Make your own free website on

3. Bifurcation and aesthetics

The whole history of art exhibits inclination towards following some regular patterns. At the beginning of an artistic epoch, the art is calm, simple, clean and easy to recognize as an original movement. In its full development, some changes started, trying to diverge from the initial rules. It causes introduction of unnecessary extraneous elements and after while, the art movement becomes too complex, with too many divergent branches that can hardly  be called  unique. This process makes a kind of "dynamics" that can be found in larger or smaller scales of time. For ex., the development of a single artist  sometimes resembles on development of the mankind art through centuries.

The author's suggestion is to distinguish a special "order" in these dynamical processes that are characterized by gradually progressing in making forms more complex. Here, we have to restrict ourselves to visual arts alone. Otherwise, the story will be too extensive, but other art branches exhibit similar phenomenology.

Constancy  Science is full of different but important constants. So, we have  c = 2.99792458×108 m/sec  as light speed, e = 9.10938797×10-31 kg  as the mass of electron or g = 9.80665 m/sec2 as the gravitation constant. Quantum physicists found that six constants determine the creation of the whole Universe. These are W= 0.03 (a ratio of the observed density of matter in universe to critical of matter required to stop the expansion of universe)l = 0.07 (a new force discovered in 1998),Q = 10-5(measure of how much force is required to brake apart structures in our universe),N=1036(a comparison of strength of gravity to strength of electric force in atoms),e0.007 (the efficiency of conversion of Hydrogen to Helium and to heavier elements their after), D=(dimensions of space).

But, the history of visual art found constants that were important for visual esthetics. Such constants are proportions or "sections". Some of them are (“Holly section”), (Paladio proportion) or  (“Golden section”). The calm sea has flat and smooth surface. Great art has tendency to be simple and still. The blueprint of constancy in visual arts is a horizontal line. Then, its dual, a vertical. Ancient Greek art is a rich source of masterpieces being produced as unity of horizontal and vertical in harmonic combination using  proportion.  Marble colonnade and human figure (Fig. 13) express similar sensation of stability due to harmony in proportions. A house is usually made using human proportion as a modulus. Even in cases when the house is just slightly sketched (Fig. 13, right) it has firm form of a rectangle. to show how important this space is for man. Some artists, up to nowadays, turn back to some constants in Nature. One of such constants is circular form. It is universal and widespread. Two examples lent from artistic heritage of Turner and Malevich are presented in Figure 14. In many examples, constancy reveals in repetition of identical objects. This formation elevates the a esthetic value of an object. In [Bori], Borisavljevic addresses to this a esthetic phenomena as the law of identity.

Figure 13.  Constancy as expression of stability. Left: Doric colonnade; middle: Kroisos, 525 BC, Athens; right: Chinese relief in granite (18. century).

Figure 14.  Constancy (circle) as expression of harmony. Left: William Turner, Buttermere Lake, 1798, oil, Tate Gallery, London; right: Kazimir Malevich, Black Circle, 1913.

Linearity The next, more complex term of art is linearity. Increasing or decreasing. Diagonal versus horizontal. It is almost a miracle how the diagonal contrasts horizontal in a fresco of Giotto (Figure 15, left). Here, the Flaming Chariot is directed diagonally upward to the skies, which stands as a sharp contrast to the firm construction of a temple that stays down. The same diagonal is used by Michelangelo in his famous Sistine fresco The Creation of Man. Both diagonals causes vivid impression of motion. The next usage of linearity is perspective, the great discovery of Renascence. It brings illusion of space that can be used in different purposes, as works of Van Eyck and De Chirico in Figure 16 shows. The first one uses perspective to describe interior of a church. The last, constructs its subjective space that does not obey the usual law of perspective. In fact, he uses false perspective to stress that his geometry is not  trivial, but it comes from internal, psychological space. Also,  linearity can be used as visual form (linear drawings etc.), and a canvas of Louis (Fig. 16, right) is a good example.

Figure 15. Left: Giotto di Bondone, Legend of St. Francis 8. Vision of the Flaming Chariot,1297-1299, fresco; right: Michelangelo, The Creation of Man (Image 1), 1511-1512, fresco, Sistine Chapel.

