Insanity #05 28 февраля 2000

Playing in the sand - story of the discovery of fractals.

                WHY scientists play in the sand



(C) Dr.Dismal



                             - Do you know what a fractal??

                              -Yes you that, lamer what?? Shtu
                               ka so funny, there's still all

                               repeated ... "


    Hmm ... But I recently pondered the question, what is at
Indeed - the fractal. Who came up with this very thing?? Say
Honestly, the first of fractals I learned a long time. Of course
foot under the table has not gone, but the school has not yet 
matured, appears 5-6 years. So, gave me a toy, interesting - 
Magic Caleidoscope. Well, a kaleidoscope, a kaleidoscope, only

Big is so beautiful - with foreign labels, snowflakes. But
it's almost to the point does not apply ...


    So, as a child curious and precocious its
shocking adults began to ask what was inside
obtained. Someone limited explanation of what was beautiful
uzorchiki, someone (probably with a technical education):)
told me that there mirrors and shards of glass and different
stuff-dryuchek form a symmetry that pleases the eye of many
children. And someone blurted out a strange word - FRACTALS. Out
in a kaleidoscope fractal snowflakes are drawn. Naturally I
not only as a curious child, but also as a curious,
wanted to reach those most fractals, because no one did not 
have them! An autopsy revealed that no fractals and does not 
exist, only mirrors yes' stuff-dryuchki "...



    The second time I've encountered with fractals, which 
defended the diploma on the occasion of receiving the 
profession. Even still remember the name thesis' Computer 
Graphics. Computed images and data visualization. " Then I had 
to casually mention fractals. It was then that I have more 
details about them. But of any rigid definition of fractals I 
have not found. Well abstract words do not count. What is 
causing chaos, symmetry of the disorder, trinkets for 
scientists-mathematicians may say even the announcer on TV. To 
me, though, and just like the general words in explaining, at 
least once to hear a good make, and simple determination and a 
little explanation of the essence. 


    Subsequently, interested in becoming, in my opinion about 
ALK'a Fractals in OBERON'e. Just recently, another article on

this topic.


    Of course you can ask - you're a kid Th
brake? Could not for a long time to find the answer to
theme?? Yes, no and maybe yes, but not - then I knew I
nearly enough to explain in general terms, it is still
someone. Well, then I'm pretty lazy person, I do not even
hide. Well broke me delve into the literature, sitting in the 
library not like I sit there, but go there to ... :)



    So that's fallen into the hands me an article that tells the
a very instructive story, which I would like you
tell.


    The most accurate in the world is in the sandbox
Technical Research Centre of the world famous company
"IBM". Here's the "play in the sand in a scientific way." Glass
bulb "sandbox" is enshrined in the terminals with a little 
motor in such a a way that makes one revolution per second. 
When you rotate the bulb grains of sand in it trickle trickles 
down a long narrow neck, which has a small, just 2 degrees 
slope. C every turn of a few grains of sand are moving on the 
inclined neck, while one iznih not drop out. Usually this

occurs once in 10 seconds.


    Thirty thousand fallen grains form a mound at 2, 5 cm
height, which is its base fills the entire area
plastic disc with a diameter of 4 cm And now comes
critical moment. Once the incident is one more grain of sand, it
knocks from the top of another, that - the third ... 
Mikrolavina begins. Part of the sand wakes up from the disc on 
the tray and ... In general, anything a special happens:) At 
this point stops mikromotorchik, because it turns off the 
signal to the computer received information from the electronic 
scales, platform which rests a disc. When mikrolavina ceases

During the computer records the weight remaining on a platform 
of sand and again includes a motor. Slide again begins to grow. 
And so on following an avalanche ...



    No, a physicist Glenn Heald, designed this intricate
device is not going to modernezirovat hourglass.
Moreover, the properties of sand themselves to have little 
interest - even though had to be thoroughly washed and calcined 
to grains of sand do not stick together in the nose bulb. 
Researcher on the basis of its "Sand model tries to confirm the 
validity of the theory, according to which the formation and 
destruction of the sand hills obey the same laws of statistics, 
and river floods, earthquake, and even fluctuations in prices 
and demand for goods in stock Exchange.



    A quarter century ago such an idea suggested has worked 
here, in "IBM", mathematician Benoit Mandelbrot. He tried to

analyze the statistics of the annual flooding of the Nile, for 
which Egyptians watch thousands of years. At first glance, the 
schedule, constructed from observational data, no particular 
trends were discovered: in some years the water in the Nile 
rises above average, more often fluctuates somewhere near the 
middle mark. Well, what of it?



