Fractal expressionism

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Fractal expressionism is used to distinguish fractal art generated directly by artists from fractal art generated using mathematics and/or computers. [1] Fractals are patterns that repeat at increasingly fine scales and are prevalent in natural scenery (examples include clouds, rivers, and mountains). [2] Fractal expressionism implies a direct expression of nature's patterns in an art work.

Contents

Jackson Pollock's poured paintings

The initial studies of fractal expressionism focused on the poured paintings by Jackson Pollock (1912-1956), whose work has traditionally been associated with the abstract expressionist movement. [3] [4] [5] Pollock's patterns had previously been referred to as “natural” and “organic”, inviting speculation by John Briggs in 1992 that Pollock's work featured fractals. [6] In 1997, Taylor built a pendulum device called the Pollockizer which painted fractal patterns bearing a similarity to Pollock's work. [7] Computer analysis of Pollock's work published by Taylor et al. in a 1999 Nature article found that Pollock's painted patterns have characteristics that match those displayed by nature's fractals. This analysis supported clues that Pollock's patterns are fractal and reflect "the fingerprint of nature". [3]

Taylor noted several similarities between Pollock's painting style and the processes used by nature to construct its landscapes. For instance, he cites Pollock's propensity to revisit paintings that he had not adjusted in several weeks as being comparable to cyclic processes in nature, such as the seasons or the tides. [8] Furthermore, Taylor observed several visual similarities between the patterns produced by nature and those produced by Pollock as he painted. He points out that Pollock abandoned the use of a traditional frame for his paintings, preferring instead to roll out his canvas on the floor; this, Taylor asserts, is more compatible with how nature works than traditional painting techniques because the patterns in nature's scenery are not artificially bounded. [8]

The perceived similarities between the processes and patterns involved in Pollock's paintings and those of nature compelled Taylor to posit that the same "basic trademark" of nature's pattern construction also appears in Pollock's work. [8] Since some natural fractals are generated by a process known as "chaos", [9] including fractals in human physiology, [10] Taylor believed that Pollock's painting process might also have been chaotic, and could therefore leave behind a fractal pattern. Taylor's hypothesis seems to be reflected in Pollock's statement "I am nature", which he made when asked if nature was a source of inspiration for his work. [11] Furthermore, Pollock is also quoted as stating "No chaos, damn it", in response to a Time magazine article that referred to his paintings as "chaotic". [12] However, chaos theory was not understood until after Pollock's death, so he could not have been referring to the chaotic systems in nature but rather its common usage to mean disorder. In the famous film footage of Hans Namuth, [13] Pollock says his paintings are no accident, and that he was able to control the flow of paint onto the canvas.

Taylor points to two aspects of Pollock's painting process that have the potential to introduce fractal patterns. The first is Pollock's motion as he moved around the canvas, which Taylor hypothesized followed a Levy flight, a type of chaotic motion that is known to leave behind a fractal pattern. [8] [14] More specifically, a number of studies have shown that the motions associated with human balance have fractal characteristics. The second source of chaos could be introduced through Pollock's pouring technique. Falling fluid has the capability of changing from a non-chaotic to a chaotic flow, meaning that Pollock could have introduced a chaotic flow of paint as he dripped it onto the canvas. [8] Although the fractal characteristics of human balance and falling liquid are generated on Pollock's painting time and length scales, Predrag Cvitanovic notes that it would be quite an artistic challenge to control them: such parameters "are in no sense observable and measurable on the length-scales and time-scales dominated by chaotic dynamics".[ citation needed ]

Since Taylor's initial Pollock analysis in 1999, more than ten research groups have used various forms of fractal analysis to successfully quantify Pollock's work. [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] In addition to analyzing Pollock's work for fractal content, some groups such as that of computer scientist Bruce Gooch, have used computers to generate Pollock-like images by varying their fractal characteristics. [17] Benoit Mandelbrot (who invented the term fractal) and art theorist Francis O’Connor (the chief Pollock scholar) are well known advocates of fractal expressionism. [27] [28]

The relationship between fractal expressionism and fractal fluency

Fractal fluency is a neuroscience model that proposes that, through exposure to nature's fractal scenery, people's visual systems have adapted to efficiently process fractals with ease. This adaptation occurs at many stages of the visual system, from the way people's eyes move to which regions of the brain get activated. Fluency puts the viewer in a ‘comfort zone’ so inducing an aesthetic experience. Neuroscience experiments have shown that Pollock's paintings induce the same positive physiological responses in the observer as nature's fractals and mathematical fractals. [29] This shows that fractal expressionism is related to fractal fluency [30] by providing motivation for artists, such as Pollock, to use Fractal Expressionism in their art to appeal to people.

