Turbulence in Paintings

Notes Under Construction

Recently I became aware of an interest in the study of turbulence in paintings. Googling turbulence in paintings leads to several sites with artists from Leonardi da Vinci to Van Gogh and Jackson Pollack. For example*, "a mathematical analysis of the works of Van Gogh reveals that the stormy patterns in many of his paintings are uncannily like real turbulence, as seen in swirling water or the air from a jet engine". 

Some Links of Interest 

• Painting with Turbulence: https://www.npr.org/sections/13.7/2015/04/01/396637276/van-goghs-turbulent-mind-captured-turbulence https://tigerprints.clemson.edu/cgi/viewcontent.cgi?article=3292&context=all_theses - a master's thesis, very technical involving simulations. But it includes information about the history of the term, turbulence, in art beginning with Leonardi da Vinci: "More than five hundred years ago, Renaissance artist and engineer Leonardo da Vinci identified turbulence as a distinct natural phenomenon and used the term turbolenza for the first time." "George Gabriel Stokes later extended the equations to what we currently know as the famous Navier-Stokes equations, which are believed to fully describe turbulent flows" 
• https://medium.com/the-omnivore/how-did-van-goghs-turbulent-mind-depict-one-of-the-most-complex-concepts-in-physics-a10d0faacdbc "Yet perhaps no element of The Starry Night is as recognizable as the air currents unfurling across the sky. This emotive effect of ethereal movement was achieved through the painter’s employment of luminance, “a measure of the relative brightness between different points,” NPR defines. “The eye is more sensitive to luminance change than to color change, meaning we respond more promptly to changes in brightness than in colors. This is what gives many Impressionist paintings that familiar and emotionally moving twinkle.” 

Ted Talks

• https://www.youtube.com/watch?v=PMerSm2ToFY (The Unexpected Math Behind Starry Nights) (for children but fun to watch)
• https://www.ted.com/talks/natalya_st_clair_the_unexpected_math_behind_van_gogh_s_starry_night?language=en (See transcript below.)
• https://www.ted.com/talks/amy_segami_everything_new_emerges_from_turbulence_may_2020 
• https://www.facebook.com/TEDEducation/videos/2473815429298315 

Physics and Mathematics of Turbulence:

One book in particular  that I found by searching for Physics and Mathematics of Turbulence,

Patterns in Nature by Philip Ball,, 2016, might be of interest. It has many beautiful images and straightforward explanations. This is written up in another post as well.

*And in another article by Philip Ball from 2006:

https://www.nature.com/news/2006/060703/full/news060703-17.html :
-Begin article:

"Vincent van Gogh painted perfect turbulence The disturbed artist intuited the deep forms of fluid flow.  

Ball stated "The Starry Night has really turbulent swirls".

"Vincent van Gogh is known for his chaotic paintings and similarly tumultuous state of mind. Now a mathematical analysis of his works reveals that the stormy patterns in many of his paintings are uncannily like real turbulence, as seen in swirling water or the air from a jet engine. 

Physicist Jose Luis Aragon of the National Autonomous University of Mexico in Queretaro and his co-workers have found that the Dutch artist's works have a pattern of light and dark that closely follows the deep mathematical structure of turbulent flow1. The swirling skies of The Starry Night, painted in 1889, Road with Cypress and Star (1890) and Wheat Field with Crows (1890) — one of the van Gogh's last pictures before he shot himself at the age of 37 — all contain the characteristic statistical imprint of turbulence, say the researchers. 

These works were created when van Gogh was mentally unstable: the artist is known to have experienced psychotic episodes in which he had hallucinations, minor fits and lapses of consciousness, perhaps indicating epilepsy. "We think that van Gogh had a unique ability to depict turbulence in periods of prolonged psychotic agitation," says Aragon. In contrast, the Self-portrait with Pipe and Bandaged Ear (1888) shows no such signs of turbulence. Van Gogh said that he painted this image in a state of "absolute calm", having been prescribed the drug potassium bromide following his famous self-mutilation.

Measured Chaos 

Scientists have struggled for centuries to describe turbulent flow — some are said to have considered the problem harder than quantum mechanics. It is still unsolved, but one of the foundations of the modern theory of turbulence was laid by the Soviet scientist Andrei Kolmogorov in the 1940s. He predicted a particular mathematical relationship between the fluctuations in a flow's speed and the rate at which it dissipates energy as friction. Kolmogorov's work led to equations describing the probability of finding a particular velocity difference between any two points in the fluid. These relationships are called Kolmogorov scaling. Aragón and colleagues looked at van Gogh's paintings to see whether they bear the fingerprint of turbulence that Kolmogorov identified. "'Turbulent' is the main adjective used to describe van Gogh's work," says Aragon. "We tried to quantify this." 

Darkness and light 

The researchers took digital images of the paintings and calculated the probability that two pixels a certain distance apart would have the same brightness, or luminance. "The eye is more sensitive to luminance changes than to colour changes," they say, "and most of the information in a scene is contained in its luminance." Several of van Gogh's works show Kolmogorov scaling in their luminance probability distributions. To the eye, this pattern can be seen as eddies of different sizes, including both large swirls and tiny eddies created by the brushwork. 

