What is fractal geometry? │ The History of Mathematics with Luc de Brabandère

The length of the Brittany coast will depend on whether you’re an ant, a rabbit or a human.

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Mathematics was a key element in the development of computer science. No mathematics, no computer. This is obvious. But somehow, the computer is a thankful object. Thanks to computer science and thanks to those powerful machines, a new 'e-era' started in mathematics. A new discipline of mathematics made possible by the development of computer science.

And one of the most famous examples of this is what we call: fractal geometry. Geometry was developed a long, long time ago. We have talked about Plato and Pythagoras and others. And thanks to the computer, a new kind of geometry became possible. And the founder is Benoit Mandelbrot, a French genius who worked as an IBM fellow.

And he started his research with a very peculiar question: ‘How long is the coast of Brittany?’ Looks obvious. Where is the problem? But the answer of Mandelbrot was unexpected. The length depends on who you are!

If you are a human being, you walk the coast of Brittany, you have how many kilometres? I don't know. But if you are a rabbit, for example, and you follow the path of the coast of Brittany, it's gonna be a longer path. Now imagine you are an ant. You have to go like ‘this’ to just follow the coast of Brittany. And that was the starting point for Mandelbrot.

Another thing is, if you look at this coast of Brittany, what the ant can see is similar to, imagine, an astronaut from outer space looking at the coast. The ant and the astronaut see the same thing. The only difference is the scale. And this is called self-similarities. Self-similarity is the property of an object, which looks the same whatever the scale. And that's how Mandelbrot started this new geometry.

There are many famous examples. Let's take two of them. The first is the snowflake. If you start with a line, like this, a segment of a line. And then you do this. Just build like a triangle.

And on the side of the triangle, you build another triangle. And then you build... You always build a new triangle in the middle of the side. You finally get a snowflake.

That's how Mandelbrot imagined the design of the geometry of nature. He was looking for the formula of nature, like the snowflake. He did the same for the clouds. A bit more complicated. And of course, you need to draw and that's why drawings are so important.

Another example is this one. If you take a segment of a line like this. And you remove what's here. Go one step below. And on the two remaining segments of line, you remove what's in the middle. And again, and again, and again... You’ll notice self-similarity. This little area here you see on the drawing is self-similar to this one. From outer space they look the same. This is self-similarity. And that was the starting point of fractal geometry. And this brings us back to something we have seen before: How to prove with drawings.

Imagine you're asked to calculate the sum of 1/3 + 1/9 + 1/27... and you have ‘the power of 3’ here. How much is it? You can use a kind of fractal formula. You start with 1, divided into three pieces. Of course, you have to keep one, one third. You can drop one and you have to keep the middle one and do it again. You now need to calculate 1/9. So, you cut it into three pieces. You have to keep one. You can drop the last one. And you have to keep the middle one. You can go as far as you want. At any time, you have to keep one piece and drop one other. So, the sum is 1/2.

Next time, we will be looking at algorithms and how they can help us with practical tasks.

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