Characteristics and aesthetic analysis of computer fractal art

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1 Development of fractal art

"Fractal art is made up of simple element symbols， and it is a surprising art to create a complex aesthetic by repeating the art of one of the laws". They can accurately reproduce a natural landscape， or humans have now used the totem， the pattern and the image， the phenomenon is common in daily life， a large part is fantastic， fantasy， these or singular or see shape is used as can be through the double iteration of the fractal element， and all follow certain mathematical relationships， showing the beauty of a rational， is people's understanding of the world and express the world new angle.

The traditional Euclidean geometry can not be described as a very complex and seemingly irregular natural image in nature. For example， the winding coastline， undulating mountains， gnarly section and the vagaries of clouds， Jiuqu River， Criss crossing the blood vessels， dazzling stars such as， they showed a very irregular or smooth， easy to change characteristics.

To explain and describe and express the nature and variety， which was originally the work of an artist. But how to make scholars in various fields of these natural phenomena can be widely understood and as an artist that saw the natural phenomenon， this is the fractal geometry research content， according to the principle of fractal geometry， using computer means， the figurative， art is the research category of fractal art.

The fractal theory can be roughly divided into the following three stages：

2 The characteristic and aesthetic analysis of fractal art

Fractal art works is an unique style， changing thousands， endless. Fractal art reflects the characteristics of many of the traditional aesthetics： balance， harmony， symmetry and aesthetics rules， but more is beyond the standard of new performance： fractal art of balance is a dynamic balance， each part of the picture in changes in the process of mutual restriction of balance. The harmony of fractal art is a kind of mathematical harmony， every shape of the change， every color of the transition is a natural flow， no hard feeling. The symmetry of the fractal art is neither about the right and the left or right， nor the upper and lower symmetry， but the local and global symmetry. In fractal art most is bifurcation， winding， irregular edge and rich change， bring the aesthetic feeling of a kind of pure pursuit of wild， a not civilized not tamed the natural taste.

2.1 Fractal art aesthetic significance

2.2.1New forms of aesthetic thought

Can be found from the history of aesthetics， human in history did not stop pursuit of beauty and performance， both of the Paleolithic cave paintings or New Stone Age Pottery Art Deco， regardless of crafting or machines in production， both the original tattoo or modern clothing， human beings are always used some form of art to decorate a their life and their own， with all sorts of aesthetic pursuit and inner thoughts and feelings expression. This from the original chaotic aesthetic subject object separated on to focused on the development of landscape， this from the pursuit of the beauty of external to internal art and culture is the spirit， from primary visual senses to high aesthetic ideal， but man's understanding of beauty in the sense of survival now.

The United States has the times and the class nature， but the understanding and experience of the United States is always accompanied by the productivity of the time and the development of technology and continuous evolution. In contemporary art and technology， the integration of sensibility and reason is the new embodiment and new development of aesthetics. As a kind of art form which is based on computer technology and rational thinking and means， fractal art can adapt to the characteristics of contemporary art and the spirit of art.

The product of fractal art is not mathematics， but the artist， human understanding of the world， the subjective emotion of novice Duan Yuxin thinking， although it also follows the basic aesthetic principles， such as balance， contrast， rhythm， rhythm， such as laws， but its evolution out the unique aesthetic form， such as the dynamic equilibrium， rational rhythm and rhythm and so on.

2.2 Unique visual form of the United States

The definition of beauty and the scope of the United States is a very complex concept， but it is recognized that the United States is the product of the integration of the subject and object. The visual form of beauty is also the result of the combination of the subjective and the objective. First of all， it is objective， and not subjective conjecture， but it is through the subjective feelings of the extraction and distillation of more objectively than more focused， more intense， more complete expression subjective feelings of "class"， for people to accept and produce resonance form of art. Moved and therefore， when we in the face of nature， our senses and the mind or not and the nature of the colorful， colorful occurred collision and friction， when the subjective feelings and natural beauty of love at first sight， when spark is generated， we will provide， relaxed and happy， linger forgotten to return. This is the formal beauty of charm and magic is we hope has been a long， hard in pursuit of the artistic expression of lifeline.

