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Sunday, March 2, 2008

Patterns that Enchant vs. Patterns that Matter

A fractal is a special kind of crystal. Before I go into this, I
will attempt a very high level definition of the concept crystal.

Crystals are patterned aggregates (accumulations) of similar parts.
These parts come together over time and assume structures that are
notable principally because they assume shapes that are repetitive at
any scale. Crystal parts do not necessarily look like the final
crystal, in fact it is rare that they do. If the crystal's shape is
a cone, when you hit it with a hammer the shards that result will
most probably not be cone shaped. Think of a crystal as a hollow box
and then think of the shape a brick or tile would have to have so
that if it was repeated over and over it would form layer after layer
of concentric skins or shells... and that will be one of the possible
part shapes for a crystal of that shape. This is true while the
crystal is growing, at each stage it is a cone, a cone that grows
into a bigger cone. Of course the parts themselves can be made of
other smaller sub-parts, (atoms, molecules, etc.) that are not
themselves shaped like the crystal or its parts. The shape a crystal
takes is dependent on inherent energetic and structural properties of
its component parts and the conditions present during each stage of
its development or growth. Another way to say this is that under
different environmental conditions the same sub-parts will self
organize into different shapes or crystals. This is why carbon finds
so many crystalline forms in nature (graphite, diamonds, and the far
more exotic Bucky Balls, and Bucky Tubes).

In simplest terms, if the crystal's parts are shaped like cubes...
the crystal that grows of these parts will naturally tend towards
larger and larger cubes. Pyramid parts will most likely be some
slightly imperfect three dimensional rhomboid shape which when
allowed to aggregate slowly in large numbers yield larger and larger
pyramids made up of concentric skins like the layers of an onion or a
self-stacking Russian doll. Of course you can build odd shaped
things from a cube... but the shape that will occur most often (cause
it takes less energy for these parts to fall into this shape) will be
a shape the mimics the shape of the original part. But sometimes the
conditions in the environment will influence which types of
aggregates will form... with which resulting shape. The graphite in
your pencil, a diamond, coal, and exotic materials like the carbon
fibers used in tennis rackets and fighter jet wings, and the even
weirder Bucky Balls or Bucky Tubes (little molecules shaped like
geodesic spheres or tubes and named after the systems scientist and
inventor Buckminster Fuller... all of these things are crystals made
of the carbon atom under various common and uncommon (extreme)
environmental conditions. Are carbon atoms shaped like cubes or
balls or tubes or strands? No. But a small variability in the
number (and orbit) of the negatively charged electrons that orbit a
carbon nucleus (positively charged) is all that is necessary to cause
accumulations of similar atoms into one of these many crystal
shapes. Be cautioned here, the conditions surrounding a bunch of
potentially crystal building parts plays a huge role over the growth
of a crystal. Under the wrong conditions, a crystal will never form.

Unlike the strict chemical or mineral shiny things that generally
come to mind, the term crystal, at a more abstract level, can
describe a looser category of patterned accumulations, even
behaviors. Mountains, for instance can be thought of as a type of
crystal. At the scale of a mountain, it really doesn't matter
whether it is made of silica rocks or basaltic rocks, if you get a
mountain's worth of rocks and stack them up, the resulting shape will
always be mountain like, a squashed pyramid or cone shaped thing.
Given earth amounts of gravity, a mountain is the shape a whole lot
of rocks takes. Now if you had the same mountain amount of mass but
instead of rocks the mass consisted of large slippery-smooth ball
bearings, it would be improbable that a mountain would ever result.
Try and stack them and their slipperiness makes them act more like a
fluid, flowing down and outward until there is nothing but a sea or
lake of little balls. It is some combination of the downward force
of gravity and the grittiness and irregularity of rock-like things
that makes mountains or mountain shaped things happen. This is why I
think it is funny when people are in such awe of the Egyptian
pyramids. Sure it took a lot of work to cut the rock from its
source, ship and stack it, but the shape is exactly the only shape
you could stack that much sandstone (the Pharos tried other shapes...
they all failed, crumbled, and don't exist except as much flatter and
less defined pyramids or mounds). Now if that same quantity of stone
had been formed into a sphere, then even I might be inclined to
believe in ancient astronauts. Planets are a spherical crystal that
results when gravity at huge planetary scales absolutely overcomes
the stickiness of any possible chemical or aggregate parts. When a
system is so large that mountains are the parts, even a mountain like
Everest becomes inconsequential to the dominant crystal, the sphere.

