EARTH/BODY-SCAPES & SELF-SIMILARITY

 Self-similarity appears to be one of nature’s most favoured design principles. The curious term was coined in the 1920s by the physicist Lewis Fry Richardson who specialized in fluid turbulence, wherein big eddies have smaller eddies and so on.  Were it not for this elegant strategy we would face at least five major problems :  

1.Our bodies would perform poorly and decay due to inadequate circulation if self-similarity did not exist.   

2. If our brains and nervous systems could not benefit from self-similar networking, our IQs would be roughly on a par with fence posts. 

3. Without the self-similar villi and microvilli in our intestines, our ability to digest food would be compromised. 

4. There would be little food to eat in the first place because, in the absence of self-similarity, the Earth would be virtually devoid of vegetation, which directly or indirectly provides most of our food sources. 

     

Left : A nearly exact self-similar fractal form in the garden vegetable ‘chou Romanesco’,    Right :  self-similarity in the human lungs

5. We would be unable to breathe without the critical self-similar architecture inside our lungs.

Self-similarity is so widespread and important in nature, the arts and society that one can no longer be considered well informed without at least a conceptual understanding of this remarkable phenomenon.  The main ideas of self-similarity are easy to understand, and good examples of self-similarity are easy to find.  Think of a lone leafless tree standing dark against a grey winter sky.  There is a single trunk that rises to a region where major branches split off.  If we follow any one of these major branches, we see that they also split into smaller branches, which split into even smaller branches, and so on.  A “unit pattern” (in this case one length splitting into two or more thinner branches) is repeated on ever-smaller size scales until we reach the treetop.  With this picture in mind, we can state a simple generic definition: the fundamental principle of a self-similar structure is the repetition of a unit pattern on different size scales.  Note that the delicate veins of leaves also have self-similar branching patterns, as do virtually all root systems.  A tree is thus the epitome of self-similarity.  

    

 A leafless tree in winter and a computer generated Maple leaf

Another intriguing self-similar pattern that was published in 1907 by the natural philosopher E. E. Fournier d’Albe asks us to think of a pair of dice and focus on a side representing the number 5.  The unit pattern involves five dots, four of which are located at the corners of a square and the fifth is located at the centre.  For simplicity we will call this unit pattern a quintet.  Now imagine that each dot is really a miniature quintet, and further, that each dot of the miniature quintet is a microscopic quintet.  This constitutes three scales of the self-similar structure, but Fournier d’Albe went much further and imagined the pattern repeating without limits.  In this case, the structure we started with would be just one of five quintets of a larger scale “whole”, which would be one of five quintets of an even larger scale quintet, and so on forever.  Likewise the quintets-within-quintets hierarchy extends endlessly to ever-smaller scales. The two examples given above both involve spatial structures, but self-similarity can also occur in temporal processes.  For example, imagine the opening theme of Beethoven’s 5th symphony (da da da, DAA) being created on a synthesizer.  One could replace the “das” of the first three notes with brief versions of the whole 4-note pattern.  One could arrange it so that these compressed 4-note patterns happened fast enough that they could be mistaken for single notes.  The individual notes of the compressed 4-note patterns could, in turn, be composed of even faster (instead of smaller) 4-note patterns, and so on.  We end up with a sequence of sounds that is self-similar in time, and playing the sequence at different speeds could reveal this.

Part of a music score demonstrating self-similarity

On smaller scales within the atmosphere, snowflakes often display self-similar branching patterns or hexagonal crystals within crystals, and aggregating dust particles have growth patterns that are statistically self-similar.  On very small scales, there is chaotic Brownian motion wherein big molecules and microscopic particles are buffeted around by the smallest and fastest moving air molecules in erratic zigzag patterns that are self-similar on micrometer to nanometer scales. Lowering our gaze, we might look out on an ocean whose surface usually has a self-similar hierarchy of waves upon waves upon waves, with heights ranging from meters to millimetres.  Bodies of water also have an abundance of self-similar turbulence: whorls within whorls within whorls, as in the case of atmospheric turbulence.  The mixing of fluid masses with differing densities, temperatures, or chemical contents often occurs in a self-similar hierarchy of interpenetrating “fingers”.  The tributary and drainage systems of rivers usually exhibit branching self-similarity, and as mentioned before, coastlines are a classic example of statistical self-similarity. If we see a distant mountain range, we could count its profile as yet another example of statistical self-similarity because of the hierarchy of peak/valley morphologies on size scales ranging from kilometres to centimetres. The growth patterns and interconnectedness of cities exhibit similar phenomena on different scales, the hallmark of self-similarity. 

The following image could be on the scale of a large piece of rugged terrain photographed from an aeroplane, or the side of a mountain, or a patch of dirt on the scale of a few meters, or a magnification of the surface of a rough rock? Whichever it is, it could also easily be imagined to be any one of the others. So one could start at the large-scale view from the air and apply successive zooms down to a microscopic scale, the surface maintains self- similarity across those scales. As with most fractal structures found in nature, the self-similarity only occurs over a range of scales. In the above example, there is no self-similarity as we zoom out to see the whole planet, or zoom in to microscopic scales. 

Rough ground image 

In the world of art, self-similarity is a common theme.  It is found in the floor and wall art of medieval churches and mosques, in the drip paintings of Jackson Pollock, in African art and sculpture, in the drawings of M.C. Escher, in the tradition of Russian dolls, and in a host of other artistic forms.   

    

Sky and Water’ by M.S. Escher       Classic group of repeating Russian Dolls 

   

‘Blue Poles’ by Jackson Pollock demonstrates repetitive use of paint marks

In mathematics, where recursive operations are common, self-similarity pops up everywhere from proofs of the Pythagorean theorem to logarithmic spirals.  By far the most impressive example is the incomparable Mandelbrot set, with its infinite hierarchy of M-sets within M-sets within M-sets.  Interestingly, in Albert Einstein’s last scientific paper, written for a ‘50th anniversary of relativity’ conference in Italy, he noted that the equations of general relativity had an intrinsic self-similarity to them.  He struggled to understand the strange implications of this finding, but his time on Earth ran out before he reached an answer.   

    

Examples of computer mathematically generated Mandelbrot  fractal forms

 The Mandelbrot set that shows self-similarity is an approximate one, that is, as one looks at the object at different scales one sees structures that are recognisably similar but not exactly so. The following image shows three successive zooms and at each level a structure similar but not exactly the same as the whole Mandelbrot set can be found.

Mandlelbrot images 

Fractals usually possess what is called self-similarity across scales. That is, as one zooms in or out the geometry/image has a similar (sometimes exact) appearance. The following example is the well-known Koch snowflake curve created by starting with a single line segment and on each iteration replacing every line segment by four others shaped as follows . As one successively zooms in the resulting shape is exactly the same no matter how far in the zoom is applied. 

Koch curve

 Sometimes the self-similarity isn't visually obvious but there may be numerical or statistical measurements that are preserved across scales. One obvious measure might be the fractal dimension, in the example below of electronically metered noise the fractal dimension is constant as one zooms in. This is known as Statistical self-similarity an example of which could be a coastline 

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