08.04.2021
Fractals. The Sierpinski triangle by the chaos method.
Today we will talk about fractals. First we need to figure out the definition of fractals. Fractal is a term used to describe geometric shapes containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension.
In ordinary life, we very often encounter fractals. In the real world, it can be snowflakes, ferns, snail shells, lightning and much more.
In ordinary life, we very often encounter fractals. In the real world, it can be snowflakes, ferns, snail shells, lightning and much more.
An example of a simple fractal is Sierpinski triangle . The first mathematical description of this fractal was published by the Polish mathematician Vaclav Sierpinski in 1915. To build a Sierpinski triangle, you need to build a large triangle, and then build another triangle, which is obtained by connecting the midpoints of the sides of a large triangle. After that, you will get four identical triangles of a smaller size. Next, you need to repeat the algorithm with the resulting triangles several times, ignoring the triangle obtained when connecting the midpoints of the sides of a large triangle. You should get something like this:
This animation demonstrates the self-similarity property of the Sierpinski triangle:
You can also draw the Sierpinski triangle by iteration or by the chaos method. Chaos theory is an interdisciplinary scientific theory and branch of mathematics focused on underlying patterns and deterministic laws highly sensitive to initial conditions in dynamical systems that were thought to have completely random states of disorder and irregularities.
Since we use the method of drawing a game of chaos, we will need to repeat the same action hundreds and thousands of times. For automation, we will use a python program using the pygame multimedia library. I will not go into the details of python programming and using the pygame library because this article is not about that, but about fractals.
Let's continue talking about the Sierpinski triangle. In order to draw this triangle, first you need to put three random points on an arbitrary plane:
Since we use the method of drawing a game of chaos, we will need to repeat the same action hundreds and thousands of times. For automation, we will use a python program using the pygame multimedia library. I will not go into the details of python programming and using the pygame library because this article is not about that, but about fractals.
Let's continue talking about the Sierpinski triangle. In order to draw this triangle, first you need to put three random points on an arbitrary plane:
Next, we need to put a random point (marked with a red arrow), it can be located anywhere on this plane, even inside the triangle formed by the three starting points, even outside this triangle:
The next step is that we have to randomly select one of the three main points and the point that we set in the previous step, and find the middle of the segment created by these two points, and put a new point there (marked with a red arrow):
The next step, we must once again select a random point from the three main ones and again find the middle of the segment between the selected point and the last set point. This step will need to be repeated many times before we draw the Serpinsky triangle:
It is very interesting that despite the fact that our construction of the Sierpinski triangle is completely based on chaos, which manifests itself in a completely chaotic, random choice of one of the main points, we will still get this fractal.
Perhaps you have a question "What if you put the starting point in the supposed empty triangle formed by the midpoints of the segments of a large triangle, then what will happen?". As mentioned earlier, we can put the first point at any place on the plane, that is, if we put the starting point in the supposed empty triangle formed by the midpoints of the segments of a large triangle, then we will get only a pair of points that will be in the voids. But the Sierpinski triangle will still work. After all, we should not look at the behavior of individual points, but at the group behavior of a large number of points:
Perhaps you have a question "What if you put the starting point in the supposed empty triangle formed by the midpoints of the segments of a large triangle, then what will happen?". As mentioned earlier, we can put the first point at any place on the plane, that is, if we put the starting point in the supposed empty triangle formed by the midpoints of the segments of a large triangle, then we will get only a pair of points that will be in the voids. But the Sierpinski triangle will still work. After all, we should not look at the behavior of individual points, but at the group behavior of a large number of points:
As we can see, we have three points that stand out from the general behavior, but if we look at the general behavior of the points, then it is completely predictable.
Fractals are very beautiful mathematical figures, it is very surprising that all this beauty can be obtained from pure chaos.
Fractals are very beautiful mathematical figures, it is very surprising that all this beauty can be obtained from pure chaos.
