A Starry Starry Night or The Unexpected Maths in a Van Gogh’s Masterpiece

An image showing Van Gogh's painting "The Starry Night" (1889).Van Gogh’s Starry Night

When Classical Physics and Post-Impressionist artists meet, few results are as hauntingly beautiful or as enchanting as one of Vincent van Gogh’s most famous masterpieces.  The Starry Night embodies the inner, subjective expression of van Gogh’s response to Nature.  And the churning night sky he depicted tells of the artist’s very unique perception of the World around him…

Painted in June 1889, the oil on canvas painting of the Starry Night depicts the view from the east-facing window of his asylum room at Saint-Rémy-de-Provence.  Vincent van Gogh wrote to his brother Theo:

“This morning, I saw the country from my window a long time before sunrise, with nothing but the Morning Star, which looked very big.”

Rooted in imagination and memory, van Gogh’s painting is not astronomically correct.  But that is not why it is amazing… and scientifically interesting.

 

Luminance

Since the early days of Impressionism, artists empirically discovered that an adequate use of luminance could generate the sensation of motion in a painting.

A diagram explaining the properties of Luminance.
Luminance is often used to characterise emission or reflection from flat, diffuse surfaces. In this case, the solid angle of interest is the solid angle subtended by the eye’s pupil.  Luminance is used in the video industry to characterise the brightness of displays. A typical computer display emits between 50 and 300 cd/m2.  The Sun has a luminance of about 1.6 × 109 cd/m2 at noon.

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.  Its psychological effect is bright and thus is an indicator of how bright a surface will appear.

The technique of equiluminance has been used by many artists to produce certain visual effects.  Although two equiluminant regions can be differentiated using contrasting colours, they have poorly defined positions, making them seem to vibrate and shimmer…

An image showing Claude Monet's painting "Impression Sunrise" (1872).
Impression Sunrise (1872) by Claude Monet – the painting that gave its name to the style. Equiluminance is a recurring technique in many of Monet’s paintings.

The biological basis behind this effect is that colour and luminance are processed by different parts of the brain’s visual system.  While shape is registered by the region of the visual cortex that analyse colour information, motion registers in the basal ganglia by the colour blind part of the brain.

But unlike those of other artist’s, van Gogh’s paintings appear alive and pulsing with the artist’s very own emotions.  Using thick, sweeping brushstrokes, van Gogh shows something strikingly real about how light moves.  The light flickers and radiates.  It is as if van Gogh captured light in motion.

Vincent van Gogh undoubtedly mastered this technique and some of his later paintings produce an even more disturbing eerie feeling.  They transmit a sense of turbulence. 

 

Fluid Dynamics and Outstanding Physics Problems

A photograph showing an airplane wake turbulence.
Turbulence in the tip vortex of an airplane wing.

Fluid mechanics is a branch of physics involving the study of fluids in motion (that is, liquids, gases, and plasmas) and the forces acting on them.

Fluid dynamics is an active field of research with many unsolved or partly solved problems, a subject which models matter from a macroscopic point of view.

While scientists do understand fluid dynamics, the irregularity of the phenomenon that causes turbulence makes it difficult to understand fully and to predict.  The agitated, irregular motion usually involves complex movement at various rates of speed, and many factors influence the dynamics of liquids and gases.

Famous physicists Feynman and Heisenberg realised the sheer complexity of turbulence.

Nobel laureate Richard Feynman described it as

“the most important unsolved problem of Classical Physics”.

Werner Heisenberg once said:

“When I meet God, I am going to ask him two questions: Why Relativity?  And why turbulence?  I really believe he will have an answer for the first.”

Russian Mathematician Andrey Kolmogorov (1903-1987) proposed the first statistical theory of turbulence, based on the notion of energy cascade and the concept of self-similarity.  As a result, the Kolmogorov micro-scales were named after him.

It is now known that the self-similarity is broken so the statistical description is presently modified.  Still, a complete description of turbulence remains one of the unsolved problems in Physics.

 

Air Turbulences, Eddies and the Kolmogorov Equations

Turbulence is a form of movement which is characterised by the irregular or agitated motion of a fluid.  Both liquids and gases can exhibit turbulence, and a number of factors can contribute to the formation of turbulence.

When a liquid or gas is moving smoothly and regularly, it is said to be exhibiting laminar flow, which is the opposite of turbulent flow.

This phenomenon occurs when a dynamic flow is suddenly interrupted or impeded by an obstacle.  For example, a river may flow smoothly until it encounters a boulder, at which point the water around the obstacle will become turbulent as it moves around or over it.