Form is composed of form elements， such as point， line， surface， body， color， texture， tone， composition， and so on，

Shape and so on by the combination of their relationship. In the fractal art， we can also find that they follow these basic forms of elements， but on the basis of the development of a unique form of visual art and its beauty.

Reflected in color， simulation performance of any wavelength of visible colors and it can achieve perfect transition between different hue of color， there is no any blunt， obscure feeling， not only that， the colorful and beautiful， with a strong visual stimulation and bright natural psychological feelings， reflected in the modeling， the elements of infinite iteration of self similarity： regardless of the enlarged many times or reduced to a certain degree can be found between the whole and partial correlation， these two points is difficult to achieve the traditional art form.

Second， this slight adjustment of the parameters can generate a great visual difference of the art works， for today's digital art design provides unlimited potential as well as a large number of data.

3. Analysis of fractal art form beauty

Formal beauty is the composition of the form of beauty， the means and methods of creation， the principle of formal beauty as the guiding ideology and the action program. Contrast and harmony is the general law of the visual form of the United states. Other elements should be unified under the overall tone of the harmonic that is to say， between all the elements of composition should form a unified tone， the tone can be color， shape， texture or other elements. Contrast is the difference between elements. The contrast and harmony between the elements form the unity of the elements. The unity of diversity refers to a common structural form and rhythm between different parts of the form， which makes people feel that there is not only change and difference in the whole work of art， but also a unified whole.

Diversity and unity is the law of nature， and it is also a common criterion for all works of art， which is the basic law of the form of beauty， and the dialectics of art aesthetics. The unity of opposites reflects the law of the unity of opposites in nature. The whole universe is a unified and harmonious whole.

Fractal art to a new style of art show to the people， said the reason for "art" is because of the following human identified the basic rules of aesthetics and aesthetic psychology， said the reason for fractal art is because. These figures are in line with the aesthetic standards of beauty of form， and give the enjoyment of beauty， so that people realize that the art and traditional art has the same symmetry， harmony， rhythm， rhythm， order and other characteristics of aesthetic standards. But it is more important to go beyond these new standards.

3.1 Symmetry and the law of equilibrium

Symmetry is a common natural phenomenon， we is not strange， regardless of the nature of things or man-made things， everywhere is full of the symmetry： Butterfly Lepidoptera is symmetrical， Rosaceae plants （peach， hawthorn， apple， pear， plum， etc.） of sepal and petal with five rotational symmetry， conch shell， ammonites has symmetric stretching， according to the mirror between object and image is the mirror reflection symmetry， building wall pattern， ancient buildings of Soviet style paintings， window arris pattern often have translational symmetry， and so on. It can be said that there is no symmetry of the basic means of decorative art.

Fractal is a subject that studies and displays the natural image， and the fractal art is a means of the cognitive world， so it is inevitable to study and represent the aesthetic law in fractal art.

Li Zhengdao said in a speech at the Academy of Fine Arts in May 1996 23： "art and science are clever combinations of symmetry and asymmetry." Completely symmetrical things， although give us a stable visual beauty， but in the psychological， but there is a rigid. In fact， in many kinds of art， all forms of art are to seek a certain balance between symmetry and asymmetry. In the law of aesthetics， there is a kind of form that is called "balance".

Balance is the balance of the visual， rather than the complete symmetry， that is， in the form of art， balance is defined in the specific space， form elements to maintain the balance between visual force. Why the people of non balance things full of interest？ Paul Rand once in "on the design idea of" put forward such views： "the symmetric strict give the audience too simple and too obvious performance， which have little or no to the audience to provide spiritual joy and enlighten， arising from non symmetric design of fun is conquer the audience mind consciously and unconsciously inhibition， so as to achieve some kind of beauty to enjoy."

