o tome pojavljuje li se ta translacijska simetrija samo u jednom smjeru ili ne. U ovom je radu opisana klasifikacija frizova i tapeta obzirom na njihove grupe simetrija. Pomoću tzv. Conwayevog magičnog teorema pokazano je da postoji točno 7 grupa frizova i 17 grupa tapeta. Za frizove smo isto pokazali i elementarnijim, geometrijskim pristupom. Primjeri uzoraka frizova i tapeta izrađeni su pomoću aplikacije Kali, koja je zbog jednostavnosti primjerena svim generacijama, te je iz tog razloga primjenjiva i u nastavi matematike vezano uz teme simetrija.

Abstract (english)

Every plane figure’s symmetries form a group with respect to composition. If a figure possesses translational symmetry, one speaks of a periodic pattern, more precisely of a frieze or wallpaper, depending on if translational symmetries occur in only one direction or not. This thesis describes the classification of friezes and wallpapers with respect to their symmetry groups. The fact that there exist exactly 7 frieze groups and 17 wallpaper groups is shown using the so-called Conway magic theorem. The result on friezes is also proved using a more elementary, geometric approach. Examples of frieze and wallpaper patterns have been generated with the Kali application which is, due to its simplicity, suitable for all generations, and as such usable in teaching symmetry topics in the mathematics classroom.

• Keywords: simetrije ravninske figure; friz; tapeta; klasifikacija frizova; klasifikacija tapeta; Conwayev magični teorem; Kali
• Keywords (english): plane figure’s symmetries; frieze; wallpaper; classification of friezes; classification of wallpapers; Conway magic theorem; Kali
• Language: croatian
• URN:NBN: urn:nbn:hr:217:746688
• 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: Closed access
• Created on: 2018-02-01 14:03:03