09. Fractal – Sabina Hyoju Anh

Sabina Hyoju Anh (KR)

Fractal

Texts by Paolo Saracco

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line”

Benoit B. Mandelbrot, 1982

By trusting Mandelbrot’s inspiring words, we cannot always expect Nature to adapt itself to Geometry.
Thus, let us explore how Geometry adapts to Nature.

A key concept in classical geometry is that a “regular” object, no matter how “complicated” it may look like at first sight, becomes “simpler” if observed close enough. As the Statue of Liberty may look like a large area of flat land for an ant walking on it. On the contrary, a fractal is an object that remains “complicated” independently of the distance we look at it.

The name ‘fractal’, from the Latin ‘fractus’ meaning broken or fragmented, was coined by Mandelbrot in 1975.

Roughly speaking, a fractal is a set that is more “irregular” than those described by means of traditional Mathematics. No matter how much the set is magnified, it preserves the richness and complexity of the whole structure and smaller and smaller details become visible. As it happens for the border of a snowflake, the surface of a Romanesco broccoli or the coasts of Great Britain.

Lungs and trees are excellent examples of natural fractals. In fact, they share the same branching pattern and it is for good reason. Both trees and lungs serve a similar function: respiration, and hence it should not be surprising that they share a similar structure (this concept in science is known as the Structure-Function Relationship). The key to their success is that they both need a large surface stored in a limited volume to function well. In fact, the amount of gas that can be exchanged through the leaves on a tree or the lungs in an animal is directly proportional to their total surface area. Although the volume of a pair of human lungs is only about 4 – 6 liters, their surface area is between 50 and 100 square meters. That’s about the same area as a tennis court.

More formally, a fractal can be defined as a set “whose dimension does not correspond with the expected one”. Look, for instance, at the famous Hilbert-Peano curve, constructed by iterating the process shown below an infinite number of times.

Being a curve, it is expected to be a one-dimensional set, but it actually fills the whole plane, so formally it is a two-dimensional one. The existence of objects for which the intuitive notion of dimension is inappropriate led the mathematical community to explore new ones: the fractal dimensions.

Nowadays, fractal dimensions are used to characterize a broad spectrum of objects, including turbulence, river networks, urban growth, human physiology, medicine, and market trends.

Both natural and mathematical fractals can be extremely beautiful, and one of the extraordinary things about them is to discover how science, math and art may be intimately related.

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