Holditchev teorem

master's thesis

• Author: Lucija Ćurić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu izloženi su različiti aspekti proučavanja teorema iz geometrije koji je engleski matematičar Hamnet Holditch objavio 1858. godine. U njegovoj izvornoj formulaciji riječ je o površini područja ravnine omeđenog zatvorenom krivuljom C i geometrijskim mjestom (putanjom) točke koja jednu tetivu krivulje C dijeli na segmente zadanih duljina \( a\) i \(b\), kad se tetiva giba (rotira) duž cijele krivulje C. Iznenađujući ishod da ta površina iznosi jednostavno ab , neovisno o obliku i veličini krivulje C, potaknula je znatno zanimanje matematičara koji su nastojali pronaći dublje razloge takvog rezultata. To je ujedno značilo i preciziranje pretpostavki Holditchovog teorema, istraživanje mogućnosti njegovih poopćenja i prostornih analogona. Prikazan je izbor nekih od važnijih rezultata te vrste. Poseban interes izazvala je i “skrivena” elipsa čija se površina prepoznaje u tvrdnji teorema. U završnom poglavlju ukratko je opisan postupak konstrukcije niza poligona čije pojedinačne takozvane Holditchove krivulje konvergiraju prema traženoj elipsi.

Abstract (english)

This thesis presents various aspects of the study of a geometrical theorem that was published in 1858. by the English mathematician Hamnet Holditch. In his original formulation it yielded the area of a planar region limited by a closed curve C and the locus of the point dividing a chord of C into two segments of given lengths \( a\) and \(b\), when that chord moves (rotates) along the curve C. A surprising result that the area is simply equal to ab , not depending on the shape or the size of the curve C, provoked a considerable interest with mathematicians who attempted to discover a deeper reason to that statement. Such an investigation also involves finding precise assumptions for the theorem to be true as well as examination of possible generalizations and analogous results in three dimensions. An especially intriguing part of the theorem is played by a ”hidden” ellipse, whose area is readily apparent in the statement. In the final chapter it is briefly described a procedure of construction of a sequence of polygons, with their associated so-called Holditch curves converging to the desired ellipse.

• Keywords: Hamnet Holditch; Holditchove krivulje; skrivena” elipsa
• Keywords (english): Hamnet Holditch; ”hidden” ellipse; Holditch curves
• Language: croatian
• URN:NBN: urn:nbn:hr:217:760041
• Study programme: Title: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education; Course: Mathematics and Computer Science Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-12-08 10:35:25