A recent study published on the journal PNAS Nexus in early February, titled "Scaling in Branch Thickness and the Fractal Aesthetic of Trees," reveals that trees depicted in paintings by renowned artists like Leonardo da Vinci and Piet Mondrian follow the same mathematical laws of natural branching as real trees.
In nature, trees exhibit a self-similar branching pattern, meaning the same structures repeat at progressively smaller scales from the trunk to the branch tips.
In the study, scientists mathematically analyzed the scaling of branch thickness in artistic depictions of trees. Leonardo da Vinci, whose guidelines for painting trees have influenced both artists and scientists, described this scaling with a parameter called α (the radius scaling exponent). He proposed that if a branch's thickness equals the combined thickness of its two smaller branches, then α = 2.
Applying da Vinci's ideas to famous artworks, researchers measured α and compared it to real trees. They found that α in paintings ranges from 1.5 to 2.8, similar to natural trees. While some artworks adhere to this pattern, others deviate intentionally for artistic effect or due to stylistic constraints.
The study examined trees in art from around the world, including the intricate tree patterns carved into the stone window screens of the late-medieval Sidi Saiyyed Mosque in Ahmedabad, India; Cherry Blossoms, an ink-on-paper painting by Matsumura Goshun (1752–1811) from Japan's Edo period and Gustav Klimt’s L'Arbre de Vie (Tree of Life), a celebrated work known for its decorative, spiraling branches.
Even in abstract works like Piet Mondrian's De Grijze Boom (Gray Tree, 1912), where the branches are portrayed as abstract clusters of dark arcs instead of traditional tree-like colors, some scaling in branch thickness still exists. Despite the lack of typical tree representation, these arcs symbolizing parts of the branches allowed the researchers to analyze the overall structure of the tree. This suggests that when artists, even unknowingly, use a mathematically based value for α, the human brain still recognizes the image as a tree.
To further explore this idea, Mondrian's later painting, Bloeiende Appelboom (Blooming Apple Tree, 1912), removes even the scaling of branch diameter. The result is that the "tree" effect vanishes. Without the natural variation in branch thickness, Blooming Apple Tree could just as easily be interpreted as dancers, fish, flowers, or a purely abstract composition.
The study implies that incorporating realistic branch scaling into art enhances our ability to identify tree depictions. This opens up new ways to appreciate both nature and art, while also offering fresh perspectives on the beauty of trees in both contexts. Moreover, it emphasizes how various forms of artwork, whether traditional carvings or abstract paintings, can be analyzed using scientific methods to reveal hidden patterns in the portrayal of nature.
This brings us to an interesting parallel: fashion, much like art, is shaped by mathematical principles - think measurements, patterns, and sizes. So, could the concept of self-similarity inspire a mathematical approach to fashion design? Yes as the principle of self-similarity could be used to create layers that follow a specific progression, generating a sense of continuity and flow in designs. However, it's crucial to distinguish between self-similar scaling and graduated or tiered designs, as these two concepts serve different purposes in shaping visual harmony.
Self-similar scaling refers indeed to a pattern where each part is a smaller version of the whole, often following a mathematical ratio. It’s the same principle seen in nature, such as the branching of trees or the structure of Romanesco broccoli, a fractal vegetable where each smaller part mirrors the whole, no matter how much you zoom in or out, adhering to a consistent ratio. On the other hand, graduated or tiered designs involve a structured arrangement where elements change in size, but they don't necessarily follow a strict proportional rule.
So, in nature, self-similar scaling follows a power law, where each segment is reduced by a consistent ratio, commonly ½ or ⅔ of the previous one. Applied to fashion, this concept requires precision, as it is easy to mistake a tiered or layered design for a self-similar one.
For instance, stacking seven shirts with progressively shorter sleeves does not create a self-similar pattern; rather, it results in a graduated or stacked effect without a strict proportional relationship. True self-similar fashion structures must adhere to a recursive scaling principle, where each element is a mathematically scaled-down iteration of the previous one, forming a continuous system rather than independent layers.
So remember, a self-similar design is characterized by strict proportionality, if each layer is exactly 70% or 50% of the preceding one, it meets the mathematical criteria. However, if proportions vary based on aesthetic preference rather than a precise ratio, the result is a graduated rather than a self-similar composition.
Consider, for instance, Cinzia Ruggeri's costume for Valeria Magli's Banana Morbide an example of a graduated design. The costume is made of multiple thin tops with one progressively shorter sleeve stacked one on top of the other, but the sleeve does not follow a strict mathematical sequence. Capucci's 1956 Nove Gonne (Nine Skirts) dress could theoretically approach self-similarity, as it is a single-piece garment with diminishing layers. Yet, its asymmetrical structure, shorter in the front, longer in the back, reveals that Capucci's methodology leans toward sculptural volume. In a nutshell, the design explores architectural repetition, but the scaling is more intuitive than formulaic.
For purer examples of self-similarity in fashion, we must look to designers who embrace mathematical rigor, such as origami-inspired couture. Some of the "haute papier" creations from Sandra Backlund's "Ink Blot Test" collection (A/W 2007–08) and Bea Szenfeld's "Sur La Plage" (2010) demonstrate the recursive logic of fractal-like structures, creating a mesmerizing visual effect. Two designs from these respective collections, with their progressively diminishing shapes, create an intricate, almost hypnotic, self-similar texture.
So, from now on, try to spot more examples of self-similarity in fashion. Exploring this concept sharpens our aesthetic awareness and pushes the boundaries of garment construction, challenging us to construct garments with the same mathematical harmony and rhythm that nature so effortlessly achieves in its patterns.
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