A Basic Study On The Application Of Fractal Theory In Costume Pattern Design

Clothing and costume pattern designs are inseparable, which plays the decoration and beautification role for clothing, especially in the modern society. People are constantly seeking beauty, focusing on beauty, advocating individuality, enjoying new unusual things, which also provides broad prospects for development of clothing designs.Fractal theory is a new branch of mathematics geometry, which is mainly used to study the complex patterns of natural things, using fractal dimension to characterize the majority of the geometric figures which does not meet the rules of Euclidean geometry in the natural world. In accordance with fractal mathematical algorithms, an unlimited number of fractal graphics which has great visual impact can be produced, which obviously has artistic boldness. Therefore, how to apply beautiful fractal graphics to the apparel design has also become a hot issue. However, the multi-fractal nature of variability and less control has brought a lot of difficulties in the application. Therefore, this article took this as a starting point, summed up and sorted out the relevant laws of the fractal graphics through the research of fractal graphics structure.Fractal is developing and an emerging discipline which attracts widely attention in recent decades. So far it does not have a strict definition. Its English interpretation shows that the fractal is used to describe and deal with rough, irregular geometry form. It has two main features, that is, self-similarity and scale invariance. The combination of different algorithms can produce a great variety of graphics. The common usage of fractal algorithms are: escape time algorithm, L systems, iterated function system IFS. Although the algorithm is limited, changes in the fractal is infinite. The primary cause lies in: an initial graph can generate different graphics through the various algorithms; different initial graphics, after enough iterations, can generate same graphics; different fractal algorithm may also generate the same graph; identical algorithm, identical function, but different parameter, also has the possibility to produce the entirely different graph. These are leading to the fractal countless changes.Overall, from the algorithm, you may discover some basic graph production characteristic of three kinds of algorithms, such as flower-shaped structure generated by the escape time algorithm, which mostly are strange abstract graphics, colorful, symmetry. L system has the nature of "growth", mostly multi-analog plant growth or nature scenery, such as plants and trees. Emphasizing the process, forming gradually, along with the increase in the number of iterations, fractal becomes more and more clear, more and more sophisticated. Iterated function system IFS focuses on its graphics randomness which is presented in non-deterministic algorithm, which has dynamic effects and can be formed different forms of the same graph by changing the parameters. In order to obtain the specific laws of the fractal graphics, this dissertation mainly discusses M-set, J-set and the Newton fractal based on escape time algorithm, and exploringly summarizes the basic rule and structural feature of fractal graphs based on the iteration function by means of fractal graph, based on the polynomial function, produced by mathematics softwares such as maple and so on, along with the corresponding iteration function from the graph, particularly the fractal graph symmetrical aspect's rule. The main conclusions are as follows:1. Structural Characteristics of Mandelbrot Fractal1) The characteristics of the shape based on M-set of function of zn + c (the high order)(1) The number of petal to generate the depends on the power of iteration functions, namely, first-order iteration function is generated graphics showing less a petal;(2) When n is odd, the graphics is on the origin of symmetry, that is symmetrical both from top to bottom and from left to right straightly; When n is even, the graphics is on the axial symmetry, that is symmetrical from top to bottom, left and right are not entirely on the axial symmetry, but a slight rotation, the whole petal is the average distribution of each other separated by the angle between the line;(3) The greater the number of n, the graphics tends to be more round, when it is n→∞the limit is a circle graph.2) The characteristics of the shape based on generalized M-set of the function f ( z)=zⁿ +z+c(1) The number of petal to generate the depends on the graph iterated function, that is, with the first-order iteration function it is generated graphics showing less a petal;(2) When n is odd, the graphics is on the origin of symmetry, that is symmetrical both from top to bottom and from left to right straightly; When n is even, the graphics is on the axial symmetry, that is symmetrical from top to bottom, left and right are not entirely on the axial symmetry, but a slight rotation, the whole petal is the average distribution of each other separated by the angle between the line;(3) The greater the number of n, the graphics tends to be one round point, when it is n→∞the limit is a circle graph with the outer circle for a color gradient ring.3) The characteristics of the shape based on M-set of the polynomial function of f ( z)= zⁿ +z^n-1++z³+z²+z+c(1) It is the successive cumulative polynomial function corresponding to the fractal graphics. There is no more clear symmetry in the first two sorts, but generally showing a flower-shaped structure. The number of petals still depends on the highest power of the iterated function, that is, with the order iterated function drawing out a graph showing one less petal;(2) The graphics only meet on the axial symmetry, that is, from top to bottom symmetry, left-right asymmetry. When the highest power increases, we can see the whole interval between petals roughly the same point of view;(3) The greater n is, the more similar the basic shape of the graph is, with the difference decreasing, and the red center zone reducing.2. The Structural Characteristics of Julia FractalJ-set of the iterative function remains, but it is different from the M-set. The difference is that z 0is unchanged in M concentration, while c of the J set is unchanged. J-set on the constant c is extremely sensitive, and we analyze the shape characteristics based on the higher-J-set, function zn + c. The conclusions are as follows:(1) The number of the petals in high Julia sets depends on the number of n in iterated function, which is to use n-order iterated function to generate graphics.(2) Their images, both the second and high-order set of Julia set, are very sensitive to initial valuec, different values will lead to different structure of Julia sets.(3) The flower-shaped graphics are very symmetrical; each petal is almost exactly the same.3. The Structural Characteristics of Newton FractalNewton fractal graphics is based on the Function f ( zn )= zn?1. When its escape radius is unchanged, with one only change of the function, its structural features are as follows:(1) a broad petal emerged in the region of the graphics Circle, the number depending on the number of iterated function n, that is to use n-order iterated function to generate graphics of a n-petals.(2) Graphics law is not very obvious, but the function can be seen when the number of times is 3, 6, 9 iterations. The symmetry of relatively is strong, graphic basic shape is relatively similar; other cases are similar, both have a single color region, other petals are nearly the same.(3) All graphics are on the axial symmetry, that is, the upper and lower symmetry.When the escaping radius changes, and with the changing of the iterated function, the structural features are as follows:(1) The number of petals shown by the iterated function still depends on the number of n, which is to use n-order iterated function to map the fractal n-petal flower.(2) When the iteration function of the number of n is among the 3 to 10, the graphics are basically the same shape and color, but a different number of petals, the red center region is gradually increasing.(3) The graphics of the flower is symmetrical; each petal is almost exactly the same.It can be seen that based on several escaping time algorithm categories of typical sub-type graphics, when the iteration function is for the polynomial function, they are generated by a typical flower-shaped structure. A specific number of petals and their iterated function have the highest power of association, in which M-petal set is the maximum number of iteration function power of minus 1, while the J-set and the Newton fractal is the highest power of iterated function. Graphic is symmetry, essentially on the axial symmetry, that is, the upper and lower symmetry. Moreover, with the increasing in power of iteration function, graphics tend to be more vivid, more alike, more symmetrical and more clearly.These conclusions are given for the further study to make better use of fractal design. It provides a powerful data, but also opened up a better grasp of the research methods of fractal graphics.Finally it is given the fractal generated graphics in a variety of apparel and clothing which applied in the design of specific cases, based on Photoshop and 3Dmax software, combined with the three-dimensional model of human body and clothing. For the combination of the fractal algorithms and computer software the study provides the preliminary exploration of analog design.