Očrtane kocke

delo diplomskega seminarja

Matej Melanšek (Author), Aleš Vavpetič (Mentor)

Abstract

V delu je predstavljena rešitev problema očrtanih kock, pogosto imenovana Kakutanijev izrek. Ta nam pove, da lahko vsaki omejeni zaprti konveksni množici v ${\mathbb R}^3$ očrtamo kocko. Prav tako je opisana teorija homotopije, fundamentalih grup in kontraktibilnosti, ki jo potrebujemo, da dokažemo da je vsaka dvakrat ovita zanka v grupi $SO(3)$ homotopna konstantni preslikavi. Dokažemo tudi par posledic, ki sledijo iz dokaza Kakutanijevega izreka, med katerimi je razširitev problema na višje dimenzije. Izkaže se, da je odgovor na ta problem prav tako pritrdilen. Torej lahko poljubni konveksi množici v ${\mathbb R}^n$ očrtamo $n$-dimenzionalno kocko. V zadnjem poglavju na kratko opišemo še zgodovino reševanja Knasterjevih problemov in dokažemo, da Knasterjeva domneva ne velja v splošnem.

Keywords

matematika;očrtane kocke;Kakutanijev izrek;fundamentalne grupe;specialne ortogonalne grupe;kontraktibilnost;Knasterjev problem;

Data

Language:	Slovenian
Year of publishing:	2022
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[M. Melanšek]
UDC:	515.1
COBISS:	122340099
Views:	983
Downloads:	54
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Circumscribed cubes
Secondary abstract:	In this paper we solve the problem of circumscribed cubes, frequently simply named Kakutani’s theorem. This theorem tells us, that we can circumscribe a cube around every closed bounded set in ${\mathbb R}^3$. Furthermore we describe the theory of homotopies, fundamental groups and contractible spaces needed to prove, that every double loop in the group $SO(3)$ is homotopic to a constant map. We also prove a few corollaries of Kakutani’s theorem, one of them being the extension of the problem to higher dimensions. It turns out that the answer to that problem is also affirmative. That means, that we can circumscribe a $n$-cube around every closed bounded set in ${\mathbb R}^n$. At the end we briefly summarize the history of solving Knaster’s problems and prove that in general Knaster’s conjecture is false.
Secondary keywords:	mathematics;circumscribed cubes;Kakutani theorem;fundamental groups;special orthogonal groups;contractibility;Knaster 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:	16506293