letting nature back in

We cannot necessarily put our finger on what makes patterns in nature so recognizable. In some natural phenomena we may see elements that are recurring, sometimes at different scales: repeating elements in shapes that are complex and irregular.

One of the more obvious examples of patterning that is repeating (self-replicating) and scalable (whether enlarged – by zooming in, or reduced in size – by zooming out) is evident in the structure of the leaf divisions of the frond of a fern.

The leaf structure of a fern frond is pretty much self-similar as its intricately detailed leaf divisions self-replicate or repeat at different scales, as can be seen in the photographs above

In addition to being scalable and self-repeating, the outlines of these shapes are irregular and wiggly and cannot be mathematically mapped using classical geometry. Classical (or Euclidean) geometry is based on geometrical shapes with smooth lines such as two-dimensional squares, rectangles, triangles and circles, and three dimensional cubes, cylinders, pyramids, cones and spheres, all being shapes that are seldom found in nature.

Many plants have forms that are geometrically irregular with rough rather than smooth outlines, comprising repeated shapes that are self-similar and scalable

So, now I am wondering why I have followed through on doing a post on fractals? Mathematics is not something I have an affinity for, and something as complex as fractals constantly eludes me. Yet, the development of fractal geometry commencing in the 1970s was not only ground-breaking in mathematics, but the use of computer-generated graphics to make data visual resulted in ‘computer art’ that impacted on popular culture and even resonated with the subculture of psychedelia.

Above is a sample of the image results found by doing an Internet search using the term ‘computer-generated fractals’

Benoît Mandelbrot, often described as a ‘maverick mathematician’, building on the work of previous mathematicians (including Gaston Julia and Pierre Fatou) and utilizing the data visualization power of computers (he worked at IBM) is credited with being a founder of fractal geometry. His name entered popular culture when complex and beautiful computer-generated imagery graphically demonstrated his discovery of a fractal shape that became known as the Mandelbrot Set.  The development of computer graphics enabled the set of coded mappings of points on the complex plane to be endlessly reiterated and presented graphically.

Samples of T-shirts found when searching for images using the search term ‘Mandelbrot T-shirts

In South Africa we have a saying ‘ja well no fine’, which indicates something like reluctant acceptance, which might approach my initial response to the Mandelbrot Set and fractal geometry, given my lack of even basic mathematical conceptual understanding.  So if you are interested, Wikipedia has detailed entries on both the Mandelbrot Set (see here) and fractals (see here).

Mandelbrot coined the mathematical term fractal (meaning broken-up, fractional or irregular) in the 1970s. He is quoted in the Wikipedia entry on fractals as summarizing fractals as “beautiful, damn hard, increasingly useful. That’s fractals”. He later defined a fractal as “a shape made of parts similar to the whole in some way”.

When I started reading about fractals, my interest was to try to learn about fractals in nature, and many shapes in nature are in fact fractal. Fractal geometry can provide modelling of the complexity and roughness (wigglyness) of fractal phenomena in nature.

Fractals in nature include fractal branching in natural phenomena such as trees, river systems, lightning bolts and in the vessels in blood circulatory systems and in the lungs. Other fractals include snowflakes, mountain ranges, clouds, heart beats, pineapples, broccoli and ocean waves!

Following  research by the more than interesting British mathematician, scientist and pacifist Lewis Fry Richardson, in an early paper, ‘How Long is the Coast of Britain?—Statistical Self-Similarity and Fractional Dimension’ (1967), Mandelbrot discusses the geographical curves of the coastline and the phenomenon that the length of a coastline increases when smaller units of measure are used.  In other words, the length of the coastline is determined by the scale of measurement, and the closer it is measured, that is the more it is magnified, the more details appear.

This photograph of a shoreline (Lake Sibaya in KwaZulu-Natal) shows a close up view of the wiggly shoreline, illustrating how a shoreline will become longer, as the unit of measure that is applied becomes smaller and so more detailed

The more wiggly (or complicated) a coastline is the faster the rate of the increase in the length of the perimeter becomes as smaller units of measure are applied. This rate of increase is reflected in the calculated fractal dimension, which is a higher number the more wriggly a coastline is.

Key concepts such as complex numbers, the complex plane and fractal dimension are beyond the scope of this post (and my ability to explain). So I refer to three enlightening introductory texts that are listed as sources below: Rose (2012), Dallas (2014) and Najera (2020).

Perhaps the fractal nature of branching in nature is easier to grasp as it is more visual. For example, like all fractals, a tree can be seen to be a rough or fragmented shape that can be broken up into small parts, which can be seen as smaller copies of the larger shape.  The branching of a tree is fractal as the branches remain approximately self-similar, repeating as they progressively decrease in size culminating in twigs.  

