Thymio and fractals
This page includes the script of the video. Sentences in italic are those presented in the clip.
Opening credits
Question about Thymio
- Is Thymio an artist?
General concept presentation
- Definition
- For starters, let's try to define what a fractal figure is, or simply a "fractal". It is a geometrical form whose pattern repeats itself indefinitely. This can be observed at every scale.
- Koch Curve
- To understand how such a figure works, let's starting by drawing a line.
- We divide this line into 3 equal pieces and then draw an equilateral triangle on the centre part. We can now remove this centre part, which has been replaced by 2 lines of the same size.
- We continue our drawing by repeating the same steps on each new part. We could continue doing this indefinitely, but we will stop here. This figure is called the Koch Curve.
- If we observe one side of our first iteration, we can see that the figure is identical to our whole figure. And if we take a small part of that figure, we can see that again, it is identical to the whole! This is what we call a fractal.
- Dimension
- To strictly describe a fractal, we must observe its dimension.
- Different kinds of dimensions exist: Euclidean, topologic and fractal. In our case, only the fractal dimension is interesting. It is defined by d = log n / log m, where n is the number of parts obtained after iteration and m is the homothetic ratio (in how many parts we divide the initial part).
- For the Koch curve, we divide our segment into 3 equal parts (so m=3) and we obtain 4 segments that are each 1/3 of the initial length (so n=4). This gives us a dimension d = ln 4/ln 3 = 1.26.
- To see if our Koch curve is a fractal, its fractal dimension must either be greater than its topological dimension, or not a whole number. In our case this is verified, so our Koch curve is a fractal!
- New definition
- We can now give a more rigorously defined definition of a fractal: it is an object which:
- is too irregular to be defined by the usual geometric vocabulary
- is self-similar (which means that each part of the object resembles its whole)
- has a non-whole dimension, or one that is greater than its topological dimension
- We can now give a more rigorously defined definition of a fractal: it is an object which:
- What is it used for?
- Now we know what a fractal is but… what is it used for? Many applications exist, but I will only cite a few.
- First of all, we can make beautiful figures like this *show a picture* or this *show a picture*. But it also gives us a way of describing physical phenomena such as a fluid's turbulent behaviour or galaxies' organisation in space, or even the geometrical shape of a romanesco broccoli! *photo of a romanesco broccoli*
- Fractals are also a way of compressing images with a constant quality for every zoom.
Concept presentation with Thymio
- Now let's use Thymio to draw a fractal composed of a big circle with in it three smaller circles, which each hold three smaller circles. We could repeat this an infinite number of times.
Closing credits