Fractal art can not only achieve strict symmetry， that is， the traditional geometry of the upper and lower， left and right and central symmetry. Also can achieve visual balance， as shown in Figure 2-3， more important is， Art Fractal self similar and show the a new symmetry， namely partial screen and a greater range of local symmetry， or the symmetry of the part and the whole. This symmetry is different from Euclidean geometry symmetry， but symmetry in proportion to the size of the， namely each element in the system and reflect the containing the nature and information of the whole system. Fractal art of equilibrium， is a dynamic equilibrium， a picture of each part of the process of change in the balance of the balance， as shown in figure 2-4. Whether you are from a scientific point of view or from an aesthetic point of view， she is so rich in philosophy， she is the United States and the United States on the scientific beauty of the organic combination.

Fractal art can not only achieve strict symmetry， that is， the traditional geometry of the upper and lower， left and right and central symmetry. Also can achieve visual balance， as shown in Figure 5， more important is， Art Fractal self similar and show the a new symmetry， namely partial screen and a greater range of local symmetry， or the symmetry of the part and the whole. This symmetry is different from Euclidean geometry symmetry， but symmetry in proportion to the size of the， namely each element in the system and reflect the containing the nature and information of the whole system. Fractal art of equilibrium， is a dynamic equilibrium， a picture of each part of the process of change in the balance of the balance， as shown in figure 6. Whether you are from a scientific point of view or from an aesthetic point of view， she is so rich in philosophy， she is the United States and the United States on the scientific beauty of the organic combination.

In the fractal graph， there is an infinite self similarity nested structure， which brings about the great richness of the content of the fractal graphics. Here is the dynamic structure of real change， "is not an art at a glance". A fractal graph can be magnified infinitely， it can be found that the different parts of the fractal image can be different.

3.2 Rhythm and rhyme rule

Ancient Greek philosopher Aristotle said： "people are born to like the rhythm." Platon think： feel the rhythm of the rhythm is a special talent. This is because the tempo is uncommon in life phenomenon， such as nature in the changes of the seasons， the alternation of day and night， especially the movement of life， physiological activities such as breathing， heart rate， pulse， is a concrete manifestation of the rhythm and beautiful rhythm through to people to experience the beauty of harmony.

In the field of plastic arts， the periodic movement of the "rhythm" is the regular pattern. In short， the same image elements appear repeatedly in the same picture of the composition of the law， called the "rhythm"". In the modern decorative art design， the rhythm of the specific performance of the form， including a single rhythm， alternating rhythm， symmetrical rhythm， launch rhythm， reverse rhythm and other forms. Repetition is an important symbol of the rhythm， because there is no repetition can not make people feel the rhythm of the.

Single rhythm： the same modeling elements through the displacement of the screen appears repeatedly in the form of rhythm， called a single rhythm". The displacement can be up and down， around and so on. Rhythmic movement and modeling features： uniform， bright pure.

Alternating rhythm： in the picture， the different modeling elements through the interval and repeatedly appeared in the picture above the rhythm of expression form， called "alternating rhythm"". Rhythmic movement and modeling features： the increase in the number of modeling elements， can reduce the sense of the image of the screen， so that more rich.

Symmetrical rhythm： the form of the image of the form of a mirror image， called the "symmetrical rhythm"". It has a sense of dignity， seriousness and stability.

According to the emission rhythm： linear emission bone constitute a picture of rhythm form， known as the "launch rhythm". According to the curve of emission bone constitute a picture of rhythm， is known as the "rotation rhythm"， the feeling of strong motion.

In short， the rhythm of the construction of the screen， as a result of small， less change， so the most likely to make a unified picture of the visual beauty of harmony. Harmony in the fractal art is a kind of mathematical harmony， every shape of the change， the transition of each color is no blunt feeling， is a natural flow.

