Fractals have come up as an important question twice
before the invention of computers. The first time was when British
map makers discovered the problem with measuring the length of Britain's
coast. On a zoomed out map, the coastline was measured to be 5,000
miles. But by measuring the coast on more zoomed in maps, it got
to be longer, like 8,000. And by looking at really detailed maps,
the coastline was over double the original. You see, the coastline
of Britain that's on a map of the word doesn't have all the bays
and harbours. A map of Britain has more of these, but not all the
little coves and sounds. The closer they looked, the more detailed
and longer the coastline got. Little did they know that this is
a property of fractals as well!
instance of pre-computer fractals was noted by the French mathematician
Gaston Julia. He wondered what a complex polynomial function would
look like, such as the ones named after him (in the form of z^2
+ c, where 'c' is a complex constant with real and imaginary numbers).
The idea behind the formula is that you take the 'x' and 'y' coordinates
of a point, and plug them into 'z' in the form of 'x + y*i,' where
'i' is the square root of negative one, square this number, and
then add 'c', a constant. Then plug the resulting pair of real and
imaginary numbers back into 'z', run the equation again, and keep
doing that until the result is greater than some number. The number
of times you have to run the equations to get out of its orbit can
be assigned a colour and then the pixel (x,y) gets turned that colour,
unless those coordinates can't get out of their orbit, in which
case they're made black.
Mandelbrot, an employee of IBM, thought about writing a programme
with a formula such as, maybe Z*(n)^2 + c, and then running it on
one of IBM's many computers. And they eventually got some pretty
pictures. Mandelbrot was the first person to get computers to do
the many repetitive calculations to make a fractal look good. And
now you know the mathematical aspects of fractals.
The basic concept
of fractals is that they contain a large degree of self similarity.
This means that they usually contain little copies of themselves
buried deep within the original. And they also have infinite detail.
Like the coastal problem, the more you zoom in on a fractal, the
more detail (coastline) you get. And this keeps going on forever
and ever, so you could make a pretty movie of a fractal zooming
One of the unique things about fractals is that they have non-integer
dimensions. That is, while you are in the third dimension, looking
at this on a flat screen which can be considered more or less the
second dimension, fractals are in between the dimensions. Fractals
can have a dimension of 1.8, or 4.12. Although fractals may not
be in integer dimensions, they always have a smaller dimension than
what they're on. If you make a fractal by drawing lines that obey
a certain rule, like Koch's Curve, that fractal can't have a dimension
higher than the paper it's drawn on, which would be 2 (it can be
assumed that paper is as good as we're gonna get to 2 dimensional).
And how exactly
does one calculate how many dimensions a fractal has? First, you
must realise that in math, dimension means much more than whether
it's a point, or if it's flat, or if it has length, width, and height.
Dimension has been dummed up for the public so they could enjoy
their 3D movies and the like. With this in mind, we can continue.
This can be
simplified with logarithms. (Not an oxymoron). If, for instance,
you take a cube and multiply its edge length by 2, then you can
fit 8 of the old cubes into the new cube. Taking these two numbers,
you can find that log 8 / log 2 equals three. So, a cube has a dimension
of 3, which we already knew. Eight is also 2 raised to the 3rd power.
Not a coincidence.
It can be assumed
that for any fractal object (of size 'P', made up of smaller units
of size 'p'), the number of units (N) that fits into the larger
object is equal to the size ratio (P/p) raised to the power of 'd',
which is called the Hausdorff dimension. Using only line segments
that are 3 centimetres long (P), you make a simple Koch's Curve,
which is just a Star of David. 12 segments; 3 centimetres per segment.
