In the last example we enter a field of more modern
mathematics. Students often hear about fractals in art and design. So it is
almost necessary to introduce the mathematical view of this topic as well. The
before-mentioned examples can also be also solved with the aid of
"nonsymbolic" calculators. However, with fractals the aid of a
computer is crucial. Students draw a few iterations of fractals and see how
they develop. This helps them when calculating their dimension.
The dimension of a fractal is very interesting. We are used
to the idea that a line is one-dimensional, a plane two-dimensional, and a
solid three-dimensional. But in the world of fractals, dimension acquires a
broader meaning, and need not be a whole number. We only study just some simple
examples, such as the dimension of the Sierpinski gasket. Fractals with
their rational dimension can be an introduction to the new topic: geometry of
2D and 3D space.
Students first encounter with fractals is not in the classroom. As their homework they have to find out everything they can about fractals. In the class we take a look at the collected data, show some interesting pictures of fractals – from the simple ones to the artistic ones and some fractals from the nature as well (trees, cauliflower, broccoli, …). After this kind of introduction the students are prepared for solving the exercise.
The worksheet is on http://rc.fmf.uni-lj.si/matija/logarithm/worksheets/fractal.htm.
Several students showed so much interest in this subject that they agreed to prepare their own "research work" in the next school year. Again we provide them with some WWW links as starting points.