Fractal Foundation Online Course - Chapter 1 - FRACTALS IN NATURE

As we saw in the previous section, nature is full of fractal patterns. Many of these structures use branch mode or spiral mode and you can see that these complex patterns can be created by a simple process that is repeated over and over again. In this section we will explore the same type of iterative process, but this time we will do it in the field of pure geometry. Through the investigation of the iterative function system, we can understand the ability to iteratively create complex fractal patterns and see some excellent characteristics of them. We even learn a special language of the L system that describes how to iterate a simple system to create a realistic plant - like structure.

There are many ways to generate geometric fractals. First, start with the deduplication process and examine Sierpinski's triangle.

Beginning with a simple triangle, the first step shown in the figure is to delete the middle triangle. This will leave three black triangles around the center white triangle (repeat 1). Then repeat the same process on a smaller scale and delete the middle one third of each of the three triangles, you can repeat the second iteration. At this level, nine small black triangles remain. On the third iteration, the minimum number of triangles is tripled and reaches 27. The small triangle of level 4 is (27 * 3) = (9 * 9) = 81.

One of the advantages of geometric fractals is that they are not finished, they can be repeated forever! By doing this, you can dig into these fractals and find out more.

Another fractal that we can do with the iterative removal process is the Sierpinski carpet. To create this shape, start with a square, divide it into nine small squares, and delete the center square. Then repeat the process with each of the remaining 8 squares

There are two ways to solve these problems. The first way is to find and use an expression that shows how many squares were created in a given iteration.

If you spend 1 second on the calculation of each square, how long it is to calculate all the white squares in the figure above, no matter how big it is. []

How long (minute) will it take to calculate all squares of the 6th Sierpinski carpet? []

Fractal geometry is a study of shapes consisting of smaller repeating patterns. These patterns called fractals are repeated using self-similarity. For example, as shown in Chapter 1, trees show a similar structure at various magnifications. Carefully examine the leaf veins and show a branching pattern resembling the whole tree. Therefore, the tree is called the "self-similar" component of the tree. As the overall pattern of the tree is carried by its genetic code, the shape of the fractal is also conveyed by the equation which is its mathematical code. In either case, we call this generated code "seed". In theory, you can predict the general appearance from DNA, trunk, extremities, leaves. In practice, however, as the structure is further removed from the original seed at spatial distance and time, the exact positioning of each item is obviously unpredictable.

Not all fractals are artificial. Nature is full of fractals. The mountains are fractal. A healthy old tree is a fractal. The human body is fractal. As mentioned above in the general meaning of the patterns of human settlements, fern leaves, earthquake fault patterns, part of the cloudy sky, that is to say, if we do not simplify the structure deeply and do not simplify (in the very strong case Etc.) microscope). A continuum is formed from the blood vessels of the aorta to the capillaries. They diverge, divide and branch again until the blood cells are so narrow that they are forced to slip. Their branching characteristics are also fractal. As a physiological necessity, blood vessels must have a certain stereoscopic effect, ie the circulatory system has to limit the huge surface area to a limited volume. In terms of physical resources, blood is expensive and space is priceless.