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Lobachevskian geometry
Neubauerová, Alžběta ; Halas, Zdeněk (advisor) ; Hromadová, Jana (referee)
Title: Lobachevskian geometry Author: Alžběta Neubauerová Department: The Department of Mathematics Education Supervisor: Mgr. Zdeněk Halas, DiS., Ph.D., The Department of Mathematics Education Abstract: The aim of this bachelor's thesis is to introduce the topic of Lobachev- skian geometry to secondary school students. In the first chapter, we focus on the history of the discovery of Lobachevskian geometry due to attempts to prove Euclid's fifth postulate. In the second chapter we explain the basic terms, in the third chapter we list and prove chosen theorems from absolute geometry. The fourth chapter deals with theorems that are equivalent to the fifth postulate. By negating them, together with the facts from the chapter on absolute geometry, we obtain several theorems from Lobachevskian geometry in the fifth chapter. In the final chapter, we introduce Poincaré's model of the half-plane and thus gain more vivid idea about the theorems that we built in the previous chapter. Keywords: non-Euclidean geometry; Lobachevskian geometry; Euclidean geome- try
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Equidecomposability
Valkoun, Matyáš ; Halas, Zdeněk (advisor) ; Rmoutil, Martin (referee)
Title: Equidecomposability Author: Matyáš Valkoun Department: Department of Mathematics Education Supervisor: Mgr. Zdeněk Halas, DiS., Ph.D., Department of Mathematics Education Abstract: This bachelor thesis focuses on the area of polygons and its definition by equidecomposability. In the plane ρ a simple polygon and its area is defined and the notion of equidecomposability is introduced. Since any two equidecomposable polygons have equal areas, a question arises if the opposite is also true: are any two polygons of equal area equidecomposable? That is the formulation of the Wallace-Bolyai-Gerwein theorem, its detailed proof is presented in this text. Thus the notions of equidecom- posability and equal area are equivalent. At the end of the thesis it is briefly examined if it is possible to use equidecomposability in the third dimension to define the volume of a polyhedron. Keywords: Area of a polygon, equidecomposability, Wallace-Bolyai-Gerwein theorem, triangulation 1
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Orientation of a real vector space
Macek, Lukáš ; Halas, Zdeněk (advisor) ; Škorpilová, Martina (referee)
In this thesis, we focus on creating a visual understanding of orientation of a real vector space and its subsequent connection to the mathematical definiton. As a result, this thesis can be used as supplementary material in higher education or serve as in- spiration for teachers. First, we develop the idea behind the equivalence of two bases, then we examine its connection to permutations of vectors in ortonormal basis, moti- vating the definition of parity of permutation. We continue by observing the behavior of the equivalence during a transition to the opposite half-space, noting the connection to volumes, and based on that, we motivate the concept of determinants. Next, we delve into the method of computing determinants, providing a complete derivation. Finally, we demonstrate how the determinant of a transition matrix between two bases relates to their equivalence and we define the orientation of vector space. 1
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Definition of vector product
Holý, David ; Halas, Zdeněk (advisor) ; Slavík, Antonín (referee)
Title: Definition of vector product Author: David Holý Department: Department of Mathematics Education Supervisor: Mgr. Zdeněk Halas, DiS., Ph.D., Department of Mathematics Edu- cation Abstract: The main goal of this thesis is to present a compelling and well- motivated definition of the vector product and to explain its properties. Torque serves as a medium through which "the rotating effect of force"is studied on simple physical examples. Elaboration leads to revealing essential properties that define the vector product. The thesis contains the derivation of Cartesian coor- dinates of the vector product. It also contains a list of its basic mathematical properties and applications. Lorentz force is presented as a concrete example of its application and is thoroughly analyzed. In the closing section, the term curl of a vector field is introduced and conceptually explained. The thesis was focused on bringing a good didactic presentation of a vector product, its concrete appli- cations in practice, and its connection to more advanced fields of mathematical inquiry. Keywords: vector product, cross product, torque, Lorentz force, vector field, curl 1
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Mathematics in the game of SET
Koblížková, Iva ; Slavík, Antonín (advisor) ; Halas, Zdeněk (referee)
This bachelor thesis provides a mathematical description of the card game of SET. The reader is introduced to the history and the rules of this game. Further off, some combinatorial aspects of the game are investigated. In the following chapter, affine geometry is used to describe the game. Thanks to this generalization, the so-called maximum cap can be calculated. The last chapter shows how linear algebra can be used to characterize the game and some of the previously introduced ideas. 1
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Important theorems of affine geometry
Kundratová, Lucie ; Šír, Zbyněk (advisor) ; Halas, Zdeněk (referee)
V této práci se seznámíme s pojmy a nástroji afinní geometrie. To jsou zejména barycentrické souřadnice, afinní zobrazení a dělicí poměr. Ty pak použijeme k dokázání afinních vět. K Menelaově větě uvidíme vícero důkazů. Dále ukážeme Menelaovu větu z Cevovy a naopak. Uvedeme projektivní tvrzení, ze kterých obě věty plynou. Nakonec se budeme zabývat překladem Menelaových Sférik z latiny a spletitou historií Menelaovy věty. 1
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Volume of solids
Tvrdá, Monika ; Halas, Zdeněk (advisor) ; Staněk, Jakub (referee)
This didactic oriented bachelor project helps to approach an origin of relations for the volumes of solids taught at high school. It is focused on high school and university students. At the beginning the project shows historical meaning of the volumes of solids and the processes which were used to enumerate them in the ancient Egypt and Mesopotamia. Further, the project deals with the definition of volume of solids; it is based on Jordan's measure. The relations for volumes of the sorted solids are derived using the integral calculus. In the end the other ways of deriving of these relations are shown. At first, it is the method that Archimedes from Syracuse invented, furthermore by the visual imaginations and the Cavalieri's principle. 1
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History of Kurzweil integral
Berková, Andrea ; Halas, Zdeněk (advisor) ; Slavík, Antonín (referee)
The presented work deals with history of Kurzweil integral. It focuses primarily on its comparison with other important integrals, namely Newton, Rie- mann, Lebesgue, Perron and McShane integral. Each of them is discussed in a separate chapter which acquaints with their authors and theories. Attention is also oriented to Jaroslav Kurzweil and Ralph Henstock. There are also mentioned the circumstances of the discovery of the Kurzweil integral. The aim is to high- light the theory of integration, which has its origins in Bohemia and despite its elementary definition, which is very general and usable in many applications.
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