In light of fractal fluency and the associated aesthetics, other artists might be expected to display fractal expressionism. One year before Taylor's publication, mathematician Richard Voss quantified Chinese art using fractal analysis. [31] Subsequently, other groups have used computer analysis to identify fractal content in a number of Western and Eastern artists, [16] [19] most recently in Willem De Kooning's work. [32]

In addition to the above analyzed works, symbolic representations of fractals can be found in cultures across the continents spanning several centuries, including Roman, Egyptian, Aztec, Incan and Mayan civilizations. They frequently predate patterns named after the mathematicians who subsequently developed their visual characteristics. For example, although von Koch is famous for developing The Koch Curve in 1904, a similar shape featuring repeating triangles was first used to depict waves in friezes by Hellenic artists (300 B.C.E.). In the 13th century, repetition of triangles in Cosmati Mosaics generated a shape later known in mathematics as The Sierpinski Triangle (named after Sierpinski's 1915 pattern).

Triangular repetitions are also found in the 12th century pulpit of The Ravello Cathedral in Italy. The lavish artwork within The Book of Kells (circa 800 C.E.) and the sculpted arabesques in The Jain Dilwara Temple in Mount Abu, India (1031 C.E.) also both reveal stunning examples of exact fractals.

The artistic works of Leonardo da Vinci and Katsushika Hokusai serve as more recent examples from Europe and Asia, each reproducing the recurring patterns that they saw in nature. Da Vinci's sketch of turbulence in water, The Deluge (1571–1518), was composed of small swirls within larger swirls of water. In The Great Wave off Kanagawa (1830–1833), Hokusai portrayed a wave crashing on a shore with small waves on top of a large wave. Other woodcuts from the same period also feature repeating patterns at several size scales: The Ghost of Kohada Koheiji shows fissures in a skull and The Falls At Mt. Kurokami features branching channels in a waterfall.

The use of fractals to authenticate art and the associated controversy

Voss's 1998 study of Chinese art was the first demonstration of using fractal analysis to distinguish between the works of different artists. [31] Following Taylor's 1999 Pollock publication, Art conservator Jim Coddington proposed that fractal analysis should be explored as a technique to help authenticate Pollock paintings. In 2005, Taylor and colleagues published a fractal analysis of 14 authentic and 37 imitation Pollocks suggesting that, when combined with other techniques, fractal analysis might be useful for authenticating Pollock's work. [33] In the same year, The Pollock-Krasner Foundation requested a fractal analysis to be used for the first time in an authenticity dispute, [34] The analysis identified “significant deviations from Pollock’s characteristics.” Taylor cautioned that the results should be “coupled with other important information such as provenance, connoisseurship and materials analysis.” Two years later, materials scientists showed that pigments on the paintings dated from after Pollock's death.

In 2006, the use of fractals to authenticate Pollocks stirred controversy. [35] [36] [27] This controversy was triggered by physicists Katherine Jones-Smith and Harsh Mathur who claimed that the fractal characteristics identified by Taylor et al. are also present in crude sketches made in Adobe Photoshop, [37] and deliberately fraudulent poured paintings made by other artists [37] [38] Thus, according to Jones-Smith and Mathur, labeling Pollock's paintings as "fractal" is meaningless, because the same characteristics are found in other non-fractal images. However, Taylor's rebuttal published in Nature [36] showed that Taylor's group's fractal analysis could distinguish between Pollock paintings and the crude sketches, and identified further limitations in Jones-Smith and Mathur's analysis.

