Simple Elegance to Wonderful Complexity – A journey (Part 1)

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 Mandelbrot, The Fractal Geometry of Nature

When I was a 5 year old kid, there used to be a popular brand of barley sold under the name ‘Purity Indian Barley’. Though barley is a very useful and widely consumed cereal and known for its health benefits, my early memory of it is dreadful because the barley soup that my mother used to feed me tasted like high quality dish washing water.

However, I do not remember ‘Purity Indian Barley’ for taste or food value. What I remember it for is a picture on the can of it – a mother holding her baby alongside a can of ‘Purity Indian Barley’. Now that was rather interesting as the picture on the can showed the same can itself that had the picture of the same mother, albeit smaller, alongside the same can. I remember that as a kid I tried hard to imagine how many such smaller yet same mothers were alongside same yet progressively smaller cans. That used to be a very interesting journey of my young mind. I had no idea of infinity, of course.

Modern mathematicians now call this ‘self-similarity’ and each such dive into another scale as ‘reiteration’. They believe that ‘self-similarity’ emerges in any naturally growing organization that has become complex – the natural growth implied here is nothing but starting with very simple rules with huge variance and billions upon billions of reiterations with slight modifications as per those rules. Though this reminds us of biological evolution, but ‘self-similarity’, that is, the part looking like the whole in any scale is pretty much everywhere and it takes just an hour’s contemplation to see it. Take a cauliflower and separate a strand of it or take a country and separate a neighborhood – I hope that you are getting my drift.

Physical reality that presents to us and the mathematical formalism that attempts to describe it seem out of sync to many of us. In fact, to most of us the mathematical abstractions seem arcane and we do not see perfect triangles, circles, squares too often neither do we see quadratic polynomials in stock markets. The Nature is replete with systems, structures and functions that are too complex to be described with simple mathematical formalism and E = mc 2 do not happen daily. However, what most of us can see with fair distinction are patterns. This appears to be our only common hope to live with complexities. Not maths. Or if maths, highly complex and difficult maths to have any appeal to us.

While the complexity and patterns rule across boundaries of ontologisms – our appreciation of reality is fragmented by studies as different as science, arts and philosophies. It may sound absurd if science has to explain poetry or arts need to make meaning of biological evolution. But as conscious beings we need to understand the whole of reality and we continue to see patterns in everything – as long as we are interacting with the complexity that confront and inspire us. And in these patterns we see ‘self-similarity’. Reality does not seem to present itself to us in a given scientific, artistic or philosophical way distinctly.

The shadow of a common recognition of the complex reality – that higher mathematics fails to deal with in physical sciences – was unveiled and placed on a sound theoretical and applied basis, at least partly, by . A maverick visionary, Mandelbrot took the pattern seeking approach to hit on a truth so profound that it put poetry and stock market, big bang and cauliflower in the same realm of human understanding. In 1975, Mandelbrot gave a treatise by the name Fractal which, as a mathematical set, described the inherent imperfection of nature and complex systems that traditional mathematical formalism failed to describe. And, most importantly, it was fun.

I do not intend to intimidate the ‘not-so-mathematically-endowed’ reader with mathematical formulae and trust me there is nothing to be intimidated by Fractals at all. As a conscious human being you are seeing and appreciating it all your life in a more fundamental way than high school maths. You are doing it because it is better and advanced description of the complexity you are immersed in, handling intuitively and trying every minute to make meaning of. Mandelbrot just gave it a name and collated it.

It started with the coastline paradox, which in brief is this. When we measure the length of a coast, say that of Great Britain, how accurate do we think we can get? It is easy to realize that the accuracy of the length of the coastline will increase with the scale of magnification of the coast details or the resolution of the map. The magnification can be infinite (theoretically), so in whichever resolution we measure the coast length it will be very difficult to assert an absolutely correct value of the coast length. This is because the coastline geometry is not Euclidean line but a fractal. Funnily, it has a dimension slightly more than 1 – making it something like a line but not quite. This is essentially the description of a complexity that defies traditional geometry or mathematics and it presents ‘self-similarity’. The smaller creeks and bays and bends of the coast are similar (pattern-wise) in whichever scale you choose to observe it. Read more .

