Posts Tagged ‘Mathematics’

We modern humans (Homo sapiens) have been around for much longer than was previously thought – 100,000 years longer in fact.

An international team of scientists recently reported the discovery in Morocco of human remains dating back 300,000 years. Previous fossil records have put the emergence of Homo sapiens in East Africa to about 200,000 years ago. It now seems probable that our species emerged not in a single East African “Garden of Eden” but in a number of places across eastern and northern Africa.

Three hundred thousand years ago, northern Africa was not the dry and arid land it is today. Because of a wetter climate, it was clothed in woodlands, forests and grasslands similar to those known to have existed in East Africa 100,000 years later.

These savannah-like environments are thought to be the ones in which modern humans evolved and to which, as a consequence, our brains and bodies are most comprehensively and efficiently adapted. In other words, we are most at home in natural environments. The savannah is for our species the Environment of Evolutionary “Adaptedness” or EEA. Each living species has its particular EEA and the total or even partial loss of the EEA means extinction unless the species can adapt to any significant environmental changes.

Compared with our ancestors of 300,000 years ago, we 21st century humans are living in very different environments.  For the more than 50 per cent of us living in towns and cities, the most obvious difference has to do with geometry. Ours is mainly a world of smooth lines, regular shapes, simplicity of form and symmetry. It is principally a rectilinear world that our forebears could never have imagined.

Theirs, in contrast, was largely a world of raggedness, irregularity, complexity and apparent chaos. The familiar Euclidian geometry that can be used to describe urban environment is much less applicable to the natural world. For that world a very different geometry – fractal geometry – is also required.

Fractals are created by patterns that recur on finer and finer scales meaning that a fractal object looks very similar whether it is viewed from some distance away, close up or anywhere in between.

Fractals are readily observed in tree branches like the ones shown in the accompanying figure (which originally appeared in a research article). The red rectangles show the same tangle of branches from three different distances. While the three images are not identical, they are remarkable similar.

A better gauge of the “self-similarity” of the three views is obtained using an analytical procedure that produces a measurement called a “D”. A smooth line, which has no fractal structure, has a D value of 1 while a completely filled space, which also has no fractal structure, has a D value of 2. Once a line begins to repeat itself, it starts to occupy space and its D value falls between 1 and 2. The D value of the three images of the branching limb is the same even though the patterns formed by the branches vary slightly.

As more fine detail is added to a fractal mix, more of the space is filled and the value of D moves closer to 2, as a photo which I received recently illustrates very nicely.

D values for some common natural features are:

Coastlines                           1.05 – 1.52clouds

Woody plants and trees  1.28 – 1.90

Waves                                 1.30

Clouds                                 1.30 – 1.33

Snowflakes                        1.70


I have risked boring you to sobs with this technical excursion into fractals because I want to share with you some recent discoveries that illustrate how wondrously our brains have been shaped by nature.

The ability to see and make sense of fractal objects in nature was central to the survival of our species. Without it, the complexity of nature would have been mentally (and emotionally) overwhelming. But millions of years of evolution produced a brain that could “decode” nature’s fractal language and extract the information needed to solve the problems of survival and reproduction.

Because the move by modern humans from natural to urban habitats started only a matter of a few thousand years ago, we remain creatures of the wild in terms of evolutionary development. As a consequence, the ability to respond to fractal objects endures as part of our make-up.

Studies of this response have provided several arresting findings:

  • Fractal objects appeal to our senses and many elicit aesthetic pleasure (or the “beauty buzz”). Such was the genius of the artist, Jackson Pollock, that he was able to create fractal masterpieces. Inspired by the fractal patterns he observed from the verandah of his house on Long Island, New York State (The house in the top image was his), he developed his drip and scatter painting technique to capture what he saw. Typically, he would proceed by creating relatively dense clusters of lines joined by longer sweeping lines. Then, often after a period of days, he would return and add finer and finer details. D analyses have confirmed that the images produced in this way are indeed fractal in nature. While Pollock’s earlier works had low D values (e.g. 1.3), his later works, like the one shown here, had higher values (in the order of 1.7 – 1.9). This is interesting because studies have shown that fractal objects in the mid-range of D values are generally found to be most attractive (the “Goldilocks” factor again). Perhaps the extra “challenge” of Pollock’s later paintings added to their artistic appeal.
  • There is an extraordinary parallelism between fractal forms in nature and the way the human eye moves when observing them. Maps of these eye movements also turn out to be fractal in structure. Why this is so is still a matter of speculation but it may have something to do with the information gathering efficiency of scanning patterns that move from larger to smaller features (Just as Pollock did when painting). Interestingly, animal grazing patterns sometimes take on the same whole-to-part, fractal organization.
  • The brain is both relaxed and busy when observing fractals. It is thought that when our brain is doing things it is wired to do, less effort and energy are involved. The concept of “fluency” is often used to describe non-demanding mental processing of this kind. This has led some researchers to predict that when our brain is processing fractals, the visual receiving and interpreting parts of our brain will be active while the parts of the brain to do with planning, executive control and concentrating will be in a more “free-wheeling”, relaxed mode. Studies using a physiological measure of stress and brain monitoring procedures report findings squarely supporting this prediction.