Rhythm is elements continuously and repeatedly caused by circumflex tone， has the sentimental factor， rhythm can give a person to taste， to meet the spiritual enjoyment to people. It plays an important role in the modeling is to taste in the form， with a lyrical. Rhythm can enhance the emotional factors and the appeal of the design works， the resonance， produce the beauty， open the art of expression.

Rhythm according to the form of the general characteristics of the following forms：

Continuous rhythm： in one or more elements of a continuous， repetitive arrangement， the formation of a variety of elements with the same basic intervals regularly repeat. Graded rhythm： rules of the gradual formation of rhythm， it can constitute the elements of the size， shape， direction， color to the law of the evolution， such as the gradual lengthening or shortening， the wider or narrower， the color of the cooling or warming， as shown in Figure6.

According to the size of the shape of the triangle， according to the size of the spiral curve as the path， and constitute a gradual rhythm. Showing a strong sense of dynamic balance and a strong sense of order.

Rhythm is the rhythm of the form of purification， rhythm is the rhythm of the form of deepening， the rhythm is rational， and the rhythm is rich in sensibility. The periodic variation of the constituent elements as the length and the strong or weak. Repeat the simple rhythm， rhythm with mechanical beauty， rhythm is not a simple repetition， it is alternate between the change， is the mood in the rhythm of the fusion， can produce unusual beauty in the whole， so， Goethe said： "beauty belongs to the rhythm."

3.3 Gradient and specific rules

In the plastic arts. Gradient is refers to the basic constituent elements between gradually change process， is a quantitative change qualitative change process. A large number of gradual phenomena exist in nature. To throttle spatio-temporal changes， four seasons alternating gradient， gradient change in a day， geographical displacement of the earth's north and south， East and West， equatorial and polar temperature changes， small to people and plants and other organisms and the growth process， maturation and embryo development to fruition， would be impossible without the gradient principle. Therefore， gradual change is not only a natural phenomenon， but also a life phenomenon and aesthetic phenomenon.

Gradual phenomenon of human visual physiology and aesthetic mind has produced such a smooth， safe， comfortable， soft， natural feeling， with strong visual affinity. Gradient can be constituting elements form of changes， changes in the size and color of change， texture changes， but also changes in the position， this a series of similar elements between the slow change constitutes the law of gradient. In the fractal art in the gradual change did not learn the law of the embodiment and the works are numerous， as shown in figure 2-7. Constitute the basic patterns of the color is through the brightness gradient to achieve the stereo sense， many basic structure patterns between the position and size of the gradient reflects the strong space Zongshen Gan. Because the fractal art is produced due to the mathematical iterative function， therefore， the gradient here is a mathematical gradient， changes in every shape， each a color transition are a natural flow， no stiff feeling.

Specific phenomenon just and gradual phenomenon relative is in many similar elements， the sudden appearance of a contrast is very strong element， the element can be a color， status quo， texture， location， size can also be is the sum of these elements. Specific image to the completely the nature of the emergent face， independent not exposure in the shape of a clone copy of all living creatures and so on， quite have the "standout" feeling.

Specific image with distinctive visual tension and strong visual impact to a unique feeling， as if in the dark night lightning， ringing clear sky in a cloud， in order to give people a sense of suddenly chilling. Such as in Figure 2-8 shows the fractal art， in numerous if picture of the star organization in a color， shape and size of the full mutation of the image the has a strong visual and mental stimulation， become the focus of the picture and visual center.

3.4 Contrast and harmony law

Contrast and harmony in decorative art is a pair of contradictory and unified aesthetic rules， reflecting the contradiction of the two states， contrast is the difference between the elements of the harmony between the elements of the composition of the homogeneity. Excessive contrast will make the picture too exciting， too many factors will lead to a lack of homogeneity of the screen， so the artist has been repeatedly debugging between the elements of the balance point". There are many elements in the fractal art， such as the current situation， color， size， texture， and also can be a gradual change， rhythm and rhythm and other aesthetic rules to achieve.