With an even looser definition of the concept crystal, one can
include the behavior of atmospheres... of weather. Once again we
start with a whole mess of self similar parts, in this case, the
molecules that make up air and sea. As these parts interact within
the context of energy applied by the sun, patterns arise at various
scales. Wind, clouds, fog, storms, thermals, precipitation, tornados,
hurricanes, jet streams, ocean currents, etc. Unlike the spacial
patterns of mineral crystals and mountains, most weather systems also
show their patterns as temporal cycles repeating over time, even if
these patterns sometimes also have a spatial component (often a
spiral or hexagonal rod).

Building bricks can also be thought of as crystals or crystal causing
parts. If you don't have any cement to glue them together you end up
with certain shaped buildings... add some grout and you can end up
with other shapes that generally mimic the original block or brick.

What makes this true, that the aggregate crystal retains the shape of
its component parts, is that the crystal is the most stable
relationships for these parts to find themselves in... it would take
more energy to get them to assume non-crystal like shapes... the
crystal is the natural or least-energy arrangement. Statistically,
the overall structural and energy state of the parts, insures that
the crystal will most often result. At the sweet spot (in scale),
where the dominant force or condition that builds parts into crystal
shapes neither overwhelms the structural integrity of the parts nor
is itself overwhelmed by another force or condition, crystals will
continue to grow and maintain their own integrity. At this in-system
scale, the more parts, the more likely the result will be crystals,
lots of crystals, and crystals that approach the perfect shape
defined by that particular crystal's growth pattern.

A more accurate categorization of crystals would be by how
autonomously they come into existence (morphological vs.
developmental). Developmental crystals would be crystals in the
traditional sense... those that develop purely as a result of the
forces innate to the crystal materials themselves. On the other end
of the scale are morphological crystals; those that have crystal like
shapes, but which got that way because of external processes or
machinery. A snails shell is morphological. Quartz is developmental.
DNA is morphological. Salt is developmental.

Now that we have an understanding of crystals, their shape and why
they are likely to have that shape at all stages of their growth...
we begin to see that the really interesting thing about crystals is
that they aren't very interesting at all. Here's what I mean. A
crystal of any size can be described by a very short statement... x
number of bricks... where the brick is like y... and is arranged in
pattern z. Doesn't matter if the diamond is microscopic or the size
of a house... the same little sentence or equation will describe
both. And the description in both cases will be exact and complete.

Compare a crystal to the text in this essay. The arrangement of
characters, words, sentences, paragraphs, etc. is not repeating in
any kind of readily apparent pattern. Even if I was to write a
condensed version of this essay... it would not be an exact
description of the original. This essay is not pretty or elegant or
shiny... will not illicit any kind of emotional outpouring... but
it's information content is much more rich than the biggest diamond.
Unfortunately... we humans are drawn to the kinds of simplicity
represented by crystals... like crows to shiny objects. I suspect we
are attracted to the profound simplicity of crystals precisely
because the rest of nature is in fact so profoundly complex. Perhaps
the attraction to this kind of simplicity reveals a deeper mechanism
of pattern finding at work deep within our big brains. But the truly
interesting (information rich) things around us describe patterns
much too complex to resonate emotionally with the same simple pattern
matching parts of brains. As a consequence, we have a tendency to
value the simple and ignore the complex and truly amazing. Why?
Because the emotional part of our brains... the old parts that have
not really changed sense we crawled out of the ooze... are very
simple... can only see simple patterns. And the very new complex
parts of our brains... the parts that can write and read things as
complex as language and its content... these parts do not directly
control emotions. Sad. The other reason we struggle with complex
patterns is because they are so much harder to see against a
background of true noise or randomness. Arrange text by blindly
picking from a box of letter blocks and you will approach the look of
actual text. Same goes for the arrangement of base pairs that make
up the ladder at the middle of DNA's famed double helix. It is
taking researchers around the world millions of man hours just to
begin to tease pattern and meaning from the seaming randomness in the
long arrangement (three billion pairs) of the four little proteins
adenine, thymine, guanine, and cytosine in the human DNA within each
of our cells. The exact why behind the truth that is the similarity
between any particular statement written in any particular language
and a random arrangement of the symbols of that language of the same
length is beyond the scope of this essay but it is worth looking into
(I will examine this critical point in a later post to this blog).

