Morikavov problem : magistrsko delo

Štrovs, Nina (author), Šivic, Klemen (mentor)

Krožnici se dotikata z zunanje strani in imata skupno tangento, ki ne poteka skozi njuno dotikališče. V vmesni prostor včrtamo kvadrat tako, da se dotika obeh krožnic in premice. Glavno vprašanje je, kolikšna je dolžina stranice najmanjšega možnega kvadrata. Zapisani problem, ki izvira z Japonske, se imenuje Morikavov problem. V delu so predstavljeni vsi možni položaji kvadrata, razlogi, da najmanjši kvadrat sploh obstaja, in geometrijske utemeljitve, v katerem položaju dobimo iskani kvadrat. Zapisana je funkcija, katere najmanjša vrednost na določenem intervalu je ravno dolžina najkrajše stranice. Število, pri katerem je dosežen minimum, je tudi ničla polinoma 10. stopnje. Na koncu je s pomočjo Galoisove teorije dokazano, zakaj zapisani polinom 10. stopnje ni rešljiv z radikali in zakaj dolžina stranice najmanjšega možnega kvadrata kot funkcija polmera ni radikalska funkcija.

• Language: Slovenian
• Keywords: Morikavov problem, krožnica, kvadrat, premica, konfiguracija, Galoisova teorija, radikalska funkcija
• Work type: Master's thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-144181
• UDC: 514
• COBISS.SI-ID: 138143235
• Publication date in RUL: 03.02.2023

Secondary language

• Language: English
• Title: Morikawa's problem

Abstract

Two circles are externally tangent and have a common external tangent. A square is placed in the area between the two circles and the line in such a way that it is touching all three objects. The main question is what the length of the side of the minimal square is. The described problem originates in Japan and is called Morikawa's problem. This master's thesis presents all the possible positions of the square, why the minimal square even exists, and geometric arguments for whether or not the minimal square can exist in a given position. An explicit formula for a function whose minimum value on a certain interval is exactly the length of the square's shortest side is also given. This length can be expressed in terms of a root of a certain polynomial of degree 10. Finally, with the help of Galois theory, we proved that the polynomial of degree 10 cannot be solved by radicals and why that is, as well as why the length of the minimal square's side as a function of radius is not a radical function.

• Keywords: Morikawa's problem, circle, square, line, configuration, Galois theory, radical function