And all these mind blowing complexities are born from rather simple rules – the complexity is manifest in numerous reiterations of these rules. It’s only advanced computer aided computations that laid bare this truth. For example take Koch’s snowflake .

Source: Wikimedia Commons

Starting with an equilateral triangle of finite perimeter, if we set a rule of removing the middle third of each side and inserting two sides of equilateral triangle having sides equal to the removed middle third to have a new perimeter and define this as the first step of possible infinite re-iterations, then we start to see a perimeter of infinite length (just like the coast) and a fractal. The fractal dimension of the perimeter of Koch’s snowflake is 1.26, so we wonder whether it is something like a line but not quite.

Our reality is that of ‘not-quiet’s. I hope you make connections with Mandelbrot’s quotes from your daily experiences. I, despite my limited faculty, see the world around me as fuzzy; shapes, processes, evolutions and renditions as nothing mathematically precise yet wonderfully complex. In this fuzzy warmth of partly feeling and partly knowing reality, I see fractals as beacons of understanding. At the heart of this fuzzy complexity lies simple elegance.

However, fractals describe more than complex shapes. They constitute a unifying framework of patterns that run sublime in human pursuits like social growth and Arts besides Nature. Centuries before invention, fractals deeply influenced African architecture, art and design. From a political perspective, Ron Eglash suggests in his book ‘ ’ that European settlers considered most African settlements to be large villages rather than cities, because instead of the Euclidean street arrangements of Europe, they found complicated fractal arrangements. “Thus fractal architecture was used as colonial proof of primitivism”.

Balla Village Archetecture - Source Mandelbrot and Frame

Balla Village Archetechture Line drawing - Source Mandelbrot and Frame

Kotoka Village Archetechture - Source Mandelbrot and Frame

The same book, as pointed out by Mandelbrot with Michael Frame as a sitting professor of mathematics in Yale University, also mentions African Art as fractally influenced. See Egyptian Column, Bamana headdress and Tuareg leatherwork for examples. I dare say that much of Indian village hierarchy and planning is fractal in nature (it’s a pity Mandelbrot did not study those)but that will be a different article in future.

Fractal Scheme in Egyptian Columns - Source Ron Eglash

Fractal design of Bamana Headdress - Source Ron Eglash

Fractal design in Tuareg Leatherwork - Source Ron Eglash

Perhaps the most fascinating fractal renditions that uncannily imitate Nature (and I strongly contend that Nature follows fractal organization) are the following:

These are computer generated fractional Brownian motion fractal art by Ken Musgrave. When he first showed these in a slide show in Yale University in 1993, many students refused to believe these as virtually created art.

Since I am much inspired by poetry, I think it will be fitting to mention that poems are fractally influenced too – often without the poet knowing it. If we choose a word as ‘root’ in a poem and plot the occurrences of the root in a box graph, we can (sometimes) see the fractal working. Pollard-Gott showed that the word ‘know’ as a root in Wallace Stevens’ poem “The Sail of Ulysses (Canto I)” follow roughly a fractal known as Cantor set.

The wonderful complexity with an elegant simple chance at its heart and proliferating with infinite reiterations is perhaps the true essence of reality. Fractals describe this profound truth showing self-similar patterns in every conscious observation. Mandelbrot called it ‘art of roughness’. I shall invite my readers to take part in an odyssey of such a journey named as Mandelbrot set. This set generates from a complex quadratic polynomial as simple as Z n+1 =Z n + c .

Fractals though are only the tip of the iceberg. They set in motion a thinking so radical as to propose a new kind of science. I shall present it the concluding part of this post.

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