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How were fractals named?
Why are fractals appealing?
Generating fractals yourself
fractal is a geometric construction that exhibits self similarity
across all scales. In mathematical terms it does not matter how
far you zoom into a fractal image there will be an equivalent level
of detail at every stage. Fractals are also found in the natural
world. Many phenomenon which are difficult to analyse using traditional
methods can be modelled using fractals eg clouds, trees, landscapes,
mathematical fractals contain an infinite level of detail with 'real
world' fractals this is not the case, for example you can magnify
the view of a coastline and at first there will be major estuaries
and larger bays. Further magnification will reveal smaller estuaries
and bays then down to coves and inlets and finanlly individual rocks
and pebbles. At some point further magnification causes the previously
observed fractal structure to breakdown. Fractals in the natural
world are subject to this sort of limit. However because they have
fractal structure over a range of scales fractal techniques can
be used to successfully model them.
also effect the computer generation of fractals. Even though the
mathematical processes used as the basis of calculation are capable
of being magnified forever without a loss of detail, a calculator,
computer or (heaven forbid!) pencil and paper are not. When an image
is calculated on screen it is limited to the resolution of the monitor
therefore this process only provides a representation of the fractal.
You could magnify and recalculate so that a small region of the
previous image is now displayed on the monitor as the new image.
At least this will show levels of detail which were not previously
apparent but nevertheless any particular render will only be a representation
of the fractal at a particular scale.
feature of fractals related to self-similarity is 'fractal dimension'.
Going back to coastlines, how long is the coastline of say, Britain?
Unfortunately, there is no straightforward answer as it depends
on the size of the ruler used to measure it. The smaller the ruler
the more details can be measured and the larger the answer will
be. Measurements for different ruler sizes can be used to calculate
the 'fractal dimension'. A coastline will have a fractal dimension
between 1 and 2, compared to the topological dimension of 1 that
lines have. All fractals will have a fractal dimension greater than
the topological dimension.
were fractals named?
term 'fractal' was coined by the Benoit B. Mandelbrot in 1975 to
collectively describe sets that although previously studied by other
mathematicians had, at that time, remained unnamed. Effectively
this was the birth of Fractal Geometry. Mandelbrot derived the term
'fractal' from the Latin for 'irregular'. Mandelbrot later asserted
that fractals deserve to be called 'geometrically chaotic'.
are fractals appealing?
is no entirely straightforward answer. Fractal geometry is very
much at odds with the Euclidean geometry which we see around us
in our man made environment; straight lines, clean curves, smooth
surfaces, perfect parallels and perpendiculars. These are not the
stuff of nature, they have no aesthetic 'depth'. No one wants to
see a park or garden consisting of cones and triangles, it might
be argued that this is because we enjoy plants and trees as an 'break'
from the simple, tidy geometry of everyday life, but does it really
go further than this? A smooth polished surface has a certain attraction
but the attraction is much greater if it is smooth polished marble
or beautifully grained wood. Not that it is suggested that complete
disorder has any appeal, untidiness appeals to no one. As with marble
or wood grain the beauty of fractals lies in their 'order within
disorder'. For my own part the most attractive fractal images are
those where the longer you look the more you see, what starts off
just looking like an irregular shape will have form which will be
repeated at smaller scales, not necessarily identical but similar.
But, as in all matters of artistic taste, individual opinions will
further appreciation of fractals comes from generating them yourself.
It is possible writing your own programs but a much easier way is
using either a freeware or shareware program. In many cases using
a such a program means you can avoid all the difficult maths stuff
if you wish and just concentrate on producing great images. But
remember, as with all things that are worthwhile, it does take a
bit of practice and perseverence to produce good results. Further
details of such programs can be found on the Links