Fractals are a family of complex crystals. Fractals result in
organic shapes that are more like living things. Examples are the
branching of trees, the shape of a cauliflower bunch, a rocky
coastline. One of the characteristics of a fractal is that the
repetitive patterns are the same at any scale. If you look at the
edge of the cauliflower you have cut in half... then zoom in to one
small region of that edge... you will find that the pattern is
identical in both large and small scales. Of course this is the same
with regular crystals. But the organic shapes in fractals seem too
complex to repeat, so we are surprised.

Though they look more complex than grains of table salt, fractals
share the same descriptive characteristic... a fractal of any size
can be described by one simple sentence (equation). Size does not
change the small number of bits it takes to describe a fractal.

Fractals illicit an even stronger shiny object response in us (than
their more common rock crystals brothers and sisters). The slight
added complexity surprises us... we are amazed that living plants and
eroded landscapes could be described by simple mathematic
equations... so fractals do for nerds what a quartz crystal hanging
in the window does for new-age healers and unicorn loving romantics.

Humans are humans. We were emotion processors long before we were
reason and logic processors. We carry all of this emotional
processing machinery forward, tucked under our much much newer
ability to reason. Not only that, the emotion processing part of our
brain is in charge, at base, in the center of everything we think and
do. Because of this we seem to be most reverent to the things that
deserve it the least. It's the classic old mind / new world
mismatch. This little essay... the meaning encoded within it... is
millions maybe billions of times more complex than any fractal. But
we are in general stupid to this kind of beauty. Sad.

Technically, all solids are crystals (or in the case of a mixed up
rock like granite, all solids are made of mixed up areas of different
kinds of crystals).

Except when they aren't. Some rock-hard things that don't move or
squish or pour or compress... are not technically solids. Glass is a
liquid! Even in its

It is just so viscous (thick) that you can't see it flow. A old
window in an old house will be thicker at the bottom than at the
top... over the years, it has flowed very very slowly as it reacts to
the gravity of the earth. Crystals resist flow because their
molecules don't slip past each other (as in a liquid)... they are
locked into each other more like puzzle pieces (each atom shares
electrons with other atoms in the matrix). Some rocks that form in
volcanic eruptions (obsidian... the stuff of Indian arrow heads) are
like black glass (though not as pure) and like glass this stuff is
technically considered to be a liquid not a crystal.

Most substances are liquids when they are hot... and form crystals as
they cool beyond a certain temperature (their freezing point).
Remember that temperature is just a measure of how violently the
molecules in a thing are jerking around. The atoms or molecules of
hot things won't settle down long enough to freeze into crystal
lattices. Most substances get smaller when they freeze, dropping
down into a crystal lattice means getting tighter with your molecular
or atomic neighbors. Except water! The crystal lattice that water
forms, ice, takes up more space than it did as liquid. Which by the
way is a really good thing... it is why ice forms on the tops of
lakes (it is bigger... less dense... then the water it used to be...
thus lighter... so it floats on top). That keeps fish and everything
else alive by insulating the rest of the water from the much colder
air above it. Water is freaky in all the right ways.

Anyway...

