Art and science illuminate the same subtle proportions in tree branches
Do artists and scientists see the same thing in the shape of trees? As a scientist who studies branching patterns in living things, I'm starting to think so.
Piet Mondrian was an early 20th-century abstract artist and art theorist obsessed with simplicity and essence of form. Even people who have never heard of Mondrian will likely recognize his iconic irregular grids of rectangles.
When I saw Mondrian's 1911 'Gray Tree,' I immediately recognized something about trees that I had struggled to describe. By removing all but the most essential elements in an abstract painting, Mondrian demonstrated something I was attempting to explain using physics and fractal geometry.
My field of research is mathematical biology. My colleagues and I try to explain how treelike structures such as veins and arteries, lungs and leaves fine-tune their physical form to efficiently deliver blood, air, water and nutrients.
Fundamental research in the biology of branching helps cure cardiovascular diseases and cancer, design materials that can heal themselves and predict how trees will respond to a changing climate. Branching also shows up in ant foraging patterns, slime molds and cities.
From 1890 to 1912, Mondrian painted dozens of trees. He started with full-color, realistic trees in context: trees in a farmyard or a dappled lane. Gradually he removed leaves, depth, color and eventually even branching from his tree paintings. 'Gray Tree' uses only curved lines of various thickness superimposed on top of one another at seemingly random angles. Yet the image is unmistakably a tree.
How did Mondrian convey the sense of a tree with so little? The science of trees may offer some clues.
One goal of mathematical biology is to synthesize what scientists know about the vast diversity of living systems – where there seems to be an exception to every rule – into clear, general principles, ideally with few exceptions. One such general principle is that evolution fine-tunes treelike structures in living things to make metabolism and respiration as efficient as possible.
The body carefully controls the thickness of vessels as they branch, because deviation from the most efficient diameter wastes energy and causes disease, such as atherosclerosis.
In many cases, such as human blood vessels, the body exerts much tighter control over diameter than length. So while veins and arteries might take circuitous routes to accommodate the vagaries of organs and anatomy, their diameter usually stays within 10% of the optimum. The same principle appears in tree branches as well.
The precise calibration of branch diameter leads to a hallmark of fractal shapes called scale invariance. A scale invariance is a property that holds true regardless of the size of an object or part of an object you're looking at. Scale invariance occurs in trees because trunks, limbs and twigs all branch in similar ways and for similar reasons.
The scale invariance in branch diameter dictates how much smaller a limb should be as it branches and how much investment a tree makes in a few thick branches versus many thin ones. Trees have evolved scale invariance to transport water, reach light and resist gravity and wind load as efficiently as possible given physical limits.
This science of trees inspired my colleague and me to measure the scaling of tree branch diameter in art.
Among my favorite images is a carving of a tree from a late-medieval mosque in India. Its exaltation of trees reminds me of Tolkien's Tree of Gondor and the human capacity to appreciate the simple beauty of living things.
But I also find mathematical inspiration in the Islamic Golden Age, a time when art, architecture, math and physics thrived. Medieval Islamic architects even decorated buildings with infinitely nonrepeating tiling patterns that were not understood by Western mathematics until the 20th century.
The stylized tree carvings of the Sidi Saiyyed mosque also follow the precise system of proportions dictated by the scale invariance of real trees. This level of precision of branch diameter takes an attentive eye and a careful plan – much better than I could freehand.
Indeed, wherever our team looked at trees in great artwork, such as Klimt's 'Tree of Life' or Matsumura Goshun's 'Cherry Blossoms,' we also found precise scale invariance in the diameter of branches.
'Grey Tree' also realistically captures the natural variation in branch diameters, even when the painting gives the viewer little else to go on. Without realistic scaling, would this painting even be a tree?
As if to prove the point, Mondrian made a subsequent painting the following year, also with a gray background, curved lines and the same overall composition and dimensions. Even the position of some of the lines are the same.
But, in 'Blooming Apple Tree' (1912), all the lines are the same thickness. The scaling is gone, and with it, the tree. Before reading the title, most viewers would not guess that this is a painting of a tree. Yet Mondrian's sketches reveal that 'Blooming Apple Tree' and 'Gray Tree' are the very same tree.
The two paintings contain few elements that might signal a tree – a concentration of lines near the center, lines that could be branches or a central trunk and lines that could indicate the ground or a horizon.
Yet only 'Gray Tree' has scale-invariant branch diameters. When Mondrian removes the scale invariance in 'Blooming Apple Tree,' viewers just as easily see fish, scales, dancers, water or simply nonrepresentational shapes, whereas the tree in 'Gray Tree' is unmistakable.
Mondrian's tree paintings and scientific theory highlight the importance of the thickness of tree branches. Consilience is when different lines of evidence and reasoning reach the same conclusions. Art and math both explore abstract descriptions of the world, and so seeing great art and science pick out the same essential features of trees is satisfying beyond what art or science could accomplish alone.
