M. C. Escher i eliptičke krivulje

master's thesis

• Author: Mia Galiot
• Mentor: Matija Kazalicki (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu objašnjena je matematička struktura umjetničkog djela „Print Gallery“ nizozemskog umjetnika M.C. Eschera. Objašnjenje se bazira na činjenici da se litografija promatra kao slika na određenim eliptičkim krivuljama nad poljem kompleksnih brojeva, te idejom da se u sredini slika sama ponavlja. Preciznije, da slika sadrži kopiju sebe koja je rotirana za \(-157,625^{\circ}\) i umanjena za faktor 22,58. U 1. poglavlju prikazana je Escherova metoda konstrukcije mreže i ucrtavanja slike u mrežu. U 2. poglavlju na generaliziranom primjeru objašnjene su kompleksne transformacije koje se koriste kako bi se postigao Escherov efekt. U 3. poglavlju opisana je teorija modularne grupe, djelovanje Möbiusovih transformacija na gornju poluravninu, te teorija kompleksnih torusa. Zatim se promatra Escherova slika na kompleksnom torusu i primjena kompleksnih transformacija iz 2.poglavlja sa ciljem da se dobije preslikavanje, koje uz pomoć posebno izrađenog softvera omogućuje završetak slike, što je prikazano u 4. poglavlju.

Abstract (english)

This thesis shows the mathematical structure of the artwork ”Print Gallery” , made by the Dutch graphic artist Maurits Cornelis Escher (1898–1972). The explanation is based on the fact that lithography is viewed as an image on certain elliptic curves over the field of complex numbers and on the idea that picture is rotated until it is almost upside down and containing another copy of itself. More precisely, the image contains a copy of itself rotated at \(-157,625^{\circ}\) and reduced by factor 22.58. In Chapter 1 Escher’s method of his grid construction is presented. In Chapter 2, a generalized example explains complex transformations that are used to achieve Escher’s effect. Chapter 3 describes the theory of the modular group, the effect of Möbius transformations on the upper half-plane and the theory of complex torus. Then the Escher’s image on the complex torus is consider and the use of complex transformations from the 2nd chapter on the image, with the aim of obtaining a mapping which, with the help of a specially made software, enables to finish the image, as shown in Chapter 4.

• Keywords: M.C. Escher; Print Gallery; Escherov efekt; Möbiusove transformacije
• Keywords (english): M.C. Escher; Print Gallery; Escher’s effect; Möbius transformations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:710980
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-04-19 09:26:07