Razrez trikotnika na sedem skladnih trikotnikov : magistrsko delo

Škufca, Erika (author), Vavpetič, Aleš (mentor)

$N$-tlakovanje trikotnika $ABC$ s trikotnikom $T$ je način rezanja trikotika $ABC$ v $N$ skladnih manjših trikotnikov. Manjšemu trikotniku $T$ pravimo ploščica. Do sedaj je bilo malo znanega o možnih vrednostih števila $N$, na katere se v tem magistrskem delu osredotočimo. Ko je ploščica $T$ podobna trikotniku $ABC$, dokažemo, da so možne tri oblike števila $N$. V primeru, ko je $N$ popolni kvadrat, lahko $N$-tlakujemo poljuben trikotnik. Če pa je $N \in \{e^2+f^2, 3n^2; n, e, f \in {\mathbb N}\}$, je ploščica $T$ pravokotni trikotnik. Ploščica $T$ ima sorazmerne kote, če je vsak od njih racionalni večkratnik števila $\pi$. Naj bo trikotnik $ABC$ $N$-tlakovan s ploščico $T$, ki ima sorazmerne kote in ni podobna trikotniku $ABC$. Če je trikotnik $ABC$ enakostranični, ima $T$ kote $({\pi \over 6}, {\pi \over 3}, {\pi \over 2})︁$ ali $({\pi \over 12}, {\pi \over 3}, {7\pi \over 12})︁$ in je $N = 6n^2$ ali pa ima $T$ kote $({\pi \over 6}, {\pi \over 6}, {2\pi \over 3})︁$ in je $N = 3m^2$. Če pa je $ABC$ enakokraki trikotnik z baznim kotom $\alpha$ in tlakovan s ploščico $T$, ki je podobna polovici trikotnika $ABC$, potem je $N$ sodo število. Prav tako raziščemo možne $N$, če ploščica $T$ nima vseh sorazmernih kotov. Naj bo trikotnik $ABC$ $N$-tlakovan s ploščico, ki ni podobna trikotniku in katere koti niso vsi sorazmerni. Tedaj pokažemo, da je $N \ge 8$. Na koncu pa iz vseh zgornjih primerov dokažemo, da ne obstaja 7-tlakovanje trikotnika s skladnimi ploščicami.

• Language: Slovenian
• Keywords: trikotnik, tlakovanje, skladnost, podobnost
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2021
• PID: 20.500.12556/RUL-127386
• UDC: 514
• COBISS.SI-ID: 65315843
• Publication date in RUL: 04.06.2021

Secondary language

• Language: English
• Title: Section of a triangle into seven congruent triangles

Abstract

The $N$-tiling of the triangle $ABC$ with the triangle $T$ is a process of cutting the triangle $ABC$ into $N$ congruent smaller triangles. The smaller triangle $T$ is called the tile. So far, little is known about the possible values of the number $N$, which is the main subject of the master's degree. When the tile $T$ is similar to the triangle $ABC$, we can prove that three forms of the number $N$ are possible. When $N$ is a perfect square, any triangle can be $N$-tiled. However, the tile $T$ is a right triangle if $N \in \{e^2+f^2, 3n^2; n, e, f \in {\mathbb N}\}$. The tile $T$ has commensurable angles if each one of them is a rational multiple of number $\pi$. Furthermore, let a triangle $ABC$ be $N$-tiled with the tile $T$, which has commensurable angles and is not similar to the triangle $ABC$. If the triangle $ABC$ is equilateral, it has $T$ angles $({\pi \over 6}, {\pi \over 3}, {\pi \over 2})︁$ or $({\pi \over 12}, {\pi \over 3}, {7\pi \over 12})︁$ and $N = 6n^2$ or it has $T$ angles $({\pi \over 6}, {\pi \over 6}, {2\pi \over 3})︁$ and $N = 3m^2$. However, if $ABC$ is an isosceles triangle with base angle $\alpha$ and tiled with the tile $T$, which is similar to one half of the triangle $ABC$, then $N$ is an even number. Moreover, the possible values of $N$ are analyzed, if not all angles of the tile $T$ are commensurable. We can prove that $N \ge 8$, when the triangle $ABC$ is $N$-tiled with the tile that is not similar to the triangle and has angles that are not all commensurable. Finally, we prove, based on above examples, that the 7-tiling of the triangle with the congruent tiles does not exist.

• Keywords: triangle, tiling, congruency, similarity