 Van Gogh seems to be the only painter able to render turbulence with such mathematical precision. "We have examined other apparently turbulent paintings of several artists and find no evidence of Kolmogorov scaling," says Aragon. Edvard Munch's The Scream, for example, looks to be superficially full of van Gogh-like swirls, and was painted by a similarly tumultuous artist, but the luminance probability distribution doesn't fit Kolmogorov's theory. The distinctive styles of other artists can be described by mathematical formulae. Jackson Pollock's drip paintings, for example, bear distinct fractal patterns. National Autonomous University of Mexico".

---End Article--

Original Research Article

https://www.researchgate.net/figure/Luminace-differences-for-different-pixel-separations-for-the-example-in-Fig-2-Diagonal_fig2_2176268 (click on download text button for pdf of full article):

Abstract

"We show that the patterns of luminance in some impassioned van Gogh paintings display the mathematical structure of fluid turbulence. Specifically, we show that the probability distribution function (PDF) of luminance fluctuations of points (pixels) separated by a distance R compares notably well with the PDF of the velocity differences in a turbulent flow, as predicted by the statistical theory of A.N. Kolmogorov. We observe that Turbulent paintings of van Gogh belong to his last period, during which episodes of prolonged psychotic agitation of this artist were frequent. Our approach suggests new tools that open the possibility of quantitative objective research for art representation. 

"Since the early impressionism, artists empirically discovered that an adequate use of luminance could generate the sensation of motion [1]. This dynamic style was more complex in the case of van Gogh paintings of the last period: turbulence is the main adjective used to describe these paintings. 

"By considering the analogy with the Kolmogorov turbulence theory, from our results we conclude that the turbulence of the luminance of the studied van Gogh paintings shows similar characteristic to real turbulence.  

"Luminance is a measure of the luminous intensity per unit area. It describes the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle [10]. Its psychological effect is bright and thus luminance is an indicator of how bright a surface will appear. In a digital image, the luminance of a given pixel is obtained from its RBG (red, green and blue) components as [11]

0.299R + 0.587G + 0.114B (1)

This formula takes into account the fact that the human eye is more sensitive to green, then red and lastly blue. The luminance value of different colors is easily obtained with this formula; we quickly infer that the color with more green is brighter to the eye than the color with more blue, and some examples are as follows. Using RGB values in 24 bits per pixel (8 bits per color), black color (0, 0, 0) has the lowest luminace (0) and white (255, 255, 255) the higher one (255). Intermediate values corresponds, for instance, to blue (0, 0, 255) → 29, red (255,0,0) → 76, green (0,255,0) → 150, cyan (0,255,255) → 179, yellow (255, 255, 0) → 226, etc. Interestingly, gray colors have the same RBG values, so (10, 10, 10) is a dark gray with luminance 10 and (200, 200, 200) is a light gray with luminance 200.

Luminance contains the most important piece of information in a visual context and has been used by artists to produce certain effects. For instance, the technique of equiluminance has been used since the first impressionist painters to transmit the sensation of motion in a painting. Notably Claude Monet in his famous painting Impression, Sunrise, used regions with the same luminance, but contrasting colours, to make his sunset twinkle. The biological basis behind this effect is that colour and luminance are analyzed by different parts of the visual system; shape is registered by the region that processes colour information (ventral pathway) but motion is registered by the colour blind part (dorsal pathway) [1]. Thus equiluminant regions can be differentiated by colour contrast, but they have poorly defined positions and may seem to vibrate [1]. It seems likely that van Gogh dominated this technique but some of the paintings of his last period produce a more disturbing feeling: they transmit the sense of turbulence. By assuming that luminance is the property that van Gogh used to transmit this feeling (without being aware of it), we will quantify the turbulence of some impassioned paintings by means of a statistical analysis of luminance, similar to the statistical analysis of velocities that Andrei Kolmogorov used to study fluid turbulence. 

====================

Van Gogh and Jackson Pollock - and Turbulence

https://whimsicalmath.wordpress.com/2015/02/10/van-gogh-jackson-pollock-and-kolmogorovs-theory-of-turbulence/ 

From the article: "Turbulence is a flow of gas or liquid, which is characterized by chaotic spatial and temporal variations in pressure and velocity. In a turbulent flow, there are interacting eddies of various sizes. Kolmogorov’s theory models a continuous spectrum of energy that spans a wide range of sizes. The word “spectrum” implies Fourier transform, so all quantities are transformed from the spatial to the Fourier domain. Using clever dimensionality analysis, one can show that energy spectrum in the Fourier domain is a power law function of the wave number." 

