** **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 5^{th}
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.

**
**

**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 ‘50^{th} 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.

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.

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