Klasični fraktali

master's thesis

• Author: Andrea Filić
• Mentor: Matija Kazalicki (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Snježana Lubura Strunjak (član povjerenstva)
• Committee member: Goranka Nogo (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-11-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Točke, pravci, geometrijski likovi i tijela su savršeni oblici i njima je prirodu ponekad nemoguće opisati. Dosta oblika u prirodi je nepravilno, neravno, rekli bismo, kaotično, stoga se u takvim slučajevima koristi fraktalna geometrija. Ova grana matematike razvila se tek krajem dvadesetog stoljeća i našla primjenu u znanosti, tehnologiji, medicini pa čak u astronomiji. U ovom radu je obrađen pojam fraktala te nekoliko klasičnih fraktala - Cantorov skup, trokut i tepih Sierpińskog, Kochova krivulja i pahuljica, Pitagorino stablo, Juliaov skup i Mandelbrotov skup. Svakom od njih opisano je njihovo svojstvo samosličnosti te izračunata dimenzija samosličnosti koja se razlikuje od pojma dimenzije kakvu poznajemo u euklidskoj matematici.

Abstract (english)

Points, lines, polygons and solids are perfect shapes and the nature sometimes cannot be described by them. Many shapes in nature are irregular, rugged, or we could say, chaotic, so in these cases we use fractal geometry. This part od mathematics began to develop at the end of twentieth century and found its application in science, technology, medicine, even in astronomy. This paper presents fractals and describes some classical fractals - the Cantor set, the Sierpiński triangle and carpet, the Koch curve and snowflake, Pythagorean tree, the Julia set and the Mandelbrot set. For every of them is described their feature of self-similarity and is calculated their self-similarity dimension that differs from term of dimension we know.

• Keywords: fraktal; fraktalna geometrija; samosličnost; dimenzija samosličnosti; fraktalna dimenzija; Mandelbrot
• Keywords (english): fractal; fractal geometry; self-similarity; self-similarity dimension; fractal dimension; Mandelbrot
• Language: croatian
• URN:NBN: urn:nbn:hr:217:397648
• 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: 2020-07-02 11:51:59