Natural Patterns
(...Phi and more...)
1. Snails and Stars. Do you like this Snail shell? I photographed
it on Yeo
Island. It is tiny, about the size of a dime and owes its spiral shape
to a simple mathematical ratio known as phi that plays a part in the animal’s
development. It is the same ratio that governs the development of galaxies
like Whirlpool Galaxy M 51, pictured below.
Phi can be found in the way rose petals grow
on a bud, the way sunflower seeds align on the head of a sunflower, and the
way that branches space themselves around the trunk of trees. It is one of
the most recognizable patterns in the natural world and exists on your own
body. Your hand and lower arm are distributed according to phi, as are the
size and arrangement of many of your facial features.
Some people go phi-wild spotting the ratio everywhere, but in fact, part of
what makes the natural world beguilingly appealing is it refusal to be reduced
to simple mathematics. The wonderful patterns in the natural world are produced
by many factors.
Chaos theory postulates Lorenz Attractors to describe how patterns emerge
that do not fall into exactly defined mathematical equations. Edward Lorenz,
who uncovered the now famous Butterfly Effect, noticed that complex dynamic
systems like the weather, when described mathematically in the right way,
reveal a kind of order which can be graphed. Even chaos can show clear patterns,
but never a pattern that exactly repeats.
Phi and Lorenz Attractors are a lattice or reticulated rubric beneath the
shapes and patterns that we see around us in nature. But into these clean
numeric predictors creep random corrugated noise. In this we have the paradox
that nature is largely made up of formula driven patterns, with predictable
parameters, muddied by factors too numerous to count. What we see and feel
in the end is similarity, never exact replication.
2. It is more than Mathematics. On a very basic level this
is obvious. There are many variations in a given species, such as a gold fish,
but we usually can distinguish the pattern “gold fish” from the
pattern “Guppy.” In fact we lump things together naturally and
give them names to help us remember the lumpings.
Nouns make up about 1/4 of the english language. Like all languages english
has patterns and there is a language of patterns, called, “pattern language.”
Those who study Pattern Language concentrate
their efforts on those arrangements of the environment that we as humans prefer
to be in. Not surprisingly the patterns we tend to like to live among are
curious blends of artificial and natural patterns. We like grids and precise
shapes such as circles and squares, but we like to have them softened so that
they appear more “natural”. If they are too precise, the environment
seems sterile. If they are too random and undefined, the environment seems
shabby or overgrown.
The natural world produces patterns over time as things grow or are moved
by the action of water and wind, or are worn away by erosion and decay. In
this dynamic interaction of factors such as surface tension in liquids, force
resistance and conduction, traveling waves, vibrations, activation-inhibition
coupled mechanisms, convection and conduction, streaming fluids, and the behaviors
of granular media, we get the diversity of patterns we see. Rain washes groves
into hillsides, trees grow from genetically predetermined codes, wind rolls
sand grains into piles we call dunes, crystals form elemental shapes, fish
swim rhythmically against the current in a stream.
Children and scientists enjoy sorting the world by its patterns.
This is how we know, by classification, by identifying the animals and plants
and stones and forces and laws. These strange squiggles of dark pixels on
a background of white ones are patterns which represent verbal patterns, which
represent physical objects, or the arrangements of those objects, or the relationships,
and so on. Our thoughts themselves are patterns based on the pattern of language
and the electrochemical routes through our brains that map meaning and can
be mapped.
3. Some interesting things about Natural patterns. Evolution
predicts that stresses on a population will result in certain individuals
within a species exploiting physiological or behavioral traits that allow
them to thrive more than others members of the same species. An example often
given is the beak size in finches. The surprise is that from a group of birds
with medium sized beaks two distinct groups tend to develop, each exploiting
a niche. Few individuals with medium sized beaks turn up in future generations.
One group can eat one kind of seed and the other can eat a different kind.
Rather than compete for the same food source, finches fall naturally to niche
occupancy, rather than niche bridging.
4. Falling into Patterns. Ian Steward, Professor of mathematics
at the University of Warwick has modeled speciation mathematically with computers
and notes that his models show that after the initial divergent split occurs
and “even though the creatures can choose mates from the other group,
the clusters tighten up and remain separate from each other. This behavior
is not fully understood even mathematically (though curiously it appears to
be related to fractal geometry), but it seems very close to what happens in
real populations.” This is how we get diversity, lots of creatures falling
into one niche or another, exploiting available resources that fit their unique
abilities and behaviors. The interesting thing is that this mirrors patterns
that result from non-living interactions such as wind, water, and crystallization.
We are intrigued with the smile lines
in older faces, warmed by their weathered countenance, drawn to their expressive
and knowing eyes.
Take for example the sand dunes I mentioned earlier. If a uniform wind blows
across a uniform flat piece of desert sand, the result is parallel ridges
of sand, or dunes. What the wind does to the sand is called symmetry-breaking
and this principle underlies the formation of these kinds of patterns in sand,
water, and other inanimate structures. The symmetry of the flat featureless
sand is broken by the wind forming it into dunes.
Another example is when a flat dish of water is heated uniformly from below.
