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Moufang plane and Spin groups
Stejskal, Dominik ; Krýsl, Svatopluk (advisor) ; Holíková, Marie (referee)
In this thesis we consider the action of the exceptional simple Lie group F4 on the so called (real) Moufang plane OP2 R. The goal of this thesis is to present a proof of the transitivity of this action, which is as complete as possible. We first define related concepts such as Clifford algebras, the groups Pin(r, s) and Spin(r, s) and the algebra of octonions O, and we prove their basic properties. The group F4 is defined as the automorphism group of the algebra J3(O) of hermitian octonionic matrices of order three. The Moufang plane is defined as a suitable subset of J3(O). In the group F4 we find isomorphic copies of the groups Spin(0, 8) and Spin(0, 9). By applying certain auxilliary results from the previous chapters we obtain the desired proof of the transitivity of the action of F4 on OP2 R. 1
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Angle in Geometry
Michálková, Iveta ; Holíková, Marie (advisor) ; Pilous, Derek (referee)
This thesis examines the role of angle in the course of history of mankind. The introductory part of this thesis presents the historically first mentions of angles. The second and the third part pay attention to the development of tools and devices allowing practical measurement of angles. The fourth part follows the issue of introducing angle in textbooks of primary and secondary schools in the Czech Republic. It observes whether the topic of angle is imported in individual textbooks correspondingly or if the process in a textbook or textbooks differs considerably. The following parts of this thesis concentrate on various types of angles and pairs of angles and operations with them. Both arithmetic and graphic operations with angles are inspected. A substantial part of this thesis deals with trigonometric functions. This work comprises different manners of implementation and defining of trigonometric functions. Based on examples, this thesis demonstrates usage and significance of trigonometric functions and their properties in practical tasks. The penultimate chapter studies complex numbers. In this chapter, this thesis focuses primarily on complex numbers in trigonometric format and calculations with them. The final chapter holds a collection of exercises. This thesis is concluded by solutions...
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Beneficial utilisation of GeoGebra software in geometric constructions at the lower secondary level of basic school
Krejčíčková, Klára ; Holíková, Marie (advisor) ; Jančařík, Antonín (referee)
Diplomová práce se zabývá využitím programu GeoGebra ve výuce ma- tematiky, konkrétně v tématu konstrukce trojúhelník·. Cílem práce bylo na- vrhnout přípravu na hodiny výuky konstrukce trojúhelník· s využitím pro- gramu GeoGebra a následně ověřit, zda využití softwaru dynamické geometrie mělo vliv na zvládnutí nového učiva. Teoretická část obsahuje použité pojmy z planimetrie. V dalších dvou kapitolách se čtenář seznámí s programem GeoGebra a GeoTest a některými výzkumy, které jsou zaměřené na použití program· dynamické geometrie ve výuce matematiky. Následuje kapitola se- znamující s úlohami, které jsou v experimentu využity, a popisuje přípravu jednotlivých hodin. Poslední část je věnovaná pr·běhu experimentu, srovnání výsledk· žák· pomocí test· s kontrolní třídou. Získaná data jsou zpracována na základě pozorování a test·. Výsledky pozorování a analýzy potvrzují, že GeoGebra je vhodným doplňkem výuky konstrukce trojúhelník·. 1
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Angle in Geometry
Michálková, Iveta ; Holíková, Marie (advisor) ; Zamboj, Michal (referee)
The theme of this work is an angle in the history of mankind. The introductory part of the work is devoted to the first surviving mention of angles in the history. In the second and the third part is the attention given to the development of tools and instruments that allow to measure angles in practice. The fourth section focuses on the issue of the introduction of the angles in the textbooks for primary and secondary schools in the Czech Republic. It monitors that in each of textbooks the angles are introduced in the same way, or if there are some textbooks show a more significant deviation in this direction. In other parts of the work devoted to the various kinds of angles and couples of angles and operations with them. Attention is paid to how the counting here, so the graphics operations with the angles. A substantial part of the work consists of trigonometric functions. The work provides multiple ways of introduction and definitions of the trigonometric functions. The examples demonstrate the application and importance of trigonometric functions and their properties in the solution of practical tasks. The penultimate chapter is devoted to the complex numbers. In this section, the work focuses primarily on complex numbers in goniometrickém shape and calculations with them. A section is devoted...
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Angle in Geometry
Michálková, Iveta ; Holíková, Marie (advisor) ; Pilous, Derek (referee)
This thesis examines the role of angle in the course of history of mankind. The introductory part of this thesis presents the historically first mentions of angles. The second and the third part pay attention to the development of tools and devices allowing practical measurement of angles. The fourth part follows the issue of introducing angle in textbooks of primary and secondary schools in the Czech Republic. It observes whether the topic of angle is imported in individual textbooks correspondingly or if the process in a textbook or textbooks differs considerably. The following parts of this thesis concentrate on various types of angles and pairs of angles and operations with them. Both arithmetic and graphic operations with angles are inspected. A substantial part of this thesis deals with trigonometric functions. This work comprises different manners of implementation and defining of trigonometric functions. Based on examples, this thesis demonstrates usage and significance of trigonometric functions and their properties in practical tasks. The penultimate chapter studies complex numbers. In this chapter, this thesis focuses primarily on complex numbers in trigonometric format and calculations with them. The final chapter holds a collection of exercises. This thesis is concluded by solutions...
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