As for all fractals, tree branches repeat at different scales, having a fine and detailed structure that reiterates. Tree structures are a factor in optimizing the transport of sap and access to light and the ability of a tree to be flexible and resist wind. 

Applications of the fractal nature of tree branching include that modelling can reveal how much carbon dioxide a tree exchanges with the atmosphere, and modelling can lead to an understanding of how much water a tree transpires through its leaves.

Other examples of branch-like structures are evident in the circulatory system of the blood, in the nervous system and in the respiratory systems of mammals.  In the respiratory system, a large surface area is needed to optimise exchange of gases (oxygen and carbon dioxide). Within the space constraints of the respiratory system, the branching system – its fractal geometry – allows the packing in of a relatively enormous surface area into a small volume. Similarly, in the blood circulatory system, the branching down to ever smaller capillaries enables the supply of blood to all cells in body tissues.

A cross-section of a broccoli head showing fractal branching

Fractal branching in nature exhibits the characteristics of self-similarity and detailed intricacy. Although within nature, fractal branching cannot be infinite, like all fractals they are self-similar over different scales; in other words they exhibit zoom symmetry.

Using fractal geometry, complex natural shapes can be encoded using simple mathematical rules. The complexity emerges through multiple iterations in an ongoing feedback loop.

Even clouds have been shown to be fractal. They are the same at all scales, and using fractal geometry to determine their fractal dimension, computer images of clouds can be generated that are indistinguishable from actual clouds.

Fractals are used in computer-generated landscapes and special effects used in movies and computer games. This cloudscape, photographed in the Western Cape, is however real

In his blog, George Dallas, an Information Engineer/Internet Social Scientist had this to say about the field of fractal geometry:

The shapes that come out of fractal geometry look like nature. This is an amazing fact that is hard to ignore. As we all know, there are no perfect circles in nature and no perfect squares. Not only that, but when you look at trees or mountains or river systems they don’t resemble any shapes one is used to in maths. However with simple formulas iterated multiple times, fractal geometry can model these natural phenomena with alarming accuracy. If you can use simple maths to make things look like the world, you know you’re onto a winner. Fractal geometry does this with ease.

Dallas (2014)

Fractals can be used to create computer-generated realistic-looking images of mountain ranges. The mountains in the photo above are real mountains in the Richtersveld National Park

The practical application of fractal geometry appears to be a growing field, with its ability to map complex and irregular objects being applied in a variety of disciplines. It has been used in astronomy to analyse galaxy structures. In Earth sciences fractal geometry is applied to data pertaining to dynamic systems such as cloud formation, weather systems, water currents, soil erosion, seismic patterns and even the migrations of animals.  In medicine, fractal techniques are used in the analysis and interpretation of CT and MRI scans in the diagnosis of cancer and other diseases. Fractal tools are also used in cardiovascular research and diagnostics.

Beyond astronomy, Earth sciences and medicine, fractal geometry is widely applied in other sciences including computer science, fluid mechanics and telecommunications.

Yet, even to non-mathematicians what remains so compelling about the fractal nature of structures is how appealing they can be to the human eye and imagination. The photograph below is of flowers on the fruiting structure of a pineapple plant that is growing in our vegetable garden. The fractal flowering structure does not have the mesmerising infinite flow of computer-generated fractals, but it is no less wondrous.

Sources:

Dallas, George. 2014. What is are Fractals and why should I care? George Dallas, UK based Information Engineer/Internet Social Scientist [blog]. https://georgemdallas.wordpress.com/2014/05/02/what-are-fractals-and-why-should-i-care/#:~:text=Fractal%20geometry%20is%20a%20field,example%20of%20a%20fractal%20shape.

Fractal Foundation. [n.d.] What are Fractals? https://fractalfoundation.org/resources/what-are-fractals/

Hemery, Gabriel. 2017. The art and math of tree fractals. Gabriel Hemery [blog].  https://gabrielhemery.com/tree-fractals/

Lesmoir-Gordon, N. 2012. The maverick mathematician: Benoît Mandelbrot and the stunning beauty of the fractal universe. Medicographia. No. 112 , Vol 34, No. 3, pp. 354-364). https://www.medicographia.com/2013/01/the-maverick-mathematician-benoit-mandelbrot-and-the-stunning-beauty-of-the-fractal-universe/

Najera, Jesus. 2020. Fractal Geometry: From Gaston Julia to Benoit Mandelbrot. August 12. Medium . Cantor’s Paradise. https://medium.com/cantors-paradise/fractal-geometry-9e516a5b244b

Rose, Michael. 2012. Explainer: what are fractals? December 11. The Conversation. https://theconversation.com/explainer-what-are-fractals-10865

Posted by Carol