Off-topic: Fractal drawings

There is a way of utilizing sets of rules which in return form a complex pattern on a 2D or 3D graph. While reading this you will be presented with 7 different examples of fractal drawings. Welcome to the beautiful world of creating drawings with the help of numbers. Don't worry, they not only cover the subject from an artistic point of view, but also examine some really strange (scientifically speaking, lol, this is a technical term) behavior. The basis of all this are iterations; you will see what I mean from the pictures. Now, let's proceed..

First off there is the Minkowski sausage. Let's start with 0 iterations:

Next, 1 iteration:

2 iterations:

You see where this is going. To make things a little more interesting; it must be considered what would happen if there was a lot of iterations. Imagine you could make an infinite number of them. Strange things would happen. It is all described in the following picture:

 

How do you like that? The length of the line enclosing the area reaches infinity while the area that is enclosed remains the same as it was in all other iterations.

Now have a look at the so-called Multibrot set. Starting with the exponent 1 (the first "iteration")

 Following is the world-wide famous Mandelbrot set. The exponent is 2:

Let's try 3 as the exponent:

 This one is for the laughs. Presenting a perception of similarity:

Returning to serious business, here is the fist 3D fractal - the Sierpiński tetrahedron. Take a note of the surface area and volume progression, starting at 0 iterations:

Next, 1 iteration:

2 iterations:

And finally we are tired of these small numbers and decide to let the number of iterations go to infinity. Here is the result:

The scroller next to that huge number would be too big/thin, so we switch to the powers of 10. The surface area is the same, while the volume goes to 0

Next in line of 3D fractal drawings is the Menger sponge. You know what to pay attention to. At 0 iterations it looks like a cube:

Here it is at 1 iteration:

2 iterations:

Enough, give me infinite iterations:

The surface area goes to infinity while its volume goes to 0. Now isn't that very unusual? hehe

Some fractals need a bit of tinkering to unlock. I present to you the curlicue fractal. The variables here are the constant and the number of iterations. If the use 0 as a constant and 2000000 as the number of iterations, we get:

Well that gave a line. Let's try 1 as the constant. The number of iterations remains the same:

Wow, that gave another line. Let's try to find something a little more different. If the constant equals the number Pi and the number of iterations is the same, then we get:

Now we are talking. It's unlocked. Let's try the number Pi divided by the square root of 2 as the constant and the same number of iterations:

That's really beautiful and unexpected. Wonder what would happen if the constant got 50 times smaller?

To be honest this fractal has too many faces to be able to reveal it's entire beauty in a couple of pictures. You are encouraged to go to Wolfram Alpha and try out at least a couple of dozen other variations.

The last fractal drawing from this series is the Pythagoras tree. It may even help us reveal some of natures self-determining growth mechanisms. Don't take my word for it, have a look at these pictures:

Notice the iteration rule. 1 is represented by the small square, while 4 is represented by the larger square. Next we have 2 and 4:

Ok, it's starting to look like something. For the final touch we use 3 and 4:

And there it is, a tree like drawing created by mathematics. Makes me wonder, how many other things can be represented and understood with the help of fractals? What ever could we do with universal pattern finding knowledge? Seriously, some things that come to mind are the universal translator, compression of data, fractal graphics, greater prediction precision of everything, etc. Special thanks goes to the Wolfram Alpha website, without it all these beautiful pictures wouldn't be here.

Anyhow, hope you enjoyed this lengthy presentation. I am curious, which patterns did you see? That's what comment box is for.