The unexpected math behind Van Gogh’s “Starry Night”

Students can learn recursion algorithms to create the Koch curve using Scratch for free. Educational technologist Dylan Ryder has also written about creating fractals.

“Recursion is the process of repeating items in a self-similar way. Recursion can be implemented in Scratch by making a block that uses itself. This can be used to create fractals. A fractal is pattern that produces a picture, which contains an infinite amount of copies of itself.”

A beautiful app worth checking out is Starry Night Interactive App by media artist Petros Vrellis. Download it to your tablet and create your own version of Starry Night.

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Turbulence, unlike painting, is mostly a time-dependent phenomenon, and after some time, breaks statistical self-similarity that Kolmogorov predicted in the 1960s. To learn more about Kolmogorov’s predictions, Terry Tao provides a great overview of Kolmogorov’s power laws for turbulence.

“Many fluid equations are expected to exhibit turbulence in their solutions, in which a significant portion of their energy ends up in high frequency modes.” – Kolmogorov’s power laws for turbulence

In fluid mechanics, since we can’t often solve the equation for flow patterns, we develop a system of scaling between the physical properties. This is called dimensional analysis.

There are a few articles that outline patterns of turbulence in Van Gogh’s Starry Night.

Flow by Philip Ball (p. 164-178) provides an excellent overview of the concept for a broad audience.



“The book describes fascinating phenomena such as turbulence, which still defies complete scientific understanding; the principles of symmetry-breaking; and how chaotic behavior emerges in systems.” – featured this article: Van Gogh painted perfect turbulence.

“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 flow.” –  by Philip Ball, published on

Discover Magazine in 2006 featured another article on this topic called A Turbulent Mind. Learn how accurate Van Gogh’s turbulence was in his paintings.

“According to José Luis Aragón of the National Autonomous University in Mexico, Van Gogh’s famous spiral brushstrokes are near-perfect renderings of turbulence. From the largest visible swaths of paint to the tiniest strokes, Van Gogh’s brushwork seems instinctually guided to simulate river eddies and cloud rotations.” – by Kathryn Garfield, published on

Finally, this article entitled: Troubled Mind and Perfect Turbulence, gives a great description of several of Van Gogh’s paintings. It also discusses how the impact of the painting on the viewer was measured using the concept of luminosity.


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