    Mandelbrot, when carefully analyzed the
statistics, we found that, regardless of the time
interval - 4OOO, 4OO or 4O years - the picture is
almost identical, consisting of a series of look-alike
deviations. Where any part of the diagram, however small it 
does not was, was a kind of miniature display more

major parts.


    And here Mandelbrot remembered that back in 1904
little-known German mathematician Eric von Koch, studying the 
work his predecessors George Cantor and Karl Weierstrass

stumbled upon a description of the country made their curves 
with unusual behavior. Strangeness as their was that anyone, 
even negligibly small segment of this curve is exactly the same 
in their properties of the whole curve.



    Intrigued, Koch took a sheet of paper and tried to
draw something like that yourself. Everything was very
simple. Need to draw an equilateral triangle. Divide
each side into 3 parts. in middle part again to build
equilateral triangle, as if pulling the initial edge
out. On small edges, again divided sodium, again
construct a triangle ... And so, step by step until there is 
room for paper. That will have a figure, known in mathematics as

"Koch snowflake."


    Initially, researchers treated the "snowflake" as a sort of
mathematical trifle: fun, but do not know which of her
proc. However, after some time the British meteorologist L.
Richardson needed to measure the length of coastline
UK. At first he tried to make measurements
by curvimeter on the map. But I soon realized: with this method
measurement result depends on the scale. With more
scale on the map are visible all the smaller capes
and coves, coastline length is extended. In general, the Coast
line showed the same properties as the Koch snowflake, the 
schedule flooding of the Nile ... So, these laws have a

practical sense and deserve to do them thoroughly.


    Mandelbrot, then tried to put on the full schedule
the number of spills given height of the Nile - one foot above
average, two feet above average, etc. In this case revealed
that the graph formed by the smoothed curve, which is quite
simply described by mathematical equations.


    Disordered form of graphics Mandelbrot called "fractal"
from the English word "fractanial" - float, thereby, as it were
alluding to the fact that within this whole lot of Fractional
parts. Themselves fractional parts that make up the mysterious 
order of within the chaos, called "frikker-noise ratio.



    Dalneshee developments Mandelbrot himself as described in
his book "The Fractal Geometry of Nature," published in 1983
year: "uchenye with great surprise and delight clarifies for 
himself that many, many forms that they still had to

gidropodobnye described as similar to algae,
tangled, branched, etc., can now be studied and
described in strictly quantitative terms. "


    And indeed, now that every day, there is increasing
information about the use of fractals in the most unexpected 
areas Science that the most beautiful melodies obey fractal -

one and the same musical phrase can vary in their
countless times, gaining all the new paint. Biologists
found that the fern, cauliflower and broccoli, and many
other plants are a good examples of fractals,
because they have individual leaves resemble in shape all
plant as a whole, and besides all this can easily be described
mathematically. Doctors and Biophysics have discovered that when
simulation of the respiratory tract, cardiovascular and nervous 
systems, they also faced with fractals. Even the thought 
processes, turns out to have a certain amount of fractality. 
That's good see, for example, encephalogram - recordings of 
brain activity. 


    More examples of fractality in inanimate nature.
Brownian motion of molecules, turbulent fluid flow and
gas, all the diversity of mountain chains and the Cumulus clouds
zigzags of lightning and thunder acoustic - just some
examples of fractals in nature.


    Even the star formation occurring in the giant
interstellar clouds, too, can be described by fractal
structures. Moreover, the formulas and methods that are used in
this description, are similar to those used in the analysis
processes occurring in the laboratory. So;
through fractals, scientists were able to clearly
observe the processes of birth of protostars, which in nature
occupy at least 20-30 million years.


    ... Here, in brief the content of articles, which dates back
92-th year. And what now?? Definitely we can say that
the study of fractals in progress. Relatively recently I
talked about the transfer of television, which explains
just the use of fractals in modern computer
technologies in the field of visualization and computer 
graphics. Many of What was told, and retold to me naturally, 
but here application of fractal filter I remember specifically. 
The bottom line that a magnified image, which is naturally

divided into large square point - the essence of piklely,
applied fractal filter, after which the detail
zoom does not differ from the original,
down to the smallest detail and shades of colors.


    In general, fractals are a very interesting topic, but here,
for some reason, little information about them ...