I read something interesting today. About fractals. If you take any three points that form a triangle, and then a fourth point anywhere, and subsequently throw a dice, the faces of which are assigned to the first three points. Next, you move from the current point towards the point corresponding to the result on the dice and place a new point halfway; this becomes the new current point. After many iterations, the points start to form the Sierpinski triangle – the one shown in the attached picture. Intuitively, you would think the triangle should be fully filled because it involves random movements in three directions from a randomly chosen point, but no. Moreover, it works even if the starting point is inside the future empty triangle (yes, a few points will disrupt the picture, but that’s it). If you start our experiment with five or six points instead of three, different shapes will form – see the attached picture. This graphical method is called the Chaos Game.
By the way, it may seem obvious, but in case you wondered — all the presented figures have zero area.
If you take two triangles and with a probability p move towards random vertices of the first, and with (1-p) towards random vertices of the second, you end up forming a Barnsley fern (picture â„–2).
I love such things because they seem like magic at first glance 🙂
(It’s a kind of problem from the same class as the synchronization of metronomes)
Hi, I'm Rauf Aliev. 