Matematika kaleidoskopa

master's thesis

• Author: Linda Sorić
• Mentor: Franka Miriam Bruckler (mentor)
• Committee member: Franka Miriam Bruckler (predsjednik povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Zvonko Čerin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-04-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Kaleidoskopi su optički instrumenti za stvaranje i promatranje vizualno atraktivnih uzoraka. Dva matematička koncepta koja su usko vezana uz kaleidoskope, tj. uzorke koje oni tvore, su simetrije i grupe pa su to ujedno i glavne cjeline ovog diplomskog rada. Ovisno o broju zrcala u kaleidoskopu te o dimenziji prostora u kojoj ga promatramo, on može tvoriti dvodimenzionalne, trodimenzionalne, ali i apstraktne višedimenzionalne uzorke. Fokus ovog rada je na dvodimenzionalnim i trodimenzionalnim uzorcima odnosno njihovim simetrijama. Cilj je iz uzoraka na temelju vrste kaleidoskopa prepoznavati simetrije, ali i na temelju simetrija uzoraka prepoznavati vrste kaleidoskopa. Simetrije uzoraka nastalih kaleidoskopima obzirom na operaciju komponiranja cine grupu, a upravo je to tema druge cjeline rada. Dan je pregled poznatih grupa simetrija dvodimenzionalnih i trodimenzionalnih objekata: grupa rozeta, grupa tapeta, trodimenzionalnih točkinih i prostornih grupa. Kaleidoskopima s dvama zrcalima možemo generirati grupe rozeta. Dvodimenzionalnim kaleidoskopima s trima i četirima zrcalima možemo generirati pojedine od grupa tapeta, a trodimenzionalnim kaleidoskopima s trima zrcalima pojedine od trodimenzionalnih točkinih grupa. ”Zatvorenim” kaleidoskopima možemo generirati pojedine od prostornih grupa.

Abstract (english)

Kaleidoscopes are optical instruments for creating and observing visually attractive patterns. Two mathematical concepts that are closely related to kaleidoscopes are symmetries and groups, so they determine two main chapters of this paper. Depending on the number of mirrors inside the kaleidoscope, and the dimension of the space we observe it in, kaleidoscope can form two-dimensional, three-dimensional or abstract multidimensional patterns. This paper is focused on two-dimensional and three-dimensional patterns and their symmetries. Our goal is the determination of the symmetries of a pattern based on type of kaleidoscope that formed it, and also the determination of the type of kaleidoscope based on symmetries of the resulting pattern. Symmetries of patterns created by kaleidoscopes, together with the operation of composition, form a group. Next, we give an overview of known symmetry groups related to two-dimensional and three-dimensional patterns: rosette groups, wallpaper groups, three dimensional point groups and space groups. Two-dimensional kaleidoscopes with two mirrors generate rosette groups. Two-dimensional kaleidoscopes with three and four mirrors generate certain wallpaper groups. Three-dimensional kaleidoscopes with three mirrors generate certain three-dimensional point groups. ”Closed box” kaleidoscopes generate certain space groups.

• Keywords: kaleidoskop; optički instrument; simetrije; grupe; prepoznavanje vrste kaleidoskopa; grupa rozeta; grupa tapeta; trodimenzionalne točkine grupe; prostorne grupe
• Keywords (english): kaleidoscope; optical instruments; symmetries; groups; determination of the type of kaleidoscope; rosette groups; wallpaper groups; three dimensional point groups; space groups
• Language: croatian
• URN:NBN: urn:nbn:hr:217:488761
• 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: Access restricted to students and staff of home institution
• Created on: 2019-02-07 11:42:37