Trees in art, as well as life, often follow simple mathematical rules, study finds

In nature, fractal patterns aren’t just aesthetically pleasing, they’re also often related to function.

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(CNN) — Trees depicted in famous artworks across a range of styles follow the same mathematical rules as their real-life counterparts, scientists have found.

The math concept hidden in this tree art — geometric shapes known as fractals — is apparent in branching patterns in nature and may be key to humans’ ability to recognize such artwork as trees, according to Mitchell Newberry, a mathematical biologist at the University of New Mexico, and his colleague Jingyi Gao, a doctoral student at the University of Wisconsin.

Like the branches, twigs and leaves of a tree, fractals repeat the same patterns across different scales. Snowflakes, lightning bolts and human blood vessels are also fractal structures, which all show a degree of self-similarity: As you zoom into the details of a fractal, you can see a replica of the whole.

“If you look at a tree, its branches are branching. Then the child branches repeat the figure of the parent branch,” Gao said in a news release.

Newberry and Gao chose to study artworks depicting single trees. Their selections, which they said spanned different times and cultures, included 16th century stone window carvings from the Sidi Saiyyed Mosque in India, an 18th century painting called “Cherry Blossoms” by Japanese artist Matsumura Goshun and two early 20th century works by Dutch painter Piet Mondrian. They also examined Gustav Klimt’s 1909 painting “L’Arbre de Vie” (“Tree of Life”).

They found that the trees depicted in the artworks, even when abstract or stylistic, mostly, but not always, corresponded to branching patterns and scale found in natural trees.

“Any kind of abstraction is a way of trying to get at natural laws, whether it’s a mathematical abstraction or an artistic abstraction. There’s a lot of different kinds of trees in the world, but this theory shows us (and) gives us some baseline for what we expect a tree to be,” Newberry told CNN.

Newberry said he had long been a fan of Mondrian’s work and how the artist depicted trees in abstract ways, removing all but the most essential elements but still clearly conveying a tree. It jibed with his own work explaining mathematically how treelike structures in human biology such as veins and arteries and lungs use their physical form to efficiently deliver blood and air.

To reach their findings, the researchers successfully came up with a method of assessing branching patterns in trees and generalized it into a simple common formula, according to Fabian Fischer, a researcher at Technical University of Munich in Germany who wasn’t involved in Newberry and Gao’s study.

“The method is based on ideas that go back to Leonardo da Vinci and have been revisited by biologists multiple times,” Fischer said. “I found it a highly stimulating read, with an interesting connection between works of art and biology.”

Scale of 1 to tree

In nature, fractal patterns aren’t just aesthetically pleasing, they’re also often related to function. For example, branching enables trees to transport fluid, harvest light and maintain mechanical stability. Since a fractal is a geometric shape, mathematicians can calculate its complexity, or fractal dimension — even when it appears in art.

“There are some characteristics of the art that feel like they’re aesthetic or subjective, but we can use math to describe it,” Gao said.

In their research published in the scientific journal PNAS Nexus on February 11, Gao and Newberry analyzed the variation in the thickness of the tree branches in the artworks they studied. They took into account the number of smaller branches per larger branch and used this information to calculate a number they called the branch diameter scaling exponent.

The study found that the trees in the artworks had a branch diameter scaling value broadly matching the 1.5 to 3 range for real trees. Outside those values, the objects depicted weren’t easily recognizable as trees.

Gao and Newberry were surprised to find the highly stylized Indian mosque carving had a value closer to real trees than the tree in “Cherry Blossoms,” which they had initially thought was more natural-looking.

Though extremely rich in detail, with over 400 individual branches, “Cherry Blossoms” exhibited a scaling exponent of 1.4, while the pair calculated the Indian carved tree has a value of 2.5.

Newberry said that having a more realistic branch diameter scaling factor may have enabled artists to take more creative risks yet still have the object recognizable as a tree.

“As you abstract away details and still want viewers to recognize this as a beautiful tree, then you may have to be closer to reality in some other aspects,” Newberry said.

Of course, artists such as Mondrian and Klimt would likely not have been aware of fractals, or the math that underpins them, but perhaps had an innate understanding of the subtle proportions all trees share, according to the researchers.

However, Fischer noted that the study was exploratory and the range of selected tree species and works of art is small and selective, therefore it’s not possible to draw strong conclusions.

Fractal pattern impacts

The authors studied a series of works by abstract painter Mondrian that depict the same tree but in increasingly less realistic ways.

His 1911 work “De Grijze Boom” (“The Gray Tree”) shows a series of black lines against a gray background, but the painting is nonetheless instantly recognizable as a tree, with its branch scaling value in the real tree range at 2.8.

“I don’t think he (Mondrian) is even trying to find the essence of trees but as he’s pulling things out, this thing that we think is really important in science ends up being one of the last things to go (away) in the art,” Newberry said.

“Clearly, he thinks it’s really important, and clearly it’s really important to human perception.” However, in Mondrian’s 1912 “Bloeiende Appelboom” (“Blooming Apple Tree”), a painting in the same series, the branch diameter scaling is gone, Newberry said, with a value of 5.4.

“Whereas most viewers of Gray Tree immediately perceive a tree, naïve viewers of Blooming Apple Tree see dancers, roots, fish, faces, water, stained glass, leaves, flowers, or nothing representational at all,” the authors noted in the study.

The researchers also examined Gustav Klimt’s 1909 painting “L’Arbre de Vie” (“Tree of Life”). Though the tree’s depiction in this artwork is highly stylized, the study’s measurements suggest it also fell into the statistical range of a real-life tree.

The study authors are not the first to apply math to trees in art. Renaissance polymath Leonardo da Vinci observed tree growth and came up with his own mathematical rule for painting trees. His work on tree physiology inspired scientists and landscape artists alike to study branching patterns, according to the new research.

The findings from the study are intriguing because they integrate artistic and scientific approaches to studying trees, said Richard Taylor, a professor of physics at the University of Oregon.

“Although focusing on trees, the article is tackling a much bigger question — why are natural patterns so beautiful — and interdisciplinary collaborations are essential for delivering the answers,” Taylor, who was not involved in the study, said via email.

His research has focused on the positive impact of viewing fractal patterns in nature, which he said could reduce stress levels.

“Studies such as this one emphasize the aesthetic power of trees. There is a Japanese tradition known as ‘forest bathing,’ Taylor added. “Based on studies such as these, a more appropriate description is ‘fractal bathing.’ We should soak up the aesthetic qualities of trees — whether this is in nature or in art.”

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