Substances that are much more much more complex than rocks, metal,
ice, and glass... like the stuff life makes... bones, cell walls,
muscle fibers, wood, sea shells, brains, chlorophyll, proteins,
hormones... these things are nano-constructed by living things one
molecule at a time by molecular machines (RNA, proteins and the like)
acting from the outside. Compare this to things that self construct
(by virtue of their own internal properties)... these tend to form
crystals. Things that are constructed from the outside by bigger
stronger more energetic machines... can force parts into less energy-
perfect shapes... can force unnatural combinations... mashed-up into
any manor of crazy information-rich constructs that have complex
behaviors and interactions with the stuff around them.

Crystals don't do much! Why? Because all of their parts are already
in the most stable relationship they can be to the parts next to
them. That is what things acting on their own power do. That is the
only thing things acting on their own power can do. But complex
systems are complex because they have been forced into un-natural,
less stable, and as a result, more reactive orientations.

This is an important concept... so pay attention here (your guru is
saying things that few people really understand the importance of)...

Life has learned to build (or facilitate the natural building) of
crystals where stability is needed... and to build non-crystals where
reactions are appropriate.

Look at a snail. The shell is crystal like... repetitive, simple,
geometric... stable and non-reactive (dead)... that is the point...
the snail uses its shell for protection and structure. Inside the
shell, you will find all of the non-crystal stuff... information
rich... built to react all day long in very complex chemical dances
that extract energy from the surroundings, store that energy in
awkward configurations that can be relied upon to do it all over
again the next day, and get rid of waste products. The most
interesting chemical in the body... DNA... has properties of both!
It's structure (the famous double helix) is highly crystalline... but
the rungs that are strung between the winding bands... where all of
the genetic information is written... is highly reactive. Very
cool! Just think how fucked things would get in a hurry if the
crystal part didn't absolutely protect the genetic information or if
the information part couldn't easily read it's pattern out to RNA (so
it could make all of the stuff that makes us... well... alive).

Crystals are repetitive, information poor, stable, and ultimately
dead. Life stuff is complex and non-repetitive. And this is why
scientists shy away from new age (and old age) theories that try to
fit complex biology, personality, or social behavior into simple
shape and crystal-based theories like hexagons or pentagons or twelve
sided horoscopes. Nice try old brain! But things as simple as
crystals are never going to be good abstractions of the truly amazing
complexity of life and culture and behavior (and the smell of sex).

Oh, I almost forgot. The difference between fractals and regular
crystals is really kind of simple. Of course simple little
differences can make all of the difference in the world (our DNA is
less than 2 percent different than that of a Chimpanzee... less than
twenty percent different than that of a tree). The statements that
describe both crystals and fractals are not really equations like 2
+4=6 which can be computed in one pass. Crystal equations are
algorithmic... meaning they must be calculated over and over again
(so the thing can grow). What sets fractals apart as a class of
crystals is that part of the equation or algorithm demands and uses
the answer from the previous calculation of the same equation. There
is a feed-back loop in the calculation of the thing... and because of
this... a fractal is technically not solvable... goes on forever.
Actually that is the same with most crystals... they just grow and
grow and grow as long as the building blocks are available and the
environment is suitable for the necessary reactions. Of course there
are always exceptions, some crystals are self limiting... they grow
like the surface of of a ball and end up running into themselves on
the other side. But with fractals... this inward facing,
masturbatorial, feedback-loop will keep building detail at smaller
and smaller scales... so it never runs out of space... so if you
start your fractal from only one point you will never get a full
fractal built... just a more and more detailed section of one.
Nature solves this problem by calculating every part of a fractal in
parallel... starting the calculation (the reaction) from many many
points at the same time (in nature everything is the computer and the
computer is everything). In a man-made computer we are forced to do
one thing at a time... one thing after the other... in one long train
of calculations... so we have to insert artificial limits into the
algorithm... you know... do this inward loopy thing only 100 times
then move on to the next large-scale section... repeat.

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