Jones-Smith and Mathur raised a valid concern applicable to all forms of fractal expressionism: are art works too small for the painted patterns to repeat over sufficient magnifications to assume the visual characteristics of fractals? In the case of Pollock paintings, the largest range used by Taylor et al. to determine each fractal parameter in a Pollock painting is less than two orders of magnitude in magnification. Nature's fractals repeat over limited magnification ranges (typically just over one order of magnitude), prompting scientists to debate what range is required to reliably establish fractal behavior. [39] Mandelbrot refused to include a required magnification range in his definition of fractals and instead noted that it is the range necessary to generate the properties associated with fractal repetition. In the case of Pollock's work, this would be the magnification range necessary for the patterns to generate the fractal aesthetics. Neuroscience experiments have shown that this magnification range is less than two orders and that Pollock's paintings do indeed induce the same physiological responses as nature's fractals and mathematical fractals [29] Mandelbrot concluded "I do believe that Pollocks are fractal." [27]

At the time of the controversy, Coddington summarized as follows: “Fractal geometry has begun to play an important role in the authentication of the work of Jackson Pollock. We believe such analyses are necessary for pushing the field forward.” [40] The most recent results, In 2015, by computer scientist Lior Shamir showed that, when combined with other pattern parameters, fractal analysis can be used to distinguish between real and imitation Pollocks with 93% accuracy. He found that the fractal parameters were the most powerful contributors to the detection accuracy [41]

Related Research Articles

Benoit Mandelbrot French-American mathematician

Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.

Chaos theory Field of mathematics

Chaos theory is a branch of mathematics focusing on the study of chaos — dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a hurricane in China.

Fractal Self similar mathematical structures

In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly small scales called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

Jackson Pollock American painter

Paul Jackson Pollock was an American painter and a major figure in the abstract expressionist movement.

Mandelbrot set Fractal named after mathematician Benoit Mandelbrot

The Mandelbrot set is the set of complex numbers for which the function does not diverge when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value. Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit Mandelbrot, a pioneer of fractal geometry.

Self-similarity The whole of an object being mathematically similar to part of itself

In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

Fractal art

Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.

Pattern

A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated like a wallpaper design.

Fractal landscape

A fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, but rather a random surface that exhibits fractal behavior.

Abstract expressionism is a post–World War II art movement in American painting, developed in New York in the 1940s. It was the first specifically American movement to achieve international influence and put New York City at the center of the western art world, a role formerly filled by Paris. Although the term "abstract expressionism" was first applied to American art in 1946 by the art critic Robert Coates, it had been first used in Germany in 1919 in the magazine Der Sturm, regarding German Expressionism. In the United States, Alfred Barr was the first to use this term in 1929 in relation to works by Wassily Kandinsky.

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.

A Lévy flight, named for French mathematician Paul Lévy, is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions.

Fractal curve Mathematical curve whose shape is a fractal, pathological irregularity, regardless of magnification. Each non-zero arc has infinite length

A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

<i>Who the $&% Is Jackson Pollock?</i>

Who the #$&% Is Jackson Pollock? is a 2006 documentary following Teri Horton, a 73-year-old former long-haul truck driver from California, who purchased a painting from a thrift shop for $5, only later to find out that it may be a Jackson Pollock painting. She had no clue at the time who Jackson Pollock was, hence the name of the film.

Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from phenomena including natural geometric objects, ecology and aquatic sciences, sound, market fluctuations, heart rates, frequency domain in electroencephalography signals, digital images, molecular motion, and data science. Fractal analysis is now widely used in all areas of science. An important limitation of fractal analysis is that arriving at an empirically determined fractal dimension does not necessarily prove that a pattern is fractal; rather, other essential characteristics have to be considered. Fractal analysis is valuable in expanding our knowledge of the structure and function of various systems, and as a potential tool to mathematically assess novel areas of study.

Fractal-generating software

Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both free and commercial. Mobile apps are available to play or tinker with fractals. Some programmers create fractal software for themselves because of the novelty and because of the challenge in understanding the related mathematics. The generation of fractals has led to some very large problems for pure mathematics.

Mathematics and art Relationship between mathematics and art

Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.

Patterns in nature Visible regularity of form found in the natural world

Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

<i>Autumn Rhythm (Number 30)</i> Painting by Jackson Pollock

Autumn Rhythm is a 1950 abstract expressionist painting by American artist Jackson Pollock in the collection of the Metropolitan Museum of Art in New York City. The work is a distinguished example of Pollock's 1947-52 poured-painting style, and is often considered one of his most notable works.

