Trees portrayed in famous works of art follow the same mathematical rules as real -life trees they discovered scientists.
The hidden mathematician concept in this tree art – geometric shapes known as fractals – is apparent in branching patterns in nature and can be fundamental to humans to recognize these works of art as trees, according to Mitchell Newberry, a mathematician biologist at the University of New Mexico, and his colleague Jingyi Gao, a doctoral student at Wisconsin University.
Like the branches, branches and leaves of a tree, the fractals repeat the same patterns at different scales. Snowflakes, lightning and human blood vessels are also fractal structures, which show a degree of autosimilarity: by expanding the details of a fract, you can see a replica of the whole.
“If you look at a tree, your branches are branching. Then the children branches repeat the figure of the father branch, ”Gao said in a press release.
Newberry and Gao chose to study works of art portraying individual trees. Their selections, which they said covered by different eras and cultures, included carvings on 16th -century stone windows from the Sidi Saiyed mosque in India, an 18th century painting called “Cherry Flowers” by Japanese artist Matsumura Goshun and two early 20th century works by Dutch painter Piet Mondrian. They also examined the painting “L’Arbre de Vie” (“Tree of Life”) of 1909 Gustav Klimt.
They found that the trees portrayed in works of art, even when abstract or stylistic, mostly, but did not always correspond to branching and scale patterns found in natural trees.
“Any kind of abstraction is a way of trying to reach natural laws, whether a mathematical or artistic abstraction. There are many different types of trees in the world, but this theory shows us and gives us a basis 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 portrayed trees in abstract ways, removing all the elements except the most essential, but still clearly transmitting a tree. This aligned with his own work explaining mathematically as trees -like structures in human biology, such as veins, arteries and lungs, they use their physical form to efficiently deliver blood and air.
To reach their discoveries, the researchers were able to develop a method of assessing trees ramification patterns and generalized it in a simple common formula, according to Fabian Fischer, a researcher at the Munich Technical University in Germany, who did not participate in the study by Newberry and GAO.
“The method is based on ideas dating back to Leonardo da Vinci and have been revisited by biologists several times,” Fischer said. “I found a highly stimulating reading, with an interesting connection between works of art and biology.”
Scale from one to three
In nature, fractal patterns are not only aesthetically pleasant, but also often related to function. For example, branching allows trees to transport fluids, capture light and maintain mechanical stability.
As a fractal is a geometric form, mathematicians can calculate their complexity, or fractal dimension – even when it appears in art.
“There are some characteristics of art that seem aesthetic or subjective, but we can use math to describe them,” said Gao.
In his research published in the scientific journal PNAS Nexus on February 11, Gao and Newberry analyzed the variation in the thickness of trees in the works of art 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 scale exponent of the branch diameter.
The study found that the trees in works of art had a value of scale of the diameter of the branch widely corresponding to the interval of 1.5 to 3 for real trees. Outside these values, the portrayed objects were not easily recognizable as trees. Gao and Newberry were surprised to find that the highly stylized carving of the Indian mosque had a value closer to that of real trees than the tree in “cherry flowers”, which they initially thought it seemed more natural.
Although extremely rich in detail, with over 400 individual branches, “Cherry Blossoms” displayed a 1.4 scale exponent, while the pair calculated that the carved Indian tree has a value of 2.5. Newberry said that having a more realistic scale factor of scale may have allowed artists to take on more creative risks, keeping the object recognizable as a tree.
“As you abstract details and still want viewers to recognize it as a beautiful tree, so you may have to be closer to reality in some other aspects,” said Newberry.
Naturally, artists like Mondrian and Klimt probably wouldn’t be aware of fractals, or the mathematics that support them, but perhaps they had an innate understanding of the subtle proportions that 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, so it is not possible to draw definitive conclusions.
The impact of fractals
The authors studied a series of works by the abstract painter Mondrian that portray the same tree, but in increasingly less realistic ways.
His 1911 work “by Grijze Boom” (“The Gray Tree”) shows a series of black lines against a gray background, but the painting is instantly recognizable as a tree, with its ramification scale value in the real tree range at 2.8.
“I don’t think he (Mondrian) is even trying to find the essence of the trees, but as he will remove elements, this thing we really consider in science turns out to be one of the last to disappear in art,” Newberry said. “Clearly, he finds this very important, and clearly is very important to human perception.”
However, in 1912’s “Bloeiende Appelboom” (“Apple in Flower”), a painting of the same series, the scale of the diameter of the branches disappears, said Newberry, with a value of 5.4. “While most observers of the gray tree immediately realize a tree, naive Observators of Apple Tree in flower see dancers, roots, fish, faces, water, vitals, leaves, flowers or nothing representational,” the authors observed in the study.
The researchers also examined Gustav Klimt’s ““ Tree of Life ”) painting“ “Tree of Life”), 1909. Although the representation of the tree in this work is highly stylized, study measurements suggest that it also fits the statistical track of a real tree.
The authors of the study are not the first to apply mathematics to trees in art. Renaissance polymate Leonardo da Vinci observed the growth of trees and created his own mathematical rule to paint them. His work on trees physiology inspired landscape scientists and artists to study branching standards, according to the new research.
The study findings are intriguing because they are part of artistic and scientific approaches in the study of trees, said Richard Taylor, professor of physics at the University of Oregon. “Although focused on trees, the article is addressing a much greater question-because natural standards are so beautiful-and interdisciplinary collaborations are essential to providing the answers,” said Taylor, who did not participate in the study by email.
His research focused on the positive impact of visualizing fractal patterns on nature, which he said could reduce stress levels. “Studies like this emphasize the aesthetic power of trees. There is a Japanese tradition known as ‘forest bath’, ”Taylor added. “Based on studies like these, a more appropriate description would be ‘fractal bath’. We must absorb the aesthetic qualities of trees – whether in nature or in art. ”
This content was originally published in trees in famous paintings follow mathematical patterns, says study on the website CNN Brazil.
Source: CNN Brasil

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