Traditional theory of art usually tryes to explain every concrete artwork, its unique features, that differ this artwork from others. Complexity theory of art reveals universal features of art, that make this artwork as genuine art. This might be compared with the study in a biology. Before XX century biology studied mostly phenotype features, describing unique traits of living organisms. But biology of XX century studies such common genotype features, inherited to the whole animated matter, as genetic code, protein folding, etc. Traditional theory of art might be compared with phenotype studies in biology, whereas complexity theory of art - with genotype studies.

Complexity theory of art is based upon complexity theory of brain functioning, whereas traditional theory of art usually doesn't rely on psychology and theory of cognition.

In perception psychology, multistabile perception of ambiguous figures is often considered as a marginal curiosity. Nevertheless, this phenomenon is one of the most investigated psychological phenomena, because it has attracted perceptual scientists since the first description of ambiguity, given by Necker in 1832. Actually, ambiguous patterns are not freak phenomena. Every pattern, in a way, is ambiguous multistable pattern, but in everyday life, using additional information, we usually resolve or avoid ambiguity [Kruse 1995]. Nikos Legothetis recently shown that resolution of ambiguity is an assential part of consciousness job [Legothetis 1999]. The objective of this section is to show that the mathematical models of the perception of ambiguous patterns can be regarded as the basic models of artistic perception.

can be modeled by using elementary catastrophe "cusp"

The switch between two interpretation could be described by elementary catastrophe "cusp" x³ - bx - a = 0, where a and b are control parameters and x is the state variable. The first parameter a quantitatively describes the change in bias in the drawing in a "shape space".

We can connect any two points in "shape space" by a streight line and create any intermidiate shape when we move along this line and describe smooth trasformation one shape into another. Figure 2 gives an example of such smooth transformation face of actress E.Taylor into face of Presidente USA John Kennedy in "face space" [Brennan 1985]

Figure 2. Example of transformation in "face space"

It is reasonable to develop these ideas on "meaning space" firstry introduced by Ch.Osgood [Osgood 1958] in order to describe a perception of double meaning situations.

The second parameter b describes how much the amount of detailes is presented in the ambiguous figure. The state variable x is presented as a scale from +10 ("looks a lot like a man's face") to - 10 ("looks a lot like kneeling girl"). For this model we could formally represent potential function V = - 1/4x⁴ + bx² + ax which depicted on Figure 1, and consider catastrophic jump from one image to another as non-equilibrium phase transition. Actually, this potential function could be regarded as J.J.Hopfield's potential function [Hopfield 1982].

l ₁ and l ₂ are time dependent attention parameters, whereas A, B , and g are constants. A solution of the these equations describes oscillations of perception.

Let us first consider visual ambiguity in art. An example of such ambiguity is Invisible Bust of Voltaire by Salvador Dali.

Figure 3. Ambiguity of Voltaire bust in Salvador Dali's painting Invisible Bust of Voltaire

The most famous example of ambiguity in painting is, of course, Mona Lisa by Leonardo. About it, in The Story of Art Ernest Gombrich said:

"Even in photographs of the picture we experience this strange effect, but in front of the original in the Paris Louver it is almost uncanny. Sometimes she seems to mock at us, and then again we seem to catch something like sadness in her smile." [Gombrich 1995]

"This is Leonardo's famous invention the Italians call "sfumato" - the blurred outline and mellowed colors that allow one form to merge with another and always leave something to our imagination. If we now turn to the "Mona Lisa", we may understand something of its mysterious effect. We see that Leonardo has used the means of his "sfumato" with the utmost deliberation. Everyone who has ever tried to draw or scribble a face knows that what we call its expression rests mainly in two features: the corners of the mouth, and the corners of the eyes. Now it is precisely these parts which Leonardo has left deliberately indistinct, but letting them merge into a soft shadow. That is why we are never quite certain in which mood Mona Lisa is really looking at us. Her expression always seems just eludes" [Gombrich 1995 p.228]

Figure 4. Ambiguity of Mona Lisa's smile

Recognition of facial expression of emotion is one of the first communicative abilities in human life, that occurs much earlier in childhood than for instance the faculty of speech. Facial expression of basic emotions (joy, fear, etc.) form structure stable patterns and H.Haken with co-workers carried out experiments on computer recognition of some facial expressions irrespective of individual person. In these experiments the success rate was about 80 percent [Haken 1996].