One: Number 31, 1950 is one of the largest and most prominent examples of a Jackson Pollock Abstract Expressionist drip style painting. The work of art was owned by a private collector until 1968 when it was purchased by Museum of Modern Art (MoMA) in New York City and has been displayed there ever since. It has At 8 Feet 10 inches by 17 feet 5 and 5/8 inche.

References

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  2. ↑ Mandelbrot, BB, The Fractal Geometry of Nature, WH Freedman, New York, 1982
  3. 1 2 [Taylor, Richard P., Adam P. Micolich, and David Jonas. "Fractal Analysis of Pollock's Drip Paintings." Nature 399.6735 (1999): 422. Print.]
  4. ↑ ["Fractals Determine Date of Paintings." Physics World 04 June 1999: n. pag. Web. <http://physicsworld.com/cws/article/news/1999/jun/04/fractals-determine-date-of-paintings>.]
  5. ↑ Taylor, Richard P., Adam P. Micolich, and David Jonas. "Fractal Analysis of Pollock's Drip Paintings." Nature 399.6735 (1999): 422. Print.
  6. ↑ John Briggs, Fractals, Touchstone Publishers, 1992
  7. ↑ R.P.Taylor, A.P. Micolich and D. Jonas, Fractal Expressionism, Physics World, 25, October 1999
  8. 1 2 3 4 5 [Taylor, Richard. "Fractal Expressionism—Where Art Meets Science." Art and Complexity. Ed. John Casti and Anders Karlqvist. 1st ed. Greenwich: JAI, 2003. 117-44. Print.]
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  11. ↑ Krasner, Lee. "Lee Krasner Oral History." Interview by Dorothy Seckler. Archives of American Art. Smithsonian, 19 May 2005. Web. 30 Dec. 2016.
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  13. ↑ [Jackson Pollock: Paintings Have a Life of Their Own. Perf. Jackson Pollock. SFMOMA. SFMOMA, n.d. Web. <https://www.sfmoma.org/watch/jackson-pollock-paintings-have-a-life-of-their-own/>.]
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  19. 1 2 D.J. Graham and D.J. Field, “Variations in Intensity for Representative and Abstract Art, and for Art from Eastern and Western Hemispheres” Perception, vol. 37, 1341-1352 (2008).
  20. ↑ J. Alvarez-Ramirez, J. C. Echeverria, E. Rodriguez “Performance of a High-Dimensional R/S Analysis Method for Hurst Exponent Estimation” Physica A, vol. 387, 6452-6462 (2008).
  21. ↑ J. Coddington, J. Elton, D. Rockmore and Y. Wang, “Multi-fractal Analysis and Authentication of Jackson Pollock Paintings”, Proceedings SPIE, vol. 6810, 68100F 1-12 (2008).
  22. ↑ M. Al-Ayyoub, M. T. Irfan and D.G. Stork, “Boosting Multi-Feature Visual Texture Classifiers for the Authentification of Jackson Pollock’s Drip Paintings”, SPIE proceedings on Computer Vision and Image Analysis of Art II, vol. 7869, 78690H (2009).
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  25. ↑ K Zheng et al Vis Comput. DOI 10.1007/s00371-014-0985-7
  26. ↑ E De la Calleja et al Annals of Physics, Vol. 371, 313 (2016)
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  28. ↑ see o'conner's website
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  30. ↑ 23 Taylor, RP, and Spehar B, Fractal Fleuncy: An Intimate Relationship Between the Brain and Processing of Fractal Stimuli, In: The Fractal Geometry of the Brain. New York: Springer; 2016.
  31. 1 2 R. Voss, Fractal Image Encoding and Analysis, Springer, 1998.
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  35. ↑ Jones-Smith et al, “Fractal Analysis: Revisiting Pollock’s Paintings”Nature, Brief Communication Arising, vol. 444, E9-10, (2006).
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  41. ↑ L. Shamar, “What Makes a Pollock Pollock: A Machine Vision Approach”, International Journal of Arts and Technology, vol. 8, 1-10, (2015)
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