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BLoCCo teMatICo a
Tecniche di programmazione
R
ecursion is used to create the famous geometric figures known as
fractals
. These are geometric objects characterized by the infinite
repetition of one motif on an ever smaller scale. In fact, if we observe
a fractal figure in one of its details through a magnifying glass, we find
that such detail does not change in aspect; the structure of the detail is similar to
that of the whole figure. This property takes the name of
self-similarity
or
self-resemblance
: one part of the object is similar to the whole.
The process that engenders a fractal figure can be repeated indefinitely, each
time a figure different from the previous one is generated. Because the process
is infinite, we will never know what the final geometric figure might be and so,
we must limit ourselves to an approximation, meaning that we will end the
generative process after a finite number of times.
One of the main characteristics of the fractals is that, even though they derive
from very simple algorithms, they display complex forms resembling what
we see in nature. But fractals are an invention of man, and the fact that they
resemble what we see in nature may simply mean that nature prefers to use
simple algorithms to create shapes and forms that are symmetrical, complex
and full of color.
Here are some examples of fractal structures found in nature.
FraCtaLs
Fern
In a fern, one part is similar to the whole fern (recursive property).
Each part is a small copy of the whole leaf. It is possible to continue
indefinitely and achieve ever smaller parts, but in the end each of
these parts will still have the same structure of the whole leaf. A fern
possesses the property of
self-similarity
.
Snowflake
A snowflake is made up of six smaller snowflakes: five
on the sides and one at the center. Each of these is in
turn made of six smaller flakes with the same structure
of the bigger snowflake. And so on. Such a structure
is known as typically recursive. A snowflake, therefore,
also possesses the property of
self-similarity
.
GLOSSARY
Helge von Koch:
(1870 – 1924); Swedish
mathematician known for the description of
the Koch curve.
Fractal:
geometric object characterized by
the infinite repetition of one motif on an ever
smaller scale.
Mandelbrot:
(1924–2010); Polish
mathematician considered the founder of
fractal geometry.
Self-resemblance:
see self-similarity.
Self-similarity:
property on the basis of
which an object is similar to one of its parts.