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.
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