Just as great literature such as 'The Overstory' and 'The Botany of Desire' show us how trees influence our lives in ways we often don't notice, the art and science of trees show how humans are finely attuned to what's important to trees. I think this resonance is one reason people find fractals and natural landscapes so pleasing and reassuring.
All these lines of thinking give us new ways to appreciate trees.
This article is republished from The Conversation, a nonprofit, independent news organization bringing you facts and trustworthy analysis to help you make sense of our complex world. It was written by: Mitchell Newberry, University of Michigan
Read more:
Art and science entwined: This course explores the long, interrelated history of two ways of seeing the world
Art illuminates the beauty of science – and could inspire the next generation of scientists young and old
I'm an artist using scientific data as an artistic medium − here's how I make meaning
Mitchell Newberry has published research on tree branching supported by University of Michigan and University of New Mexico. He volunteers with Cool It Burque, a tree-planting group in Albuquerque, NM.
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CNN
14-03-2025
- CNN
Trees in art, as well as life, often follow simple mathematical rules, study finds
Summary Scientists have discovered that trees in famous artwork follow the same mathematical fractal patterns as real trees. Researchers analyzed tree art across cultures, finding consistent branch scaling values matching those in nature. A University of New Mexico mathematical biologist says these patterns may help humans recognize stylized images as trees. In Mondrian's increasingly abstract tree paintings, the recognizability disappeared when fractal patterns were abandoned. Fractal patterns in art and nature are both functional and aesthetically pleasing, according to the study published in PNAS Nexus. 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.'
Yahoo
12-02-2025
- Yahoo
Scientists uncover hidden maths depicted in tree branch patterns in da Vinci and Mondrian artwork
Trees depicted in the artwork of famous painters like Leonardo da Vinci and Piet Mondrian follow the math behind their branching pattern in nature, a new study says. This hidden math in some abstract paintings may even underlie our ability to recognise such artwork as depictions of trees, according to the research, published in the journal PNAS Nexus. Trees in nature follow a 'self-similar' branching pattern called a fractal, in which the same structures repeat at smaller and smaller scales from the trunk to the branch tip. In the new study, scientists mathematically examined the scaling of branch thickness in depictions of trees in artworks. Researchers derived mathematical rules for proportions between branch diameters, and the approximate number of branches of different diameters. 'We analyse trees in artwork as self-similar, fractal forms, and empirically compare art with theories of branch thickness developed in biology,' researchers explained. Leonardo da Vinci observed that tree limbs preserve their thickness as they branch. The Italian Renaissance artist used a parameter called α to determine the relationships between the diameters of the various branches. He asserted that if the thickness of a branch is the same as the summed thickness of its two smaller branches then the parameter α would be 2. Researchers analysed trees in art from several parts of the world, including those in the 16th-century Sidi Saiyyed Mosque in Ahmedabad, India, Edo period Japanese painting, and 20th-century abstract art. They found that the values of α in these artworks range from 1.5 to 2.8, similar to the range of this value in natural trees. 'We find α in the range 1.5 to 2.8 corresponding to the range of natural trees,' scientists wrote. 'While fractal dimension varies considerably across trees and artwork, we find that the range of α in case studies of great artworks across cultures and time periods corresponds to the range of real trees,' they said. Even abstract paintings such as Piet Mondrian's 1912 cubist Gray Tree, which doesn't visually show treelike colours, can be identified as trees if a realistic value for α is used, researchers say. 'Abstract paintings with realistic α are recognisable as trees, whereas an otherwise similar painting is no longer distinctly recognisable as a tree,' they say. The new study offers a perspective to 'appreciate and recreate the beauty of trees,' scientists say. It also highlights that art and science can provide complementary lenses on natural and human worlds, they added.
