Trikotniki, vložljivi v celoštevilsko mrežo

delo diplomskega seminarja

Ana Šenica (Author), Aleš Vavpetič (Mentor)

Abstract

V diplomski nalogi si bomo ogledali karakterizacijo vložljivosti trikotnikov v celoštevilske mreže ${\mathbb Z}^n$ za $n \geq 2$. Pri tem bomo rekli, da je trikotnik vložljiv v ${\mathbb Z}^n$, če je podoben kakšnemu trikotniku v ${\mathbb R}^n$, ki ima oglišča s celoštevilskimi koordinatami. Videli bomo, da je trikotnik vložljiv v ${\mathbb Z}^2$ natanko tedaj, ko so tangensi vseh treh kotov trikotnika racionalna števila ali $\infty$. Enakostranični trikotnik je primer trikotnika, vložljivega v ${\mathbb Z}^3$, ne pa tudi v ${\mathbb Z}^2$. Dokazali bomo, da je trikotnik vložljiv v ${\mathbb Z}^3$ natanko tedaj, ko je vložljiv v ${\mathbb Z}^4$. Kriterij za vložljivost trikotnika v ${\mathbb Z}^4$ (in s tem v ${\mathbb Z}^3$) je, da so tangensi vseh njegovih kotov oblike $\tan{\alpha_i} = q_i \sqrt{k}$, kjer je $k \in {\mathbb Z}$ vsota treh kvadratov celih števil in $q_i \in {\mathbb Q} \cup \{\infty\}$. Izpeljali ga bomo na dva načina, pri čemer si bomo pomagali s podobnostnimi preslikavami, kvaternioni in trikotniškimi enačbami. Obstajajo trikotniki, vložljivi v ${\mathbb Z}^5$, ne pa tudi v ${\mathbb Z}^4$. Za višje dimenzije pa velja, da je trikotnik vložljiv v ${\mathbb Z}^n$ za $n \geq 5$ natanko tedaj, ko je vložljiv v ${\mathbb Z}^5$.

Keywords

matematika;vložljivost;celoštevilska mreža;trikotniki;trikotniška enačba;kvaternioni;podobnostna preslikava;n-simpleks;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[A. Šenica]
UDC:	514
COBISS:	75800067
Views:	1347
Downloads:	71
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Triangles embeddable in integer lattice
Secondary abstract:	We give a characterization of the triangles embeddable in ${\mathbb Z}^n$ for $n \geq 2$. A triangle is embeddable in ${\mathbb Z}^n$ if it is similar to a triangle in ${\mathbb R}^n$ whose vertices have integer coordinates. A triangle is embeddable in ${\mathbb Z}^2$ if and only if tangents of all its angles are rational or $\infty$. Equilateral triangle is embeddable in ${\mathbb Z}^3$ but not in ${\mathbb Z}^2$. We show that a triangle is embeddable in ${\mathbb Z}^4$ if and only if it si embeddable in ${\mathbb Z}^3$. A triangle is embeddable in ${\mathbb Z}^4$ (and ${\mathbb Z}^3$) if and only if tangents of all its angles are rational multiples of $\sqrt{k}$, where $k$ is a sum of three squares, or $\infty$. We show this by using similarities of ${\mathbb R}^n$, quaternions and triangle equations. There are triangles embeddable in ${\mathbb Z}^5$ but not in ${\mathbb Z}^4$. A triangle is embeddable in ${\mathbb Z}^n$ for $n \geq 5$ if and only if it is embeddable in ${\mathbb Z}^5$.
Secondary keywords:	mathematics;embeddability;integer lattice;triangle;triangle equations;quaternions;similarity;n-simplex;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	29 str.
ID:	13349322