Contrast basically can be summed up in the form of contrast and feel the contrast between the two areas. Compared to the form， size， thickness， strength and brightness， and how much control to strengthen the visual effect. Comparison of the picture effect is clear， exciting， loud， strong sense of intensity. Sense of contrast， refers to the psychological and physiological feelings. Much from the static and dynamic， soft and hard， light and heavy， steel soft， fast speed and so on to give people a sense of texture and pleasure of a deep impression.

Harmony is the coordination of the form of expression， and the important method of achieving unity from the difference is the consistency and coordination of the elements. Harmonic， is the emphasis of approximation， is to make more than two elements of each other have the common character， the formation of a unified visual effect. Harmony is the principle of symmetry， balance and proportion of aesthetics， also includes a variety of artistic means， from a change in seeking the unity， cleverly elements of harmony， can satisfy the people's hearts potential on the order of pursuit. Harmony can be obtained from the similar structure of the picture： such as the similarity of the size， the similarity of the shape， the similarity of the color， the similarity of the position， the similarity of the direction， the similarity of the speed and so on.

Contrast and harmony is a pair of contradictory concepts， but often can be perfectly unified in the same picture， fractal art is no exception.

In contrast to the way： Brown lightness contrast， green and brown hue of contrast， with the figure shows the spatial position of comparative. And harmony is the important way for segmentation harmonic shape and harmonic： dark gold four color with the pattern segmentation into 5 parts， at the same time psychological mechanism and the half a body tends to be complete， the shape and the color harmony， as showed in fig.9 and fig10.

4. Characteristics of computer fractal art

It is pointed out that the theory of information psychology and aesthetics， the layout， balance， symmetry and so on in the meaning of the information theory of all the principles， are easy to understand and make the art works to leave a deep impression on the reasons. But modern art research points out that only to meet the definition of the United States is not a classic definition of art works， a real work of art should be able to stimulate interest， inspire thinking. Obviously， these stimuli from "innovation"， that is， our visual organs to see a new phenomenon in the past did not see the phenomenon of a feeling. From this sense， fractal where what theory is proposed and spread， is forming a new aesthetic ideal and a new aesthetic taste

Fractal art as a kind of new and unique art form. It not only has basic principle of traditional aesthetics thought of aesthetics， also in the visual and psychological created a new and unique construction rules， showing a different from other art forms of the new art characteristics， and these artistic features is by fractal art unique self similarity， winding and nested to achieve.

4.1 Self similarity

Peking University Center for nonlinear science director Zhao says： "the complexity of fractal patterns from simple mathematical relationship of iterative， its appeal resides in the self similarity of infinite levels of the part and the whole， understanding the aesthetic value of the need to have a certain depth of thinking." [Zhao the scientific exploration is of beauty pursuit "see， the fractal art" sequence] the fractal is self similarity， or self similar is fractal art is different from other art most basic， the core elements.

What is the self similarity？ Self similarity is in fact a similar or similar between the whole and the part. In nature there are a large number of self similar case， such as a single tree of heaven and its branches and the value tree to the right， in shape and not much difference， the tree and its branches are the relations in the geometry for self similarity.

The self similarity of fractal art can be divided into strict self similarity and self similarity in statistics. Fractal art pattern generation and design of the basic principles， in addition to and traditional art patterns with the same rules and principles， the most prominent is the use of self similarity of fractal， in shape or composition in the process of introducing recursion or iteration， and the local random disturbance. Basic skills include：

A The main function of the texture is iterative and the repeated application algorithm based on geometric process （or rule）.

B The coloring technology of the combination of the algorithm and the human-computer interaction.

C Image layout using the combination of traditional and fractal algorithm mapping technology. For example， the overall fractal composition， the rules of the skeleton map and a variety of flexible algorithm distribution of composition， etc..

