delo diplomskega seminarja
Andraž Pustoslemšek (Author), Aleš Vavpetič (Mentor)

Abstract

Gorniška lema je uporabno orodje v matematiki, ki je hkrati tesno povezano z včrtavanjem določenih likov v krivulje. Gre se za to, da dva plezalca stojita na različnih koncih hriba in ga želita prečkati tako, da sta v vsakem trenutku na isti nadmorski višini. Če nadmorsko višino hriba predstavimo kot graf funkcije f, bomo v prvi polovici tega dela pokazali, da gorniška lema velja, čim f na definicijskem območju ne spremeni predznaka in se sestoji iz končnega števila monotonih kosov. V drugem delu si bomo pogledali geometrijsko uporabo te leme. Seznanili se bomo namreč z znamenitim problemom včrtavanja kvadrata v sklenjeno krivuljo, ki je star že več kot sto let in še do danes ni rešen v povsem splošni obliki. S pomočjo gorniške leme bomo včrtavali rombe v mnogokotnike, pred tem pa bomo dokazali še marsikatero presenetljivo dejstvo o včrtavanju trikotnikov in pravokotnikov v mnogokotnike.

Keywords

matematika;gorniška lema;problem včrtavanja kvadrata v sklenjeno krivuljo;

Data

Language: Slovenian
Year of publishing:
Typology: 2.11 - Undergraduate Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [A. Pustoslemšek]
UDC: 517
COBISS: 122279939 Link will open in a new window
Views: 804
Downloads: 170
Average score: 0 (0 votes)
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Other data

Secondary language: English
Secondary title: Mountain Climbing Problem
Secondary abstract: The mountain climbing lemma is a useful tool, that is at the same time tightly related with inscribed figures in closed curves. It is about two mountaineers, both standing on different sides of a mountain range. They want to cross it, but their elevation must remain equal at all times. If we replace the mountain range with a function f, we shall show in the first part of this work that if f does not change sign and consists of a finite number of monotone non-increasing or non-decreasing pieces, then the mountain-climbing lemma holds and the mountaineers can indeed cross it in the above explained way. In the second part we will see the geometric usage of this lemma. We will introduce the problem of inscribing a square into a closed curve, also known as the square peg problem. It exists for over a hundred years and still has not been solved in a completely general form. With the help of the mountain climbing lemma we will find inscribed rhombi in polygons, but along the way we will discover many surprising facts about inscribed triangles and rectangles in mostly planar polygons.
Secondary keywords: mathematics;mountain-climbing lemma;square peg problem;
Type (COBISS): Final seminar paper
Study programme: 0
Embargo end date (OpenAIRE): 1970-01-01
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages: 29 str.
ID: 16487814