A graph showing the flow of an airplane turbulence. Source: MIT
Modelling Airplane Turbulence  Source: MIT.edu

Many people interact with some variety of turbulence on a daily basis.  Air is quite turbulent, because it constantly moves at different rates of speed and pressure across the Earth.

Air turbulence may be caused by ground obstacles, from mountains to buildings. 

Turbulence can also be caused by such things as the collision of two weather fronts, or the formation of a storm.  Actually, this is the reason why turbulence on an aircraft can be so difficult to predict, and why aircraft can move in different ways due to turbulent patches of air, depending on where they are.

Turbulence causes the formation of eddies of many different length scales.  Most of the kinetic energy of the turbulent motion is contained in the large-scale structures.  The energy cascades from these large-scale structures to smaller scale structures by an inertial and essentially inviscid mechanism.  The process continues, creating smaller and smaller structures which produces a hierarchy of eddies.  Eventually, this creates structures small enough that molecular diffusion becomes important and viscous dissipation of energy finally takes place.

 

The Kolmogorov Micro-Scales

 

The scale at which this happens is the Kolmogorov length scale

\eta = {\left( \frac{\nu^3}{\epsilon} \right)}^{1/4}.

where ε is the average rate of dissipation of turbulence kinetic energy per unit mass, and ν is the kinematic viscosity of the fluid.

In his 1941 theory, Andrey Kolmogorov introduced the idea that the smallest scales of turbulence are universal to any turbulent flow and that they depend only on ε and ν.

The definitions of the Kolmogorov microscales can be obtained using this idea and dimensional analysis.  Since the dimension of kinematic viscosity is length2 over time, and the dimension of the energy dissipation rate per unit mass is length2 over time3, the only combination that has the dimension of time is the Kolmogorov time scale

\tau _\eta = {\left( \frac{\nu}{\epsilon} \right)}^{1/2}.

 

A photograph showing a lit-up cigarette with whirling smoke.
Smoke rising from a cigarette gives a visible example of a turbulent flow. To begin with, the smoke rises up steadily with laminar flow.  As the rising hot air accelerates upwards, it then becomes unstable.  The rate at which it becomes turbulent is determined by the Reynolds number.

Turbulent flows may be viewed as made of an entire hierarchy of eddies over a wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in velocity fluctuations for each length scale (or wavenumber).  The scales in the energy cascade are generally uncontrollable and highly non-symmetric.

Most of the time, people cannot see turbulence in action, because the air is clear, but the turbulent movement becomes apparent when particles fill the air and highlight its irregular motion, or when an object is launched into the air, in which case the movement of the object through space will reflect changing speeds and pressures in the air it travels through.

People can also observe this interesting property of fluid dynamics when they watch smoke rising from a fire.  At first, the smoke starts out straight, then quickly starts to swirl and eddy as it rises.  The heat of the fire can cause this turbulence, by affecting the air above the fire, and things like breezes can also influence the movement of smoke.

 

A Turbulent Genius

But Van Gogh’s perception of the World around him was unlike any other.

So, it is possible that the mind of the undoubtedly troubled genius perceived something more about turbulence in Nature,  and this is most clearly represented in his famous painting – the evocative masterpiece known as the Starry Night.

A self-portrait by Vincent van Gogh.
Vincent van Gogh – Self-portrait, 1889

Over the years, the reason for Vincent Van Gogh’s illness and its effect on his work has been the source of much debate.  150 psychiatrists attempted to label it, with over 30 different diagnoses, including schizophrenia, bipolar disorder, syphilis, chemical poisoning from swallowed paints, temporal lobe epilepsy, and acute intermittent porphyria.

Any of these conditions could have been possible, and may have been aggravated by malnutrition, overwork, insomnia, and consumption of absinthe. 

Undoubtedly, there is a fine line between genius and madness.

One of the TEDEd videos looks at “The unexpected math behind Van Gogh’s Starry Night.”  The short lesson considers the enduring mystery that is the turbulence we see in any kind of flows in the natural world and how the human brain can recognise and make sense of the chaotic random patterns that turbulence describes.

As difficult as turbulence is to understand mathematically, we can use art to depict the way it looks. Natalya St. Clair’s video illustrates how Van Gogh captured this deep mystery of movement, fluid and light in his work.

 

 

Everything in the last period of van Gogh’s painting appears to be moving.  Turbulent is the main adjective used to describe these works.  And scientists have shown that the impassioned artworks of Vincent van Gogh, which he painted at times of prolonged psychological agitation, notably compare mathematically to the description of turbulent flows predicted by the statistical theory of Kolmogorov.

 

It takes a special mind to see the World in a special way…