Trikotniki, vložljivi v celoštevilsko mrežo : delo diplomskega seminarja

Šenica, Ana (author), Vavpetič, Aleš (mentor)

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 ${\mathbb Z}^5$.

• Language: Slovenian
• Keywords: vložljivost, celoštevilska mreža, trikotnik, trikotniška enačba, kvaternioni, podobnostna preslikava, $n$-simpleks
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2021
• PID: 20.500.12556/RUL-129883
• UDC: 514
• COBISS.SI-ID: 75800067
• Publication date in RUL: 09.09.2021

Secondary language

• Language: English
• Title: Triangles embeddable in integer lattice

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 ${\mathbb Z}^5$.

• Keywords: embeddability, integer lattice, triangle, triangle equations, quaternions, similarity, $n$-simplex