Figure 16. Left: Linearity as instrument of illusion: Van Eyck, Madonna (1432-1433, central panel of a triptych, oil on panel); middle:  Linearity as instrument of surreal expression: Giorgio De Chirico, Piazza d'Italia con Statua Equestre, (1918, oil on panel); right: linearity in composition: Morris Louis, While, (1960, acrylic on canvas).

In all above examples, linearity reveals in repeating of  similar objects that are transformed or by perspective or by color or by location. This repetition increases esthetic value of an object. This subject is also argued in [Bori], where Borisavljevic calls it law of similarity.

Periodicity   If the wind increases, the calm sea becomes wavy. Periodic patterns are visible in many natural objects. Living organisms, shells, formation of leafs and organs in fitotaxy. These forms were transferred to art. In painting of Hieronimus Bosh, The Adoration of the Magi, there is a detail showing a stone bridge and a row of trees behind it (Fig. 18, left). The arch of a bridge is repeated in the shapes of trees' crowns as an echo, but not in a linear manner, like in Borisavljevic's law of similarity.

Figure 17. Periodicity in Nature: a. Dynamics in fluids; b. Shell forms; c., d. Plant structures.

Rather, one can use word rhythm. Such a rhythmic compositions are characteristic for many modern painters, sculptors or architects. Robert Delauneay is known by his circular periodic rhythms (Fig. 18, middle). Giorgio De Chirico uses periodical repetition of levels in his tower (Fig. 18, right) in order to convey his metaphysical idea of infinity. Rhythmic compositions are frequently used by post-impressionistic movements, like cubism, futurism or dada. The Plasticity and Rhythm of Things  by  M. Sironi is a composition on the intersection of cubism and futurism (Fig. 19, left). A very known futuristic artwork by M. Duchamp, where movements of a female figure was analytically developed over the space (Fig. 19, middle). Here, periodicity is a bit more complex than in Sironi's still life. Periodic rhythms need articulation, otherwise, it become too complex and chaotic. This articulation can be inspired by different sensations. Sironi finds such rhythms in static things, Duchamp in moving objects, but some artists can be inspired by more abstract categories. Figure 19, right shows a walka (walka = drawing, picture) done by a member of Australian Aboriginal tribe that express sensation of fire.

Figure 18. Periodicity in art: Left: H. Bosch, The Adoration of the Magi (detail, 1500, oil on panel, Prado, Madrid); middle: R. Delauney, Homage to Bleriot, (1914, oil, Kunstmuseum, Basel); right.  De Chirico, The Big Tower (1918,oil on panel, privatecollection).

Figure 19. Left. Mario Sironi, Plasticity and Rhythm of Things, 1914, tempera and pencil; middle. Marcel Duchmp, Nude Descending a Staircase, No. 2, 1912, oil on canvas; right. Waru (fire),  walka drawing, Ernabella community, Australia.

Complexity  The next stage in visual appearances is over the edge of periodicity. Complex forms represent complex dynamics. Water surface exhibits such dynamics  when the wind become strong enough to overcome simple periodical movements of the waves. The motion of water starts being irregular (Fig. 20-a). Wind leaves its signature on desert sands (Fig. 20-b). Tectonic forces combined with wind activity arrange beauty of  icebergs (Fig. 20-c). Complex forms can be more or less irregular. Thermonuclear processes in the Sun are far from having any regularity, but still the form of Sun is more or less spherical (Fig. 20-d). So, the matter of the Sun is not in total chaos. Some call this state deterministic chaos. But, in patterns of plant Broccoli Romanesco (Fig. 20-e) order is much more visible, although it is not periodic at all. This geometry is known as self - affine. What is art correlative? The complex forms too. Starting with primitive Indonesian art, the Mask (Figure 21, left) is everything but geometric regular piece. It fails to represent periodic structures, but it is not amorphous too. It possesses a strange degree of irregularity, probably caused by the form of naturally cut pebble stone. But, of the other side, this irregularity favors more expressive face of the mask, probably some evil spirit. This kind of complex irregularity is inherent to primitive art (Fig. 21, middle) and it is discovered again in 20-th century by modern artists, like Klee (Fig. 21, right).

