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Fractals Challenge
Logo Procedures: Ionut
Trif
Computer Logo Graphics: Cosmin
Tanase
Text: Otilia Stanciu and
Gabriela Brancovici
Students of "Duiliu Zamfirescu"
School
Focsani, Romania
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| Fractals are a class of complex
geometric shapes that commonly exhibit the property of self-similarity.
They are capable of describing the many irregularly shaped objects or spatially
non-uniform phenomena in nature that cannot be accommodated by the components
of Euclidean geometry.
These complex shapes were named fractals
be Benoit B. Mandelbrot, meaning fragmented or broken. This new concept
has had an important impact not only on Mathematics but also on such diverse
fields as Chemistry, Physics, Biology or Modern Arts because this phenomenon
can be seen in such objects in nature as snowflakes or tree barks. Fractal
algorithms have made possible to generate lifelike images of complicated
natural objects, such as intricate branch systems of trees. Fractal geometry
with its concepts has made possible to study distribution galaxy cluster
throughout the universe or has contributed much to computer graphics, geomorphology,
human physiology, economics, and linguistics. Specifically characteristic
"landscapes" revealed by microscopic views of surfaces in connection with
Brownian movement, vascular networks, and the shapes of polymer molecules
are all related to fractals.
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| Pathological curves
A pathological curve is a fractal
too because it lacks certain properties of continuous curves. The curve
may enclose a finite area but be infinite in length or its curvature cannot
be definable. Some of these curves may be regarded as the limit of a series
of geometrical constructions. Their lengths or the areas they enclose appear
to be the limits of sequences of numbers and the scientists decide to name
them paradoxes rather than optical illusions.
For example Von Koch's Snowflake
Curve is the figure obtained by trisecting each side of an equilateral
triangle and replacing the center segment by two sides of a smaller equilateral
triangle projecting outward, then treating the resulting figure the same
way, and so on.
The first three stages of this process
are shown in Snowflake or Koch's Curve:
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Another kind of fractals is the
space-filling curves. When it is completed the curve will pass through
every point in the square. In fact, by similar reasoning, the curve can
be made to fill completely an entire cube.
Giuseppe Peano (1858-1932) was an
Italian mathematician that studied a curve that today is called by his
name. He demonstrated that any curve, which passes through all points of
the unit square in two dimensions is an analogs space-filling curve.
So, the Peano’ Curve contains every
point interior to a square, and it describes a closed path. As the process
of forming the curve is continued indefinitely, the length of the curve
approaches infinity. This is a famous paradox that preoccupied scientists
long time ago.
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