Malfattijev problem

master's thesis

• Author: Mateja Šašo
• Mentor: Juraj Šiftar (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Mirko Polonijo (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U radu su izloženi glavni rezultati vezani uz klasični Malfattijev problem iz 1803. godine izrezivanja tri valjka iz uspravne trostrane prizme tako da njihov ukupni volumen bude maksimalan. Ovaj stereometrijski zadatak očito se svodi na planimetrijski problem smještavanja tri kruga unutar zadanog trokuta tako da njihova ukupna površina bude najveća moguća. Malfattijeva slutnja da optimalan raspored čine tri kružnice takve da se svake dvije dodiruju međusobno, a svaka pritom dira dvije stranice trokuta pokazala se pogrešnom, ali je potaknula razne konstrukcije takve konfiguracije kružnica, kao i izračunavanje njihova položaja primjenom algebarskih i geometrijskih metoda. Problem najveće površine pokazao se vrlo teškim te je cjelovito rješenje postignuto tek potkraj 20. stoljeća, rezultatom da pohlepni algoritam uvijek daje optimalan raspored kružnica, površine veće nego za Malfattijevu konfiguraciju.

Abstract (english)

In this diploma thesis we exhibit the main results related to the classical Malfatti’s problem (1803.) of cutting out three cylinders with maximal total volume out of a triangular prism. This stereometric riddle can obviously be reduced to the planimetric problem of placing three circles inside a given triangle so that they cover the largest possible area. Malfatti’s conjecture that the optimal arrangement is achieved for three circles such that each one is tangent to the other two and to two sides of the triangle turned out to be wrong. However, this assumption initiated various constructions of such a configuration of circles, as well as calculations of their position using algebraic and geometric methods. The problem of maximizing the area proved to be very hard and the complete solution was accomplished in the late 20th century. The result shows that a greedy algorithm always yields the optimal arrangement of circles, with the corresponding area larger than the one obtained by the Malfatti’s configuration.

• Keywords: klasični Malfattijev problem; izrezivanje tri valjka iz uspravne trostrane prizme tako da njihov ukupni volumen bude maksimalan; stereometrija; problem najveće površine
• Keywords (english): classical Malfatti’s problem; cutting out three cylinders with maximal total volume out of a triangular prism; stereometric; problem of maximizing the area
• Language: croatian
• URN:NBN: urn:nbn:hr:217:149362
• 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: 2019-02-08 12:51:39