Figure 20. Left. Mario Sironi, Plasticity and Rhythm of Things, 1914, tempera and pencil; middle. Marcel Duchmp, Nude Descending a Staircase, No. 2, 1912, oil on canvas; right. Waru (fire),  walka drawing, Ernabella community.

Figure 21. Left. Dayak primitive, Mask, Indonesian archipelago; middle. Arrival of the Sun,Eskimo art; right. Paul Klee, BeforetheSnow, 1929watercolorcollectionAllenbach,Bern.

Complexity of forms was the topic of study of cubists (Figure 22, left) but, in composition sense, complexity was used for mass genre scenes, characteristic for Bosch and Bruegel.

Figure 22. Left. Picasso, Seated Bather1930, oil on canvas, Museum of Modern Art, New York; right. Bruegel: Netherlandish Proverbs1559, oil on canvas.

Chaos  Further increasing of wind over an irregularly wavy water surface will cause chaotic motion. Water droplets will be put in accelerating and chaotic motion (Leonardo' s Flood is an example). Forms of such dynamics are hardly to predict, nevertheless if one speaks about sea in storm, lightning dendrites or nebula formation in far interstellar space (Figure 23).

Figure 23. a. Chaotic fluid motion; b. Lightning has unpredictable form; c. A Space nebula.

Now, we are back to the "blotting-fratting"  painting method from previous section. What is bigger challenge for an artist than to explore and reproduce extremely complex patterns of grass, dry leaves, moss, rocks...  Fantastic stones in Caspar's Chalk Cliffs on Rügen (Fig. 24. leftmost), a snowstorm in Hokusai kakemono picture (Fig. 24. middy-left), rubber forms in early Kandinsky (Fig. 24. middy-right) or de Kooning's expressionistic figures (Fig. 24. rightmost) have chaos in common. These different artworks are attacks to the smooth, boring world of  ordinary  Euclidean objects. It is revolution of complexity in its higher level. If isolated, parts of these figures show no recognizable or  perceptible objects. Once chaos appears on art panels, it newer stops. Abstract expressionism of Mark Tobey (Fig. 25. left and middle) and Jackson Pollock (Fig. 25. right) now belongs to the classic heritage. 

Figure 24. To the right: Kaspar David, Chalk Cliffs on Rügen (detail), 1818-1819, oil on canvas; Katsushika Hokusai, Eagle in a Snowstorm, 1848, kakemono scroll, ink and somber color on paper, Pacific Asia Museum Collection; Vasilii    Kandinsky, Green figure, 1936, oil on canvas, Galerie Maeght, Paris; Willem de Kooning, Woman, Sag Harbor,1964, oil on wood, Hirshorn Museum, Smithsonian Institution, Washington.

Figure 25. Left: Mark Tobey, American Visitors, 1954, mixed technique; middle: Mark Tobey, Head, 1957; right: Jackson Pollock, Eyes in the Heat, 1946, oil on canvas, collection Guggenheim.

Let us go back to the title of this section. Bifurcation that is mentioned there is connected with the universal phenomena that described a route that lead from absolute calm (constancy), over linearity, periodicity and complexity to chaos. The usual bifurcation diagram is connected with the logistic mappingfl(x) =lx(1 - x)  and numerical iterative procedure for finding roots of  logistic equation x  =  fl(x)   where  l is not less than 0. It is easy to see that  solution depends on parameter l. For 0 < l 1 and = 1, this equation has zero as double root, for l > 1  it has two simple roots, a = 0 and b = (l - 1)/ l.  Simple iteration method generates the sequence xk+1=lxk(1 - xk), k = 0,1, 2,..., with starting point  0 < x0 < 1. According to the behavior of this sequence, one can study dynamics that range from constancy to chaos. The Table 1 classifies types of the bigger fixed point, that exists if l > 1. The values 0 <=l < 1 bring only  trivial solution of x  =  fl(x), an this is a = 0 . We can address to this case as constant dynamics. Indeed, nothing has changed. For1 < l < 2, the sequence {xk}k>0 approaches to b uniformly from the left, regardless the position of 0 < x0 < 1 (Figure 26, left). Diagrams that graphically show how the iteration procedure develops (Fig. 26 and 27) are known as web-diagrams.  For 2 < l < 3, the sequence  {xk}k>m (m is big enough) approaches to the solution  b = (l - 1)/ l  by oscillating  around it at the same time (Figure 26, right). The intersection point between graphs of functions x and lx(1 - x) for 2 < l < 3, moves along the line y(x) = x  as l increases from 2 to 3. By this reason, we call this dynamics linear.

