Fractal Foundation Online Course - Chapter 1 - FRACTALS IN NATURE

Click the mouse to select the value "C", click the mouse to zoom in (any panel) - click to zoom out

What we can easily observe is how sensitive this system is. Zoom in several times to find an interesting place and place the cursor close to the end of the Mandelbrot collection. Uncheck the "Animate" checkbox. If you move the cursor one pixel at a time using the arrow keys, you may see sudden fundamental changes in the track map in the right panel.

Although we compare two very similar C value trajectories, obviously the results of the orbit may vary greatly, regardless of how close the two starting points are. Indeed, it is infinitely complex so you can not zoom to the edge. As you move further deeper, the points have finite orbits, infinite small orbits can have infinite orbits. The more you explore the edge, the more iterations are needed before deciding the fate of the starting point.

Such behavior is an example of sensitivity to initial values, important concepts of dynamic systems, fractal foundations, and changes suddenly when the starting value is slightly smaller. One of the revisits in the confusion

Note: If you stop the premature iteration and you give up before the point reaches a certain threshold and spiral outwards, the starting point is misunderstood as being inside the set, so it actually belongs to the set It should be painted black though it should be. . Even at any magnification, computing the Mandelbrot set with a finite number of iterations (inevitably), our image can only be an ideal perfect fit approximation. Our estimate is always a bit bigger than the actual Mandelbrot group. Because there are always points outside the set in practice, they can not escape until giving up repetition.

This means that the deeper you search, the more you need to calculate the iteration, which slows down. (Therefore, these numbers also become slower due to the number of digits after the decimal point.) Even if the computer's speed is high, you can always generate a fractal that exceeds the limit. Because the complexity and beauty of patterns often increase dramatically with depth, artistic motivation and curiosity are deepening us. Therefore, the faster the fractal calculation, the better.

Grasp the surface of the incredible Mandelbrot fractal at the last minute. In the next chapter I will continue exploring the Mandelbrot Magic as I understand the splendid Julia set, change the index of the equation, and completely abandon the Mandelbrot collection of other equations.

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.

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.