Origami geometrija

magistrsko delo

Tjaša Zore (Author), Matija Cencelj (Mentor), Eva Berdajs (Co-mentor)

Abstract

Origami je stara japonska umetnost prepogibanja papirja. Z matematičnega stališča list papirja obravnavamo kot model ravnine in proučujemo lastnosti geometrijskih objektov – točk in daljic, ki s prepogibanjem nastanejo. V magistrskem delu bomo raziskovali geometrijo origamija. V prvem delu se bomo posvetili vprašanju, kaj vse lahko konstruiramo s prepogibanjem papirja. Predstavili bomo Huzita-Hatori aksiome, ki predstavljajo temeljne postopke prepogibanja papirja. Predstavili bomo konstrukcije s šestilom in neoznačenim ravnilom, katerih temelj predstavlja pet Evklidovih postulatov, jih primerjali z aksiomi origamija in nato dokazali, da lahko s prepogibanjem papirja naredimo vse evklidske konstrukcije. V drugem delu magistrskega dela se bomo posvetili temeljnemu razlogu, zakaj je matematični origami močnejše orodje od neoznačenega ravnila in šestila. Dokazali bomo, da nam aksiomi origamija omogočajo konstruiranje rešitev kubičnih enačb. Predstavili bomo delo Margharite P. Beloch in njene origami konstrukcije dolžine ∛2 ter z iskanjem ničel rešili nekaj kubičnih enačb s pomočjo grafične Lillove metode. S prepogibanjem papirja so tako rešljivi tudi nekateri starogrški problemi. Predstavili bomo Abejevo in Justinovo trisekcijo – dva različna postopka tretjinjenja kota ter postopek podvojitve prostornine kocke s pomočjo Messerjeve konstrukcije števila ∛2.

Keywords

origami geometrija;prepogibanje papirja;origami konstrukcije;aksiomi Huzita-Hatori;kubične enačbe;Belochin kvadrat;Lillova metoda;konstrukcije z ravnilom in šestilom;tretjinjenje kota;Abejeva trisekcija;Justinova trisekcija;Meserjeva konstrukcija;

Data

Language:	Slovenian
Year of publishing:	2022
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[T. Zore]
UDC:	514(043.2)
COBISS:	109971715
Views:	1620
Downloads:	181
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Origami geometry
Secondary abstract:	Origami is an old Japanese art of paper folding. From a mathematical point of view, we consider a sheet of paper as a model of a plane and study the properties of geometric objects - points and line segments, which are formed by folding. In the master's thesis we will explore the geometry of origami. In the first part, we will focus on the question of what we can construct by folding paper. We will present the Huzita-Hatori axioms, which represent the basic procedures of paper folding. We will present constructions with a compass and an unmarked ruler based on five Euclidean postulates, compare them with the axioms of origami, and then prove that we can make all Euclidean constructions by folding paper. In the second part of the master's thesis, we will focus on the fundamental reason why mathematical origami is a more powerful tool than the unmarked ruler and compass. We will prove that the axioms of origami allow us to construct solutions of cubic equations. We will present the work of Margharita P. Beloch and her origami constructions of length ∛2 and solve some cubic equations by finding real roots using the graphical Lill’s method. By folding the paper also some ancient Greek problems can be solved in this way. We will present Abe's and Justin's trisection - two different procedures of forming a tertiary angle and the process of doubling the volume of a cube using Messer's construction of the number ∛2.
Secondary keywords:	Matematika;Geometrija;Origami;Aksiomi;Univerzitetna in visokošolska dela;
File type:	application/pdf
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak, Poučevanje, Predmetno poučevanje
Pages:	57 str.
ID:	15502803