Yahoo
11-02-2025
- Yahoo
Art and science illuminate the same subtle proportions in tree branches
Do artists and scientists see the same thing in the shape of trees? As a scientist who studies branching patterns in living things, I'm starting to think so. Piet Mondrian was an early 20th-century abstract artist and art theorist obsessed with simplicity and essence of form. Even people who have never heard of Mondrian will likely recognize his iconic irregular grids of rectangles. When I saw Mondrian's 1911 'Gray Tree,' I immediately recognized something about trees that I had struggled to describe. By removing all but the most essential elements in an abstract painting, Mondrian demonstrated something I was attempting to explain using physics and fractal geometry. My field of research is mathematical biology. My colleagues and I try to explain how treelike structures such as veins and arteries, lungs and leaves fine-tune their physical form to efficiently deliver blood, air, water and nutrients. Fundamental research in the biology of branching helps cure cardiovascular diseases and cancer, design materials that can heal themselves and predict how trees will respond to a changing climate. Branching also shows up in ant foraging patterns, slime molds and cities. From 1890 to 1912, Mondrian painted dozens of trees. He started with full-color, realistic trees in context: trees in a farmyard or a dappled lane. Gradually he removed leaves, depth, color and eventually even branching from his tree paintings. 'Gray Tree' uses only curved lines of various thickness superimposed on top of one another at seemingly random angles. Yet the image is unmistakably a tree. How did Mondrian convey the sense of a tree with so little? The science of trees may offer some clues. One goal of mathematical biology is to synthesize what scientists know about the vast diversity of living systems – where there seems to be an exception to every rule – into clear, general principles, ideally with few exceptions. One such general principle is that evolution fine-tunes treelike structures in living things to make metabolism and respiration as efficient as possible. The body carefully controls the thickness of vessels as they branch, because deviation from the most efficient diameter wastes energy and causes disease, such as atherosclerosis. In many cases, such as human blood vessels, the body exerts much tighter control over diameter than length. So while veins and arteries might take circuitous routes to accommodate the vagaries of organs and anatomy, their diameter usually stays within 10% of the optimum. The same principle appears in tree branches as well. The precise calibration of branch diameter leads to a hallmark of fractal shapes called scale invariance. A scale invariance is a property that holds true regardless of the size of an object or part of an object you're looking at. Scale invariance occurs in trees because trunks, limbs and twigs all branch in similar ways and for similar reasons. The scale invariance in branch diameter dictates how much smaller a limb should be as it branches and how much investment a tree makes in a few thick branches versus many thin ones. Trees have evolved scale invariance to transport water, reach light and resist gravity and wind load as efficiently as possible given physical limits. This science of trees inspired my colleague and me to measure the scaling of tree branch diameter in art. Among my favorite images is a carving of a tree from a late-medieval mosque in India. Its exaltation of trees reminds me of Tolkien's Tree of Gondor and the human capacity to appreciate the simple beauty of living things. But I also find mathematical inspiration in the Islamic Golden Age, a time when art, architecture, math and physics thrived. Medieval Islamic architects even decorated buildings with infinitely nonrepeating tiling patterns that were not understood by Western mathematics until the 20th century. The stylized tree carvings of the Sidi Saiyyed mosque also follow the precise system of proportions dictated by the scale invariance of real trees. This level of precision of branch diameter takes an attentive eye and a careful plan – much better than I could freehand. Indeed, wherever our team looked at trees in great artwork, such as Klimt's 'Tree of Life' or Matsumura Goshun's 'Cherry Blossoms,' we also found precise scale invariance in the diameter of branches. 'Grey Tree' also realistically captures the natural variation in branch diameters, even when the painting gives the viewer little else to go on. Without realistic scaling, would this painting even be a tree? As if to prove the point, Mondrian made a subsequent painting the following year, also with a gray background, curved lines and the same overall composition and dimensions. Even the position of some of the lines are the same. But, in 'Blooming Apple Tree' (1912), all the lines are the same thickness. The scaling is gone, and with it, the tree. Before reading the title, most viewers would not guess that this is a painting of a tree. Yet Mondrian's sketches reveal that 'Blooming Apple Tree' and 'Gray Tree' are the very same tree. The two paintings contain few elements that might signal a tree – a concentration of lines near the center, lines that could be branches or a central trunk and lines that could indicate the ground or a horizon. Yet only 'Gray Tree' has scale-invariant branch diameters. When Mondrian removes the scale invariance in 'Blooming Apple Tree,' viewers just as easily see fish, scales, dancers, water or simply nonrepresentational shapes, whereas the tree in 'Gray Tree' is unmistakable. Mondrian's tree paintings and scientific theory highlight the importance of the thickness of tree branches. Consilience is when different lines of evidence and reasoning reach the same conclusions. Art and math both explore abstract descriptions of the world, and so seeing great art and science pick out the same essential features of trees is satisfying beyond what art or science could accomplish alone. Just as great literature such as 'The Overstory' and 'The Botany of Desire' show us how trees influence our lives in ways we often don't notice, the art and science of trees show how humans are finely attuned to what's important to trees. I think this resonance is one reason people find fractals and natural landscapes so pleasing and reassuring. All these lines of thinking give us new ways to appreciate trees. This article is republished from The Conversation, a nonprofit, independent news organization bringing you facts and trustworthy analysis to help you make sense of our complex world. It was written by: Mitchell Newberry, University of Michigan Read more: Art and science entwined: This course explores the long, interrelated history of two ways of seeing the world Art illuminates the beauty of science – and could inspire the next generation of scientists young and old I'm an artist using scientific data as an artistic medium − here's how I make meaning Mitchell Newberry has published research on tree branching supported by University of Michigan and University of New Mexico. He volunteers with Cool It Burque, a tree-planting group in Albuquerque, NM.
