Fractals part 3

Part 1 | Part 2

Fractals are not interesting only for mathematicians, nor are they recognized merely for their mathematical beauty. They appear in nature. While less precise and less iterated, understanding natural objects as having fractal properties gives a new way to look at phenomenon. When studying clouds, rocks, plants, animal forms, urban growth processes, or stock-exchange dynamics, looking at them as fractals can help with analysis.

Fractality in this broader sense is characterized by mathematically less rigorous self-similarities and algorithms. In fact, both are often not precisely known. In addition, the number of self-similar dimensions is usually not infinite in nature, but restricted to a few ranges, as for instance in the often quoted fractality of a fern leaf. [...] Fractal morphology in this broad sense seems to be an almost ubiquitous phenomenon in nature and culture, presumably because it represents one of the most economic methods for producing an enormous richness of forms by a minimal number of algorithms.

-Losa, Gabriele, Ed. Fractals in Biology and Medicine IV.

An understanding of fractals is especially important in biology. Biology does not yet understand how life works, or how it creates itself. Fractals offer a method of organizing by a simple rule many pieces. This gives much insight:

Mandelbrot’s concepts such as scale invariance, self-similarity, irregularity and iterative processes as tackled by fractal geometry have prompted innovative ways to promote a real progress in biomedical sciences, namely by understanding and analytically describing complex hierarchical scaling processes, chaotic disordered systems, non-linear dynamic phenomena, standard and anomalous transport diffusion events through membrane surfaces, morphological structures and biological shapes either in physiological or in diseased states.

-Losa, Gabriele, Ed. Fractals in Biology and Medicine IV.

Mandelbrot went to a conference where the talked about "the fractal geometry of trees and other natural phenomena among which were the structure of natural boundaries which are never simple, and the systematic structure of trees that abound in nature, in animals for example in the form of blood vessels and airways" (Losa). When biologists saw Mandelbrot's descriptions of the fractional dimensions of the coast of England, they understood "how to describe cell membranes without obtaining paradoxal results" (Losa).

In living systems, transport structures have fractal properties. The point of transport structures is to have a very large surface area within a finite space, so that as much transferring can happen as possible. These structures include the lungs, arteries and capillaries, the liver's endoplasmic reticulum, cellular membranes, and the lining of the intestines. These structures "show a level of complexity that is best described by fractal geometry [...] The concept of fractal geometry is valuable in understanding the design of biological structures at all levels of organization" (Losa).

From Fractals in Biology and Medicine

Before being introduced to fractal geometry, biologists had difficulty determining the surface area of the lungs. They kept measuring at different microscopic resolutions (similar to using different measuring sticks; see part 1 about Mandelbrot and the coast of England), and could not get their measurements to narrow down towards a number. Instead, the surface area increased with increasing resolution. When the lungs were compared to fractals, biologists could determine that "lung's internal surface is a space-filling fractal surface whose dimension is estimated at 2.2" (Losa).

From Fractals in Biology and Medicine

The problem is that some people take the comparison with fractals too far. By definition, biological things are not fractals, because they are not strictly self-similar and infinite. So the question is: Is it useful to classify biological things as fractals?

A fractal analysis is an ideal method for quantification of the branching patterns of dendritic trees, returning data not available by other methods that are based on Euclidean geometry. Fractal analysis can have three separate goals. 1. determination whether or not neurons are fractal, 2. classification of cells, 3. identification of biological meaning associated with D other than inherent in the notion of fractality.

-Losa, Gabriele, Ed. Fractals in Biology and Medicine IV.


Note the self-similarity on many levels. Romanesco is not a true fractal -- it does not iterate infinitely -- but it certainly has fractal-like properties in several iterations. Is it useful to define this as a fractal?

If you want to know more, an interesting text from which much of this information was taken, Fractals in Biology and Medicine, can be found here [sadly, best viewed in internet explorer]

Fractals today can be an art form, a display of mathematical beauty and a picture of the infinite, as well as an insight to how our world works. Try a google image search for fractals to see some of the really wonderful pictures out there, or take a look at these.

Fractal art

Katie Dektar, September 2008