zeroes (fixed points)
type of fixed point b
0<=l < 1
a = 0
l = 1
a = b = 0
1 < l < 2
a = 0,  b = (l - 1)/l
attractive (staircase in)
l = 2
a = 0,  b =1/2
super attractive
2 < l < 3
a = 0,  b = (l - 1)/l
attractive (spiral in)
l = 3
a = 0,  b =2/3
3 < l<=4
a = 0,  b = (l - 1)/l
repelling (spiral out)

Table 1. Classification of the dynamic behavior of fixed point b of the logistic equation. 

Figure 26. Web diagrams for  l = 1.742 (left) and  l= 2.728 (right).

But, the most interesting part is when lranges from 3 to 4. For ll1 = 3 one has first bifurcation. It means that the sequence {xk}k>m (m is big enough) fail to converge any more and its oscillations around the fixed point become stationary. The lower limit of {xk} is  called low quasi-solution, the other is high quasi-solution. Thus, the value l1 = 3 is the value of first bifurcation. The web-diagram (Figure 27, left) shows this oscillatory regime with period 2. If l  reaches  l2 = 3.449489... the sequence {xk}k>m continues to oscillate but now has two "frequencies" or period 4. This phenomena is caused by new bifurcation of each "quasi-solution". After l3 = 3.544090... oscillating period becomes 8, after l4 = 3.564407... it is 16 (Figure 27, right) etc. The sequence l1l2,l3l4, ... has the limit lf= 3.5699456... so called  Feigenbaum point. So, the dynamics that occur when runs the sub-range (3, lf ) we can call periodic dynamics.

Figure 27. Left: Web diagram for  l = 3.4 (period 2);  right: Web diagram for  l= 3.569 (period 16).

If increases beyond the Feigenbaum point, the sequence {xk} exhibits complex behavior. In fact, we enter a zone in which it has chaotic behavior but for some values of some repeated "windows" of  different periodicity. The numbers of periodicity of these "windows" are another strange fact. Let O be the set of all odd positive integers, and kis the set of all products of  k with members of O.  Then, the set of numbers of periods is  P = {2j ; j fomN0}U 2jO. It happens that P is ordered i.e. it makes a chain,  called Charkovsky sequence [Peit2]. The web diagram that corresponds to chaotic behavior is shown in Figure 28 (left).

Almost literal similarity between  web-diagram and the picture by Frank Stella (Fig. 28, right) is really astonishing.

Figure 28. Left. The web-diagram for logistic mapping l= 3.820; right. Frank Stella, Les indies galantes V, 1973, offset lithograph.

The integral information about bifurcation route to chaos is usually comprises in what Chaos experts call bifurcation diagram. It is shown in Figure 29, in red color. This diagram represents graphical behavior of accumulation points of the sequence {xk} (ordinate) via the parameter (abscissa). Now, we can somehow place particular artworks of all epochs over bifurcation diagram regarding to their degree of complexity. The single branch of the diagram (< 3) offers locations for constancy and linearity in art, in the sense as it is explained above. After bifurcation we have increasing degree of periodicity, until the Feigenbaum point. After that comes part of diagram that corresponds to art pieces of higher complexity that can increase up to the chaotic works. Such analysis, let us call it bifurcation analysis, can help in classification or even characterization of different products of visual art. We note that it is possible to make this classification according to fractal dimension of each painting, graphics or other piece of visual art that can be determined experimentally [Barn].

Figure 29. Bifurcation diagram and approximate locations of some works of visual art on it.

Such analyses may reveal that some classical painters did masterpieces that contain parts with different degree of chaoticity (fractal dimension). For ex., in some oil from Titian, one may find constant or linear passages, resembling to Mondrian, periodic textures like in neo-impressionistic Signac or chaotic configurations of  Mata Echaurren (Fig. 30).

Figure 30. Bifurcation analysis of a Titian painting. Extracting constant-linear, periodic and chaotic components.