| Abstract [eng] |
The goal of this master's degree thesis is to analyze properties and generation algorithms of simple fractals and to implement these algorithms into Logo programming language suggesting methodology of teaching fractals using Logo in schools and providing some examples of Logo programs for fractal generation. The thesis consists of two parts. The first part provides basic theory on fractals. It begins with a simple explanation of what a fractal is using examples of self-similarity and recursive process and going into a more mathematical definition of fractals, introduced by B. Mandelbrot. After a brief history of fractals, a more in-depth analysis of Mandelbrot and Julia sets, the two well-known fractals arising from very simple sequences of complex numbers defined by the relation z_{n+1} = z_n^2 + c is given. The last chapter of the first part points out the reasons how fractals are useful and why they should be taught at school – fractals are fun; fractals are beautiful; anyone can play with them; fractals promote curiosity; computers, when used to explain fractal theory, enhance learning. The second part focuses on using Logo to generate fractals. It provides a few Logo programs of various complexity ranging from simple recursive functions to handling operations with complex numbers. Examples of Logo programs include generation of fractal trees, Koch snowflake and Sierpinski gasket, implementation of chaos game and iterated function systems, and manipulating complex numbers to draw Mandelbrot and Julia sets. The programs are arranged in such a way that they suggest a brief methodology of integrating fractal theory and its Logo implementation into the mathematics curriculum or after-school activities. |