The ambiguity of Mona Lisa's smile one can compare with ambiguous images like "kneeling girll - man's face". The oscillation in the perception of that painting can be described by Ditzinger-Haken's model.

We see that brain resolves a visual ambiguity by means of oscillation. A semantic ambiguity (the ambiguity of meaning) is a result of ambiguous words or whole sentence [Kruse 1995]. Semantic ambiguity, wide spread in comic situations, also resolves by oscillations (like visual ambiguity).

Let us consider semantic ambiguity occurring in double-meaning comic situations. Example of such ambiguity is the following note:

In ordinary speech, and especially in scientific communication, in general we try to avoid ambiguity; by contrast, in humor, one of the aims is to create ambiguous situations to provoke a laughing. Double meaning of a word makes us laugh.

Another situation of perception of ambiguous patterns occurs in a parody of a famous person by some actor. On one hand, we can recognize the manners, gestures, style and voice of that famous person. On the other hand, we see quite a different person. Same method is used in literary and poetic parodies. Every time, we are dealing with bimodal, double-meaning situation. As a result, we have the oscillation of perception, and laughter is one of the external manifestations of this oscillation [Yevin 2000]. Evidently, a laughing person every time mentally oscillates from one meaning of double meanig word into the second meaning and vice versa, by comparing them. As a result, the rhythmical laughter is generated by the nervous system. Anecdotes, jokes and sketches deliberately are created as short as possible (laconic), in order to reduce the time needed for the saturation of attention in the process of recognition.

Tonality is a hierarchy (ranking) of pitch-class. If only one pitch-class is stressed more than others in a piece of music, the music is said to be tonal. If all pitch-classes are treated as equally important, the music is said to be atonal.

Three stable steps of tonality: tonic, mediant and dominant are keynotes or attractors of neural network. Others steps of tonality play the role of recognizable patterns, gravitating to some or other keynote.

After 1900, some composers abandoned tonality; but even today much of the music we hear is tonal. Figure 4 depicts Hopfield's potential function for major tonality.

Figure 5. J.J. Hopfield's potential function for major tonality. Horizontal axis -frequency of sounds.

We can depict exactly only distances between minims, but not depths of these minims. It is reasonable to suggest that the minims depth of tonal potential function is extremely personal for human beings and reflects a musical abilities (giftedness) of a person. The more a person has a gift for music, the more depth of valleys has appropriate Hopfield's potential function. A person who is devoid of music ability has energetic function with shallow valleys. It is interesting to create artificial neural network with the same energetic function as a tonality and evaluate the dependence of minims depth against the size and number of connections between neurons.

S.Kelso shown that the variation of magnetic field generated by intracellular dendritic currents in the brain are described by deterministic chaotic attractor [Kelso 1995].

Hubler [Hubler 1989], E.Ott, and others [Ott 1990] formulated the problem concerning controlling chaos. Due to important applications this problemhave attracted considerable interest in different fields of science.A.Pattel and E.Balaban [Patel 2000] was able to show that magnetic brain patterns from certain neuronal cell assemblies would follow the pitch contour of the tone sequences.

Birbaumer and others [Birbaumer 1996] revealed that music tends to lower the degree of chaos in brain waves in that the dimensional complexity is reduced.

One can suggest that any musical score might be considered as a program of controlling chaos in the brain. This program control the degree of synchronization of chaotic attractor in the brain making it more or less chaotic.

I would like to thank G.Mayer and J.Mikes for helpful discussions.