These are particularly intriguing discoveries in my view because they testify to the exquisite detail, subtlety, economy and efficiency with which evolutionary mechanisms have matched the human brain to the natural world. They also serve as a powerful reminder that if we are fully to understand ourselves and our behaviour, we need to understand the full scope and depth of nature’s imprint on the functioning of our brain. And we are not simply talking about “survival” behaviour. Just as Pollock’s art demonstrates, this imprint is to be found in the most sophisticated forms of human cultural, social and ethical behaviour. We cannot ignore the legacy of our species’ sojourn in nature – in its EEA – nor should we want to. It is a legacy to be embraced wholeheartedly because, as I argue in my book, it is a precious legacy.


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I have been distracted again. Sorry. I will keep my promise to write about the ripple effect of the beauty buzz, but first I have a discovery to share with you. Actually, I did not make the discovery myself but one of my bushwalking friends, Cecilia Goon, deserves the credit.

And this is it –

– not the Pouched Coral-fern (Gleichenia dicarpa) itself,  but the hexagon formed by its branching fronds. As I said in my last post, the hexagon is one of Nature’s most structurally efficient and aesthetically pleasing shapes.

For the Pouched Coral-fern to produce this shape, each pair of new fronds has to form an angle of 120 degrees. As you can see, the plants manage this with amazing, if not quite perfect, precision.

I have not found an explanation for why coral ferns grow this way but I noticed that when you look down on the leaves, you can see that each hexagonal array provides partial shelter for the layers below. The result is a protective canopy – a bit like a rainforest canopy, but with an important difference. The vertical spaces between the layers allow air and moisture to circulate.

The Pouched Coral-fern grows in sheltered, well-watered areas. The specimens I was looking at were growing where there was a lot of seeping water. I suspect that the plant has evolved to form a natural umbrella (it is sometimes called the Umbrella Fern) in order to preserve moisture around its roots. As the plants can grow to 4 metres in height, a dense patch of them forms a kind of forest with its own humid microclimate. By preserving soil moisture in this way, the clever fern maintains favourable growing conditions during dry times.

Ringtail possums love to use coral-fern to make their drays or nests in the forks of trees. They somehow know that the hexagonally arrayed leaves bind together to make a particularly strong and dense structure. Even these cute little marsupials have discovered a way of incorporating hexagons into architecture just as bees (in their honeycomb) and humans have.

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I have had a change of plan. Instead of talking about the ripple effects of the beauty buzz in this post, I want to share some further thoughts about the beauty buzz itself.

I have been watching an SBS TV documentary series called “The Code”. According to the presenter, mathematics professor, Marcus du Sautoy, everything in the Universe can be described mathematically. “He would say that”, I hear some cynics say, but he has convinced me that there is a mathematical code – read equations, ratios, numbers and geometry – underpinning all there is in the universe. This applies to the seemingly chaotic complexity of trees, forests, mountain ranges, clouds and coastlines as much as it does to nature’s more orderly structures such as planets, crystals, water droplets and snowflakes.

From time to time in his presentation, du Sautoy drops in references to aesthetics and beauty. He did this, for example, when talking about the Golden Ratio, the relationship (1: 1.618). This ratio, which is commonly associated with the curved elegance of a Nautilus shell and Leonardo da Vinci’s “Vitruvian Man”, crops up many times in art and architecture as well as in nature. The Organ Pipes, Mt Kaputar

Beauty figured again in his exploration of shapes in nature. One of the most remarkable (and potentially aesthetically pleasing) of these shapes is the hexagon – the choice of bees for the cells in honeycomb. The hexagon, or more correctly hexagonal prism, occurs commonly in lava beds such as the Giant’s Causeway in Ireland and the Organ Pipes near Mt Kaputar in northern NSW.

As the close-up shot of the “pipes” shows most of the columns are hexagonal in cross-section.

The columns took on this shape as the lava cooled, contracted and became stressed. The resulting hexagonal pattern of cracking represents the best way of containing these stresses with the minimum of energy. The Universe is lazy says du Sautay and will always save on energy and materials.

Back to the beauty buzz:  du Sautay has helped me to understand more clearly that before the human brain can experience beauty it must first detect the forms, patterns and relationship that give the universe order and meaning. It must be sensitive to symmetry, balance and proportion (e.g., Golden Ratio), for example, and to “good form” as exemplified in the hexagon, the cube, the triangular pyramid and the other three perfectly symmetrical Platonic solids.

I believe that the human brain is hard-wired to make sense of the universe’s handiwork – to discern the forms and patterns of nature spontaneously and intuitively. The fact that we humans are able, without effort, to make sense of what would otherwise be mind-blowing complexity is nothing short of wondrous.

In a particularly fascinating segment, du Sautay takes us behind the art of Jackson Pollack to reveal the depth of that man’s genius.

This is just part of a Pollack canvas but this section is remarkably similar to the canvas a whole. In this and similar works, Jackson captured the essence of the most commonly occurring structure in nature. This structure, technically called a fractal, looks rough and fragmented but it has a defining property that brings order to chaos. A part view of the structure, large or small, is very similar to the view of the whole. For example, the leafy silhouette of a tree is similar whether you are 100, 50 or 10 metres away.

Pollack’s genius lay in creating fractal designs. He probably never heard of fractals but he understood them intuitively. People find beauty in his work in the same way that they find beauty in the rough raggedness of nature. Our brains have been programmed by evolution to do so. What sets Pollack apart is his gift to produce as well as appreciate fractals. He did this without the aid of mathematics and computer generated imaging, just with his brush and his movements around a canvas spread over the floor.

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