Transcript of Ted Talk: Van Gogh and Starry Night 

https://www.ted.com/talks/natalya_st_clair_the_unexpected_math_behind_van_gogh_s_starry_night?language=en:

Begin transcript-

"One of the most remarkable aspects of the human brain is its ability to recognize patterns and describe them. Among the hardest patterns we've tried to understand is the concept of turbulent flow in fluid dynamics. The German physicist Werner Heisenberg said, "When I meet God, I'm going to ask him two questions: why relativity and why turbulence? I really believe he will have an answer for the first." 

• 00:40 As difficult as turbulence is to understand mathematically, we can use art to depict the way it looks. In June 1889, Vincent van Gogh painted the view just before sunrise from the window of his room at the Saint-Paul-de-Mausole asylum in Saint-Rémy-de-Provence, where he'd admitted himself after mutilating his own ear in a psychotic episode. In "The Starry Night," his circular brushstrokes create a night sky filled with swirling clouds and eddies of stars. Van Gogh and other Impressionists represented light in a different way than their predecessors, seeming to capture its motion, for instance, across sun-dappled waters, or here in star light that twinkles and melts through milky waves of blue night sky. The effect is caused by luminance, the intensity of the light in the colors on the canvas. The more primitive part of our visual cortex, which sees light contrast and motion, but not color, will blend two differently colored areas together if they have the same luminance. But our brains' primate subdivision will see the contrasting colors without blending. With these two interpretations happening at once, the light in many Impressionist works seems to pulse, flicker and radiate oddly. That's how this and other Impressionist works use quickly executed prominent brushstrokes to capture something strikingly real about how light moves.
• 02:13 Sixty years later, Russian mathematician Andrey Kolmogorov furthered our mathematical understanding of turbulence when he proposed that energy in a turbulent fluid at length R varies in proportion to the 5/3rds power of R. Experimental measurements show Kolmogorov was remarkably close to the way turbulent flow works, although a complete description of turbulence remains one of the unsolved problems in physics. A turbulent flow is self-similar if there is an energy cascade. In other words, big eddies transfer their energy to smaller eddies, which do likewise at other scales. Examples of this include Jupiter's Great Red Spot, cloud formations and interstellar dust particles. 
•  02:57 In 2004, using the Hubble Space Telescope, scientists saw the eddies of a distant cloud of dust and gas around a star, and it reminded them of Van Gogh's "Starry Night." This motivated scientists from Mexico, Spain and England to study the luminance in Van Gogh's paintings in detail. They discovered that there is a distinct pattern of turbulent fluid structures close to Kolmogorov's equation hidden in many of Van Gogh's paintings. 
•  03:27 The researchers digitized the paintings, and measured how brightness varies between any two pixels. From the curves measured for pixel separations, they concluded that paintings from Van Gogh's period of psychotic agitation behave remarkably similar to fluid turbulence. His self-portrait with a pipe, from a calmer period in Van Gogh's life, showed no sign of this correspondence. And neither did other artists' work that seemed equally turbulent at first glance, like Munch's "The Scream."
• 03:57 While it's too easy to say Van Gogh's turbulent genius enabled him to depict turbulence, it's also far too difficult to accurately express the rousing beauty of the fact that in a period of intense suffering, Van Gogh was somehow able to perceive and represent one of the most supremely difficult concepts nature has ever brought before mankind, and to unite his unique mind's eye with the deepest mysteries of movement, fluid and light. 

Amy Segami: Turbulent Flow in Fluid Dynamics. Dance with Turbulence (Ted Talk) 

Amy Segami: Sumi Nagami Painting.

Flowing Ink. Pebbles in water, ink on sumi painting. Fluid Mechanics, the study of air and water. 

Eddies occur when two currents are going in the opposite direction and it creates swirls Chaos and turbulence.

Probability of Randomness. Encourage employees to meet randomly. Ancient philosophy of dali chain. Two kinds of flow: Yin and Yang.

Slow dance with turbulence Embrace turbulence. Dance with Turbulence. Paintings on Water

Segami's works are in the permanent collection of the Museum of New Mexico in Santa Fe.Luminescence: the intensity of the light in the colors on the canvas. The light in many impressionist works seems to pulse, flicker and radiate oddly. Quickly executed, prominent brushstrokes to capture something strikingly real about how light moves. Turbulent flow is self similar if there is an energy cascade. Big eddies transfer their energy to smaller eddies. (See the Ted Transcript below. from a Ted Talk - on Facebook). 

My Paintings: Golden Luminescence

Golden Luminescence, 12x16, oil

Another Version: Van Gogh on my Mind

This was painted primarily with palette knife but with the same palette prismatic colors.

Van Gogh on My Mind, 12x16, oil

These were both 12x16. Next painting to try will be an 18x24 and perhaps I will try one even larger this time with wrap-around edges. I think of these paintings as expressing turbulent hillsides (in fact my original title for one). Thees are both now in my gallery, the Marigold Arts Gallery on Canyon Road, Santa Fe. (Jan 2022: Gallery now closed).

I intend to create another post, an excerpt from this to document my progress toward the larger painting. (3/29/2022: To be done yet, but am thinking about it now.)