The unexpected math behind Van Gogh's "Starry Night" - Natalya St. Clair

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

Lesson by Natalya St. Clair, animation by Avi Ofer.

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."

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.

60 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.

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.

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."

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.