Fractal art with the aid of the computer to carry on the creation. Therefore， his works have great random and arbitrary， but also often unexpected， novel and chic， strange and changeful， refreshing， with a strong sense of the times.

4.2 Winding property

In fractal art pattern more is bifurcation， winding， irregular edge and rich transformation， she gives people a kind of pure wild beauty， a not civilized， not tamed the natural taste. The artistic characteristics of this kind of art are closely related to the characteristic of the fractal art.

According to E. h. Gombrich "aesthetic pleasure from of a medium ranged from tedious and messy between patterns of ornamental， monotonous pattern to attract people's attention， overly complex patterns will then stop to watch our perceptual system overload." Fractal structure is extremely complex， between locality and it always has infinite twisted together and each local have more changes in； however， it is complex and not chaos， it has inherent order， a self similar structure， both locally and on the overall said. This symmetry to avoid the Euclidean geometry symmetry brings inflexible feeling， the whole picture from the dynamic looking for the balance of beauty， make in the eager excitement and in deep， it has generally corresponds with the constraints， so that the wild free so as not to lose control tend to chaos. From the point of view of fractal structure， the aesthetic ideal of the fractal theory is not only in the simple， nor in the chaos， but in the order of chaos.

4.3 Nested

Bennan-Wang in the science and art of marriage， says： "no matter is in the nature of the individual fractal morphology， or mathematical methods for generating fractal patterns， infinite nested， segment and segment of self similar geometric structure."

Many great works of art have such features， such as the art palace in Paris， the sculptures and monsters， angle and measuring column， full of vortex pattern of arch wall and equipped with canopy structure adorn tooth is eaves， Dunhuang Mogao Grottoes in the decorative arts，Fig14.

Gombrich discussed the design principle of the decorative pattern as art， which is exactly what we are saying. "Now，" he wrote， "we might be useful to simplify the design principle to a single subject. Draw an equilateral ten shape （i.e.， fractal） and then painted on both sides of the figure ten figure ten. As long as the pen can find tapering， as long as the paper is not broken， the program can always down to （this is the iterative method of fractal composition， here said the paper and pen difficult， the computer can be solved. The continuation of the outside is also possible， you can put another ten characters as the first ten - shaped frame， the two programs can be infinitely continue. Obviously， the first kind of painting， namely inward method for connecting to any remaining area， the closest to the style of decoration， decoration driven who continue to fill the blank remaining power is generally described as fear of leaving voids， many non classical style is supposed to possess the fear of leaving a void characteristics. Perhaps it is more appropriate to describe the dynamics in terms of the word love."

Fractal art graphics contains infinite nested structure， the structure of nested brought picture of great richness， and contains the infinite creativity， the appreciator needs careful heart to all intension grade picture. As the French Impressionist Renoir said "art is not clear".

This kind of fractal art and the traditional art model have a common feature that is not a constant scale， because you can find new scales in their different local structures. Observer from any distance looked to see a proportionally to details， when you get closer， its structure is in the scale change， also in changes， the only constant is： strong decorative means and new structure factors show， new details.

Fractal of this infinite nesting of source in manufacturing process of iterative algorithm， whether it is how complex fractal pattern. In fact， you can use a simple fractal element by certain rules of the iterative generation， resulting in details in different scales.

Compared with the traditional art patterns， fractal art pattern is undoubtedly a great innovation， while the fractal art itself is constantly innovation. We can have an infinite number of coloring scheme using many mathematical functions， as well as some kind of generation method and different combinations， so fractal art has almost endless possibilities， coupled with artificially created. Fractal art world wide.

Study of traditional aesthetics principle and fractal art of unique characteristics， its significance lies in to these formal beauty rule flexibly applied to fractal art creation to， and to demonstrate the unique charm of the fractal art， so that the form can better express the contents of visual art， the art form of beauty and beauty of the content of a high degree of unity.