Apparently, trees are ‘roughly’ fractal. As for fractals themselves, there’s this from the Fractal Foundation’s What are Fractals? webpage,
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A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.
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Caption: Piet Mondrian painted the same tree in “The gray tree” (left) and “Blooming apple tree” (right). Viewers can readily discern the tree in “The gray tree” with a branch diameter scaling exponent of 2.8. In “Blooming apple tree,” all the brush strokes have roughly the same thickness and viewers report seeing fish, water and other non-tree things. Credit: Kunstmuseum Den Haag
While artistic beauty may be a matter of taste, our ability to identify trees in works of art may be connected to objective—and relatively simple—mathematics, according to a new study.
Led by researchers from the University of Michigan and the University of New Mexico, the study investigated how the relative thickness of a tree’s branching boughs affected its tree-like appearance.
This idea has been studied for centuries by artists, including Leonardo DaVinci [Leonardo da Vinci], but the researchers brought a newer branch of math into the equation to reveal deeper insights.
“There are some characteristics of the art that feel like they’re aesthetic or subjective, but we can use math to describe it,” said Jingyi Gao, lead author of the study. “I think that’s pretty cool.”
Gao performed the research as an undergraduate in the U-M Department of Mathematics, working with Mitchell Newberry, now a research assistant professor at UNM and an affiliate of the U-M Center for the Study of Complex Systems. Gao is now a doctoral student at the University of Wisconsin.
In particular, the researchers revealed one quantity related to the complexity and proportions of a tree’s branches that artists have preserved and played with to affect if and how viewers perceive a tree.
“We’ve come up with something universal here that kind of applies to all trees in art and in nature,” said Newberry, senior author of the study. “It’s at the core of a lot of different depictions of trees, even if they’re in different styles and different cultures or centuries.”
The work is published in the journal PNAS [Proceedings of the National Academy of Sciences] Nexus.
As a matter of fractals
The math the duo used to approach their question of proportions is rooted in fractals. Geometrically speaking, fractals are structures that repeat the same motifs across different scales.
Fractals are name-dropped in the Oscar-winning smash hit “Let it Go” from Disney’s “Frozen,” making it hard to argue there’s a more popular physical example than the self-repeating crystal geometries of snowflakes. But biology is also full of important fractals, including the branching structures of lungs, blood vessels and, of course, trees.
“Fractals are just figures that repeat themselves,” Gao said. “If you look at a tree, its branches are branching. Then the child branches repeat the figure of the parent branch.”
In the latter half of the 20th century, mathematicians introduced a number that is referred to as a fractal dimension to quantify the complexity of a fractal. In their study, Gao and Newberry analyzed an analogous number for tree branches, which they called the branch diameter scaling exponent. Branch diameter scaling describes the variation in branch diameter in terms of how many smaller branches there are per larger branch.
“We measure branch diameter scaling in trees and it plays the same role as fractal dimension,” Newberry said. “It shows how many more tiny branches there are as you zoom in.”
While bridging art and mathematics, Gao and Newberry worked to keep their study as accessible as possible to folks from both realms and beyond. Its mathematical complexity maxes out with the famous—or infamous, depending on how you felt about middle school geometry—Pythagorean theorem: a2 + b2 = c2.
Roughly speaking, a and b can be thought of as the diameter of smaller branches stemming from a larger branch with diameter c. The exponent 2 corresponds to the branch diameter scaling exponent, but for real trees its value can be between about 1.5 and 3.
The researchers found that, in works of art that preserved that factor, viewers were able to easily recognize trees—even if they had been stripped of other distinguishing features.
Artistic experimentation
For their study, Gao and Newberry analyzed artwork from around the world, including 16th century stone window carvings from the Sidi Saiyyed Mosque in India, an 18th century painting called “Cherry Blossoms” by Japanese artist Matsumuara Goshun and two early 20th century works by Dutch painter Piet Mondrian.
It was the mosque carvings in India that initially inspired the study. Despite their highly stylized curvy, almost serpentine branches, these trees have a beautiful, natural sense of proportion to them, Newberry said. That got him wondering if there might be a more universal factor in how we recognize trees. The researchers took a clue from DaVinci’s [sic] analysis of trees to understand that branch thickness was important.
Looking at the branch diameter scaling factor, Gao and Newberry found that some of the carvings had values closer to real trees than the tree in “Cherry Blossoms,” which appears more natural.
“That was actually quite surprising for me because Goshun’s painting is more realistic,” Gao said.
Newberry shared that sentiment and hypothesized that having a more realistic branch diameter scaling factor enables artists to take trees in more creative directions and have them still appear as trees.
“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.
Mondrian’s work provided a serendipitous experiment to test this thinking. He painted a series of pieces depicting the same tree, but in different, increasingly abstract ways. For his 1911 work “De grijze boom” (“The gray tree”), Mondrian had reached a point in the series where he was representing the tree with just a series of black lines against a gray background.
“If you show this painting to anyone, it’s obviously a tree,” Newberry said. “But there’s no color, no leaves and not even branching, really.”
The researchers found that Mondrian’s branch scaling exponent fell in the real tree range at 2.8. For Mondrian’s 1912 “Bloeiende appelboom” (“Blooming apple tree”), however, that scaling is gone, as is the consensus that the object is a tree.
“People see dancers, fish scales, water, boats, all kinds of things,” Newberry said. “The only difference between these two paintings—they’re both black strokes on a basically gray background—is whether there is branch diameter scaling.”
Gao designed the study and measured the first trees as part of her U-M Math Research Experience for Undergraduates project supported by the James Van Loo Applied Mathematics and Physics Undergraduate Support Fund. Newberry undertook the project as a junior fellow of the Michigan Society of Fellows. Both researchers acknowledged how important interdisciplinary spaces at Michigan were to the study.
“We could not have done this research without interaction between the Center for the Study of Complex Systems and the math department. This center is a very special thing about U of M, where math flourishes as a common language to talk across disciplinary divides,” Newberry said. “And I have been really inspired by conversations that put mathematicians and art historians at the same table as part of the Society of Fellows.”
Caption: Leonardo da Vinci’s sketch of a tree illustrates the principle that combined thickness is preserved at different stages of ramification. Credit: Institut de France Manuscript M, p. 78v.
The math that describes the branching pattern of trees in nature also holds for trees depicted in art—and may even underlie our ability to recognize artworks as depictions of trees.
Trees are loosely fractal, branching forms that repeat the same patterns at smaller and smaller scales from trunk to branch tip. Jingyi Gao and Mitchell Newberry examine scaling of branch thickness in depictions of trees and derive mathematical rules for proportions among branch diameters and for the approximate number of branches of different diameters. The authors begin with Leonardo da Vinci’s observation that trees limbs preserve their thickness as they branch. The parameter α, known as the radius scaling exponent in self-similar branching, determines the relationships between the diameters of the various branches. If the thickness of a branch is always the same as the summed thickness of the two smaller branches, as da Vinci asserts, then the parameter α would be 2. The authors surveyed trees in art, selected to cover a broad geographical range and also for their subjective beauty, and found values from 1.5 to 2.8, which correspond to the range of natural trees. Even abstract works of art that don’t visually show branch junctions or treelike colors, such as Piet Mondrian’s cubist Gray Tree, can be visually identified as trees if a realistic value for α is used. By contrast, Mondrian’s later painting, Blooming Apple Tree, which sets aside scaling in branch diameter, is not recognizable as a tree. According to the authors, art and science provide complementary lenses on the natural and human worlds.
