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2. Fractals in history of painting

A classic Greek author Philostratos, in his dialog between Apolonius and Damis, argues about "images in the clouds". His conclusion is that help of observer’s mind makes these images only while an artist use both mind and hands to mimic the Nature. Plinius noticed that a sponge, impregnated by color and hit to the wall sometimes produce nice effects. These are first examples of mentioning complex or fractal shapes in visual arts. According to [Homb], the Chinese painter Sung Ti (XI Century) suggested the method known as Chinese occlusion to invent nice landscapes. A piece of white silk should be placed over an old, decrepit wall. Looking through it early morning or in evening, one can see "landscapes" that can remember and then transfer to its own artwork. It is possible that Wang Ximeng, painter of North Song Dynasty, used occlusion technique (see Fig. 1).

Figure 1. Left: Wang Ximeng (c.1196-c.1120), A Thousand Li of Rivers and Mountains, hand scroll painting on silk; right: Leonardo da Vinci: Flood, sanguine on paper.

The Italian renaissance painter Andrea Mantegna experimented with clouds having form of human faces [Homb]. But Leonardo da Vinci in his famous Treatise of Painting gave the most extensive treatment of what now is called Leonardo method or blotting method. Might have been inspired by Pietro del Kosimo, da Vinci emphasized power of "messy forms" like stains on old walls, clouds or muddy water in "favoring mind on various discoveries".

Hoogstraeten, the XVII century writer, testified that the Dutch painter Jan Van Goyen (1596-1656) was capable of getting out a picture with "small efforts" from stains of colors. So, we come to XVIII century, the period of cutting with tradition of "genuine genius" that was characteristics of renaissance. Here we found a landscape painter, Alexander Cozens, author of a textbook of drawing "A New Method of Assisting the Invention in Drawing Original Compositions of Landscape" that uses and further developed blotting method. He was aware of importance of having "a priori" knowledge of different schemes of skylines. These schemes include precise copy of mountain ranges. If the ink stains can be seen as fractal forms in painting, these skylines are fractal curves first isolated and studied by Cozens. The upper part of Figure 2 shows one of its drawings. From the skyline, ten points were selected and a fractal interpolation curve was constructed, by using interpolating method given in [Barn]. Without doubt, the 17th-century French landscape painter Claude Lorrain has big influence in such freely approach towards landscapes.

Figure 2.  Van Goyen clouds vs. fractal IFS "cloud". Left: Van Goyen, Two Men on a Footbridge over a Stream, 1655, oil on panel (detail); right:  The "cloud" generated by Iterated Function System.

Importance of the Cozens method becomes obvious by its influence on famous John Constable. Alexander’s' relative, John Robert Cozens also uses this method (Figure 3, a). 

Figure 3.  Above: Alexander Cozens, Streaky Clouds at the Bottom of the Sky (c.1785), drawing on paper; below: Fractal interpolation. The fractal curve interpolating the Cozens skyline in 10 points.

Not only painters use blots of ink or color to improve their inspiration. A German poet Julius Kerner uses stains (germ. Klecks) of ink on folded paper to recognize live figures of "ghosts", and then he wrote poems about these figures (Figure 4). Note that some fractal sets, obtained as basins of attraction of certain nonlinear dynamic processes (Figure 5), that are, fractal sets.

Figure 4.  a. John Robert Cozens (1752-1797), Satan Summoning his Legions, circa 1776, watercolor on paper, Oppé Collection; b., c., d. Justinus Kerner, ink stain figures from Klecksographien (Stutgart, 1853)

Figure 5.  Fractal sets that resemble ink stains. These sets were created by UltraFractal 2.04 software using the following formulas: a. and c. B. Margolis; b. Carr2821b UF version by Bob Carr modified and optimized by Sylvie Gallet; d. Music by D.H. Van den Berghe.

Joseph Mallord William Turner (1775-1851), English landscape painter, was the next big artist that uses amorphous stains of color to gain vibrant and dramatic treatment of natural light and atmospheric effects in land and marine subjects (Fig. 6, left). It is widely accepted that his work had a direct influence on the development of impressionism. In fact, impressionists also use freely formed colored spots to depict subtle variations in color of sky, water or vegetation.

Figure 6.  Left: William Turner, The Morning After the Deluge 1843. Tate gallery, London; right: Claude Monet, Marine View, Sunset 1874, Philadelphia Museum of Art.

By atomizing color grains, impressionism went to its pointillist stage, with Georges Seurat (1859-1891), and Paul Signac (1863-1935) as main representatives (Figure 7, left). A specific and original approach in using stochastically distributed blots of color was performed by French painter Odilon Redon (1840-1916), see Figure 7 (right).

Figure 7.  Left: Paul Signac, The Bonaventure Pine in Saint-Tropez 1892, Museum of Fine Arts Houston; right: Odilon Redon, Red Boat, 1906-7.

