## Mathematics |

*Fractals are the abstract made gloriously visible.
And now they're becoming useful shrinking images,diagnosing madness,even
finding gold. It turns out that fractals are the very stuff of the
universe*

A couple of years ago they were merely a craze, appearing
everywhere from art galleries to T-shirts. Their intricate multicoloured
whorls and spikes, at once bizarre yet strangely familiar, adorned album
covers, posters and book jackets.

These extraordinary images were fractals, patterns within patterns within
patterns, ad infinitum. They inspired artists to create images of alien
landscapes and sent mathematicians soaring into exotic flights of theory
to uncover their secrets and ramifications.

But now they are much more than just a buzzword for a bizarre abstract image.
Their unique properties are being put to use in an astonishing range of
applications.

Detectives are using them to track down stolen antiques. Geologists employ
them to predict earthquakes and reveal buried minerals. Astronomers see them
as a key to cosmic mysteries. Even psychiatrists are turning to fractals,
believing that they may cast new light on mental illnesses.

Each of these applications focuses on the fundamental property of fractals:
they are generated by repeating a simple pattern over and over again .Take
the most famous of all fractals the so-called Mandelbrot Set, named after
IBM mathematician
Benoit
Mandelbrot. Sometimes described as the most complicated object ever
discovered the Mandelbrot Set has a literally infinite amount of detail -
look closely, and you embark on an endless journey through spirals, spikes
and zigzags. Yet all this detail can he summed up in a single, simple
mathematical recipe Z_{n + 1} = Z_{n }x Z_{n} + C.

It takes some work to turn such a formula into an image, but with lateral
thinking, you can convert it into a major moneyspinner. For example, if a
small mathematical routine can be used to generate extremely detailed images,
then extremely detailed images can be turned into a small mathematical routine.

The Mandelbrot Set is an infinite object that always looks similar but never repeats

And anyone with a computer knows the advantages of that. Just one full-colour
image can easily mop up a megabyte of storage on a hard disk. Moving images
are worse: one second of video can account for 30 megabytes. Compacting all
this information has become one of the biggest challenges facing the computer
industry. Which is where fractals come to the rescue. Virtually all raw data
can be boiled down to some extent: the phrase "I sw a gd ftbll mtch n Strdy"
is still recognisable, despite being "compressed" by over 25 per cent. The
reason is that information invariably contains superfluous data which can
be chopped out without any loss of content. In text, vowels can be removed.
In images, repeated patterns can be identified.

For black and white pictures and text, data compression algorithms - that
is, sequences of mathematical operations - find such patterns and change
them into a shorthand form. Matching decompression algorithms can then be
used to turn the shorthand back into the original.

Z |
|||

The simple mathematical
equation above is responsible for the amazing, indeed neverending nature
of the Mandelbrot Set - the infinite detail of which is apparent above as
each picture zooms in on the last The equation is an
"iterative"
process - that is, an ongoing calculation in which each step is calculated
from the previous step. In the case of the Mandelbrot Set, the numbers concerned
are "complex" - ie, made up of two parts: a "real" part (eg, 1,2,3,4...),
and an "imaginary" part (a real number multiplied by the square root of -1) |
If real numbers
form a line (1,2,3, 4...) then complex numbers extend this line into a flat
plane; this plane forms the canvas on which the Mandelbrot Set is painted.
To draw a Set, take any complex number as a starting point. Run that number
through the sequence above (where Z_{n} is the complex number and
C is a constant number). If the sequence heads toward zero, colour the point
black. If its values bob about unpredictably, paint it another colour. And
if it gets larger and larger, use yet another colour. Repeat for every point
on the canvas, and the Mandelbrot Set will materialise before you. |

## Not only are cauliflowers, brains and landscapes fractal - the entire universe is too
Colour images are more problematic: there is a far wider range of colours
than there are shades of grey. As a result, colour image compression algorithms
so far developed either lose some information from the original or do not
compress very tightly. In the late 1980s, however, British mathematician
Mike
Barnsley discovered fractals to be particularly good at capturing the
essence of images.
As mathematical recipes for fractals use far less memory than the original
data, the images can be compressed by huge amounts - a factor of 100 or more.
Better still, although working out the right fractals to compress a particular
image might take a while, decompressing the image would be fast, as fractals
are just mathematical formulae - bread-and-butter
work to any computer. Thus no super-sophisticated hard- ware or software
would be needed to "unpack" a fractal image.
Fractals may soon be used to expand the world's supply of oil and minerals.
Geologists have found them to be a handy way of determining the extent to
which layers of rock are broken up - a task which may prove invaluable for
finding buried riches. At the heart of this technique is a feature again
highlighted by Benoit Mandelbrot: fractals' curious "dimensionality". The
term may be unfamiliar, but the concept isn't: a point is an object with
no dimensions, while a line has one dimension, an area has two, and a solid
has three. A fractal, in contrast, is an object occupying a strange hinterland
between these whole-number dimensions. The greater the fractal dimension,
the greater the "jaggedness" and convolution. Mandelbrot created a formula
for calculating the dimensionality of a given fractal, along with some intriguing
examples of its use. For example, the "jaggedness" of a coastline can be
summed up by its fractal dimension: for the coast of Britain, it is around
1.25, while for the much smoother edges of South Africa it is around 1.0.
For example, a high level of quartz veins in drill cores have been linked
to good quality gold seams. By counting the varying thickness of veins in
such cores, David Sanderson and colleagues at Southampton University have
been able to work out the fractal dimension of the veining - and link it
to the likely concentration of gold in the area. They discovered that the
lower the fractal dimension. the better the gold deposit.The explanation,they
suspect, is that the thicker veining implied by the low fractal dimension
has larger gaps through which water can flow. More water means a greater
chance of tiny gold particles being carried in and trapped.
If this is true, measuring the fractal dimension of core samples may help
exploration companies find other resources; the Southampton University team
is currently using the same techniques in the search for tin and tungsten.
By analysing earthquake data, scientists at the US Geological Survey have
worked out the fractal dimension for quakes in the Los Angeles area. Publishing
their results recently in Science, they concluded that there should be around
six powerful quakes of magnitude 6.6 about every 250 to 300 years. The bad
news for LA is that historical records suggest the area is behind on its
quota... Fractals are even finding applications in deep space.
Astronomers have found that galaxies tend to be gathered into clusters, and
that these are themselves arranged into superclusters. In other words,the
entire universe is fractal in nature - its fractal dimension is, apparently,about
1.2. |

Jul95