Fractal Program Implementation

In recent years, we have been studying fragment and have written several fragment-related programs. This is one of them. The program contains nearly 20 algorithms for generating fragment images. (1) Koch snowflake (2) Levy curve (3) dragon curve (4) line C (5) sherbinski (Sierpinski) triangle (6) Sierpinski (Sierpinski) carpet (7) Sierpinski triangle (8) Split triangle (9) fragment tree (10) fragment binary tree (Binary Tree) (11) Hilbert-Peano curve (12) crown curve (13) flower basket curve (14) Square Line (15) minkowski curve (16) coastline (17) the color of the graph in the cantor tri-collection program is white, and the background color is black. The most basic black and white are used to represent the fragtal graph. Software: http://files.cnblogs.com/WhyEngine/Fractal.7z

Software instructions:

• Keyboard 0 ~ 9, respectively set level 0th to level 9th.
• This is a 3D program. keyboard o is used for switching between 2D and 3D perspectives.
• You can change the angle of view by right-clicking and dragging.
• Keyboard X is used to restore to the default angle of view.
• The keyboard F11 is used for full screen switching.

 

Finally post an article on the introduction of the fractal article: original address: http://sxyd.sdut.edu.cn/gushi/gushi2.htm

Fragment ---- natural ry
I. Limitations of Euclidean ry
It has been more than 2000 years since the formation of Euclidean ry in the third century. Although many kinds of ry and differential ry are generated in the internal development of mathematics, compared with other ry, euclidean ry is more involved in production, practice, and scientific research. The importance of Euclidean ry can be proved from the history of human civilization. Euclidean ry is mainly based on the relationship between points, lines, and surfaces on small and medium scales. This concept is compatible with the human practice and understanding level in a specific period. The history of the development of mathematics tells us what kind of knowledge is and what kind of geometry. When people concentrate on mechanical motion, most of the images in the mind are cone curves and line segments. Given the limitations of understanding the subject and object, Euclidean ry has a strong "human" feature. In this case, we should not deny the glorious history of Euclidean ry, but recognize that Euclidean ry is a tool for people to understand and grasp the objective world, but not the only tool.
After the 20th century, science developed extremely rapidly. Especially after World War II, a large number of new theories, new technologies, and new research fields have emerged. People's views on the material world and human society are quite different from the past. As a result, it is difficult to describe some research objects using Euclidean ry. For example, the description of plant morphology, the study of crystal cracks, and so on.
American mathematician B. Mandelbrot once raised a famous question: How long is the coastline of England? In mathematics, can this problem be understood as: is the use of Line Segment fitting and any irregular continuous curve certain effective? The question is actually a challenge to the traditional ry with the Euclidean ry as the core. In addition, people find that traditional ry is powerless in terms of turbulence research and natural image description. The development of the field of human understanding calls for a new type of ry that can better describe natural graphs. Here, we may call it natural ry.
Ii. Generation of Fragment
Some mathematicians have discussed a type of special set (graphics), such as the Cantor Set, Peano curve, and Koch curve, in the process of in-depth study of real and complex analysis. These "pathological" sets under continuous ideas often appear in different forms of counterexamples. At that time, they were mostly used to discuss the strength of Theorem conditions, and their deeper meanings were not recognized by most people.
In 1975, Mandelbrot introduced the concept of fractal in his book "Fractal ry in nature. Literally, fractal is the meaning of fragments. However, this does not summarize the concept of manelbrot. Although there is no concept of fragment satisfactory to all parties, in mathematics, we all think that fragment has the following characteristics:
(1) An infinitely fine structure;
(2) Proportional self-similarity;
(3) generally, its fractional dimension is greater than its topological dimension;
(4) It can be defined by a very simple method and generated by recursion and iteration.
(1) and (2) indicate the inherent regularity of the fragment in the structure. Self-similarity is the soul of the fragment, which makes any fragment of the fragment contain the information of the entire fragment. Item (3) illustrates the complexity of fragment. Item (4) describes the generation mechanism of fragment. The Koch curve is continuous everywhere, but it can be exported everywhere, and its length is infinite. From the perspective of Euclidean ry, this curve is broken into an alternative, from the form of each curve in the Process of approximation, we can see the various manifestations of the four properties. Looking at the previously mentioned "pathological" curves based on the concept of fragment, we can see that they are just all kinds of fragment.
We compare the representative Euclidean ry of traditional ry with the fractal ry of the research object. We can conclude that the Euclidean ry is a logical system built on the principle, it studies various constant quantities under rotation, translation, and symmetric transformation, such as angle, length, area, and volume. its applicability is mainly artificial objects. However, the history of fragment is only 20 years old. It is generated by recursion and iteration, and is mainly applicable to the complex shapes of objects in nature. Instead of looking at the points, lines, and surfaces in the fragment from a separate perspective, the fragment is regarded as a whole.
Iii. Natural geometric view and Its Application
A straight line or a Cone Curve on a plane requires only a few conditions. So how many conditions does it take to determine a leaf? If we think of the fern leaf as a combination of line segments, then the number of conditions for determining this fern leaf is considerable. However, when people look at this fern from a fractal perspective, they can think of it as the result of a simple iterative function system, and the number of conditions required for determining the system is much less than that required. This shows that it is more effective to use the undetermined fragment to fit the fern leaves than to use the line fitting and the fern leaves.
The introduction of the concept of fragment is not just a change in descriptive techniques. In essence, fragment reflects some regularity in nature. Taking plants as an example, plant growth is a process in which plant cells develop and split according to certain genetic laws. This process of regular split can be considered as a recursive and iterative process, which is very similar to the generation of fragment. In this sense, we can think that a plant corresponds to an iterative function system. People can even simulate the mutation process of plants by changing some parameters in the system.
Fractal ry is also used in the area of coastline plotting and chart making, Seismic Forecasting, image coding theory, and signal processing. It has made remarkable achievements in these areas. As the intersection of multiple disciplines, the new interpretation of the "pathological" curve of the past Euclidean ry is the inevitable result of the constant development of the object of human understanding. At present, people urgently need a kind of geometry that can better study and describe various complex natural curves. However, fragtal ry can be used here. Therefore, the fragtal ry, that is, the natural ry, can be viewed from the perspective of the combination of fragment and fragment.

 

Fractal Program Implementation