 Besides exploring more or less amorphous and chaotically scattered particles of color, there is another, more "mechanical" method for producing effects of natural patterns. This is  “frottage” or “fratting” method, known from children's game of reproducing a coin relief on a piece of paper, by  pencil rubbings over the paper keeping fixed over the coin. This technique, applied to different surfaces, like wood grain, fabric, or leaves brings to visual effects that is different than one produced by "blotting method". Frottage  was used by Gustave Moreau (1826-1898) and some surrealists: Max Ernst, Salvador Dali and many others. Ernst also used "decalcomania"  method, which is transferring of paint from one surface to another by pressing them together. These techniques are readable in  a series of drawings (Histoire naturelle from 1926) and in many paintings such as The Great Forest (1927) and The Temptation of St. Anthony (1945). Salvador Dali is a particular story. His opus is literally a huge mixture of realistic traditional techniques and avant-garde experiments. In his early works, Dali used pebble, glued on canvas, layers (Fig. 8, right),  pieces of wood, bones, metal objects etc. Blotting and different frottage effects followed his style permanently. Dali also developed a few new techniques that can be viewed as imaginative variations of blotting method. For ex., he used explosions to scatter nails or metal particles all over the surface, like in  Pieta - From The Apocalypse of St. John, from 1959. Then, he used technique of scorching and smoking paper by candle (Sfumato, 1972 [Desc] ). Dali invented a fancy variation of da Vinci method by "recognizing" figures and genre scenes in white empty spaces between newspaper lines [Dali ]. In period between 1934 and 1938, he uses blots of Indian ink as well as frottages in many designs (Portrait of Rene Crevel, The vertebrate Grotto - Transfer Series, Woman's Head with Shoe, Gradiva, Fantastic Beach Scene with Skeleton and Parrot, etc.) A lot of his blotting-fratting experience was took part in his famous Tuna Fishing, from 1967. 

Figure 8.  Left: Gustave Moreau, Sappho, 1884, watercolor; middle: Max Ernst, Napoleon dans le desert,1914, oil; right: Salvador Dali, Big Thumb, Beach, Moon and Decaying Bird, 1928, combined technique.

But, Salvador Dali is apparently the first artist that explicitly painted a fractal! It was his Visage of War, oil from 1940. It shows a hallucinating vision of skulls nested inside skulls using "Russian doll" geometry. A slight analysis reveals that the fractal set that corresponds to Dali's work is so called Cantor dust. It is generated by three contractions with approximate contractive factor about 0.21. Using well-known formula [Barn] the box dimension (in our case it is identical to Hausdorff - Besicovitch dimension) of Visage of War is about 0.705. This fractal is relative to Sierpinski triangle (Sierpinski gasket) and falls to the family of so called Cantor dusts [Mand]. Author has no information about the history of this painting, but for sure he was not aware of fractal geometry at this time.

Figure 9.  First "fractal" in visual arts. Left: Dali,Visage of War, 1940, oil n canvas; right: Cantor dust fractal set with Hausdorff-Besicovitch dimension  approximately 0.705.

Different variations of blotting-fratting technique were used by many other surrealist painters: Andre Mason, Oscar Dominiguez, Raoul Ubac,  and Jacques Herold etc. Even "objects" got their "fuzzy" structure. The nice example is "La tasse en fourrure" an object from 1936 made by  Oppenheim (see [Alex]) (Figure 10, left) that extends the subtle flickering sensibility of sculptures, introduced by  Boccioni (Umberto Boccioni, Unique Forms of Continuity in Space, 1913, bronze,  Museum of Modern Art, New York). There are different techniques in Computer graphics that introduce "fuzziness"  into smooth objects. But, none is as expressive as fractal one. The same figure (Fig. 10, right) shows a "furry" variation of Sierpinski triangle. The tiny fur hair is a consequence of having at least one unstable component added in the known Sierpiski fractal construction.  In fact, Sierpinski triangle is a fixed point of the operator that is union of three homothetic contractions, each having 0.5 as contractive factor. First, note that one can reduce the number of mappings by replacing two homotheties with one rotation around the barycenter of the triangle. This rotation is a orthonormal affine transformation with Lipschitz factor 1. Now, introducing a slight extension (about 1%) in this rotation, one gets a non-hyperbolic IFS. This means that fixed point does not exist. But, if one performs only a limited number of iterations  the object obtained is a kind of "attractor" being "decorated" with partial spiral trajectories. This is a source of "fur effect". By differently adjusted parameters, other effects are also possible, like "rays of light", "lines of fast movement", "explosions" etc. 

Figure 10.  Left. Meret Oppenheim, Fur covered cup, saucer, and spoon,1936;  right: furry version of Sierpinski gasket.

During the second half of 20-th century, we were witnesses of a huge range of variation of the Philostratos ideas. Thus, we are not surprised whether the comics-like texture of brushstrokes  inspired  Lihtenstein  to make an acceptable picture ( Roy Lihtenstein, Little Big Painting,1965,oil, Whitney Museum of American Art, New York). 

By inspection of what is said above, one thing is clear: it does not matter what is our complex object that influences our eye and brain. It is just an initial impulse to the creative mind  that makes its own vision (Figure 11) that yields a Creation. This creation, in statu nascendi sends a feedback that impacts both the mind and vision. In this "magic circle" an Artwork is born. And once again - it really does not matter what is the object of inspiration: it may be a cloud, an old wall (with or without silk cover), an ink stain, and a dry leaf frottage or - a fractal set.

Figure 11.  A functional diagram of "blotting-fratting" method. The role of ink stain might be taken by a fractal set.