At a certain temperature the uniformity of the surface of the water is broken
into a pattern of convection cells. These cells are usually hexagonal with
some pentagons as well and they are roughly the same size. 
The
flat symmetry of the surface of the water is broken into a symmetrical lattice
of hexagonal cells. Speciation follows this same pattern. A relatively uniform
group, when stressed, falls into two or more groups of relatively uniform
individuals. Those groups continue in a certain symmetry until a stress further
divides them.
5. Spice of Life. Within any group of more or less similar
individuals there will be variation. Such variation is tolerated to a degree,
but when stressed, the group finds new patterns of symmetry based on these
variations. The new groups tend not to interact, even though they were formerly
united. Isn’t it interesting that symmetry-breaking produces new symmetry?
Like the finches, we humans generally pick for mates those most similar to
ourselves. But unlike finches, this tendency is not as strong.
One thing that is consistent is our choice for members of the opposite sex
who have very symmetrical bodies. When it comes to voting on what we consider
to be the 
most
beautiful faces, for example, symmetry is a large factor in our decision.
There is a lot of speculation about why this is. Symmetrical faces in people
might predict health. Disease, parasites, and malnutrition all break symmetry
signaling a compromised individual. Young faces are more symmetrical than
older ones and young people are less tolerant of people who vary from the
mean.
Some young people are able to get past their built-in wiring and see beyond
physical abnormalities. Perhaps they have a grandparent they love who has
failing health, perhaps they notice the bravery of a challenged child, or
the steadfast love of an friend with a chronic illness. The truth is that
beyond the level of biological attraction there is a different attraction
to faces that have character. 
We
are intrigued with the smile lines in older faces, warmed by their weathered
countenance, drawn to their expressive and knowing eyes.
Those of us who have experienced this shift in values, this appreciation of
symmetry breaking and the formation of new symmetries, or richer patterns,
or more subtle beauties, start to wonder what it is that we are drawn to.
As our palette matures we can tolerate the bitter olives, the pungent fungi,
the aged cheese, we never could have stomached before. With time the neat
rows of vegetables and carefully aligned flower beds are allowed to ramble
a bit. We notice the beauty in falling down barns, fallow fields, and crooked
fences. Walking in the woods we become aware of the lovely soldier trees wounded
in the battle for survival. The Arbutus with one living strip of bark winding
up the otherwise dead tree to support a branch of shinny leaves. Even the
ones that have lost the fight impress us with their character. It doesn’t
matter to us that the dead tree weathered by the storm is no longer of any
use, we love its bleached grain and twisted skeleton.
still, in the stream.
6.
Finding the Balance. This attraction to things with no obvious practical
value is mysterious. One time while walking home late at night across a university
campus I was struck by the beauty of wet Ivy leaves growing on the side of
a tree. Ivy is not a food source, neither is it useful in any way known to
me, but it was very beautiful on that cool evening. The same is true of the
flaky shelves of shale I observed one time from the window of my car. Yet
I pulled over and got out just to look at them. The apprehension of beauty
in the patterns we observe is a further puzzlement.
Take a hand full of stones or chestnuts or other natural objects and set them
down on a piece of wood. Move them around and observe that some arrangements
are more pleasing than others. Why should this be? They are just stones on
a piece of wood.
Another interesting question has to do with what is really
at the heart of these patterns. Authors such as Philip Ball who wrote “The
Self-Made Tapestry: Pattern Formation in Nature,” suggest that the
formation of patterns reveals a tendency for the natural world to produce
organization, and even order. Self-ordering systems and self-made patterns
might explain the factors that produced life itself, the most complex and
varied set of natural patterns we know of. Is it possible that patterns attract
us because we know, at some level, that self-ordering is occurring, that life
is emerging, that against the leveling action of the elements, order is happening?
I have noticed that what I like most is when there is a balance between order
and disorder. The dead snag standing in the wind, the deserted temple with
only one wall still standing, the concrete wall stained with lichen and moss,
all exist between the two extremes. Is this an example of the middle path?
Perhaps. We must not deny that a rain forest, steamy with life is beautiful,
or a prairie, flat and expansive, has appeal. How much more so though when
we find a ruin amidst the vines, or a stone monolith on the plain? 
One thing is clear. Once you start to notice wabi sabi beauty in the interaction
of patterns around you, you will experience a wealth of emotions ephemeral
and difficult to name. Some people describe the sight of leaves strewn under
an autumn tree as depressing and messy. But if we look past the loss of green,
the fading glory of summer, and accept the value of humus and the beauty of
those branches and bows so well hidden during the growing season, then the
feeling that emerges is a sort of pleasurable acceptance, a somber longing
for more patterns, for more awareness and clarity. This is the greatest pattern
of all, the spiral progression up the ladder of maturity like a spark of recognition
riding the DNA chain to its full expression.
© Richard
R. Powell - November 2004
Notes:
Ian Steward’s discussion of speciation is published in New
Scientist Vol. 11 October 2003 pg 35.
For a more in-depth discussion of Phi and Natural Beauty, see chapter ten
of Wabi Sabi Simple.
© All Photo's by the author, except Whirlpool Galaxy M 51 which
is available from The Hubble Heritage Project: http://heritage.stsci.edu/gallery/gallery.html