If you take
that to the next level and use line segments which are 1 centimetre
long (p), you use 48 line segments. By cutting the length of the
line segments by one third (P = 3, p = 1, P/p = 3), the number of
line segments used (N) goes up four times (48 segments for 'p' divided
by 12 segments for 'P' equals 4). That means N = 4, P/p = 3, so
d = log 4 / log 3. Using a little help from a calculator, we find
that Koch's Curve has a dimension of 1.2618595071429. Amazing but
What good are mathematical pictures that aren't even whole
dimensions? As mentioned before, nature is full of fractal-like
stuff. Twigs on trees look like the branches which they grow on,
which look like the tree itself. It's the same thing with ferns
and so many other living things. Self-similarity is part of this
world, so fractals can make pretty good copies of it. Artists have
created very realistic looking landscapes composed of just a few
have technological applications. Antennas have always been a tricky
subject. Many antenna engineers have been reduced to using trial
and error because of the complex nature of electromagnetism. The
usual long, thin wire isn't the best way.
another approach, consist of thousands of small antennas which are
either placed randomly or regularly spaced. Fractals provide the
perfect mix between randomness and order, and with fewer components.
Parts of fractals have the disorder, while the fractal as a whole
provides the order. By bending wires into the shape of Koch's Curve,
more wire can be fit into less space, and the jagged shape also
generates electrical capacitance and inductance. This eliminates
the need for external components to tune the antenna or to broaden
its range of frequencies.
started using fractal antennas in many of its cellular phones, and
reports that they're 25% more efficient than the traditional piece
of wire. They're also cheaper to manufacture, can operate on multiple
bands, and can be put into the body of the phone. The journal fractals
showed why fractals work so well as antenna. For an antenna to work
equally well at all frequencies, it must be symmetrical around a
point and it must be self-similar, both of which fractals can provide.
Sent in by:
In the recent past, PC gaming has rapidly grown
in popularity among the younger generation. Recognising the need
to bring out the best of such gamers, the Computer Club of S. Thomas'
College, Mount Lavinia has organised an Inter School LAN Gaming
Tournament for the first time in Sri Lanka.
pioneers of multiplayer gaming in our country are assisting the
club in this endeavour. Local Area Network (LAN) gaming is new to
Sri Lanka and this is the first time that a competition of this
nature has been organised. Many leading schools from in and around
Colombo will be participating in this tournament. The event will
take place on March 29 at the state of the art MindHead gaming arcade
in Dehiwala (opposite Cargills Food City).
Buddhima on 867054 / 075-360822, Sachith on 732719 or Ishan on 723367
or fax us on 737173 (attention: Computer Club) or e-mail us at firstname.lastname@example.org
or visit our website at http://www.stcmount.org/gaming.htm for more
details or confirmation.
in the limelight
Fleetwood Mac will be releasing a new album on April 28.
Mick Fleetwood, John McVie, Stevie Nicks and Lindsey Buckingham
worked on the new album Say You Will for nearly 18 months. The new
project will be Fleetwood Mac's first studio album since Tango In
The Night released in 1987.
The first single
to be lifted from the new album will be Peace Keep! The band has
made the song available on the official website fleetwoodmac.com.
There will be guest spots by former member Christine McVie and Sheryl
The other tracks
on Say You Will will be What's The World Coming To, Murrow Turning
Over His Grave, Illumne (9/11), Throw Down, Marinade, Red River,
Come, Smile At you, Running Through The Garden, Silver Girl, Steal
Your Heart Away, Bleed To Love Her, Everybody Finds Out, Destiny
Rules, Say Goodbye, Goodbye Baby and the title track.
was formed as a blues band in 1967 before hitting it off as a pop
rock act. They had many line-up changes some leaving the band, new
comers and old hands rejoining over the years.
They last came
together a few years ago to record an unplugged album.
topped the UK singles chart this week with Spirit In The Sky. The
single is in aid of Comic Relief or Red Nose Day. It has been the
fashion to get the services of a popular act of the day to record
the charity track. This year it was the turn of Gareth Gates.
credit for this No.1 hit is the Kumars. They feature on the A-side
of the single adding various comedy asides in between the verses.
have been released as Comic Relief charity singles since 1986. Of
these, five including the current release have reached the top of
the chart. The songs and artistes being 1986 - Living Doll - Cliff
Richard and the Young Ones, 1995 -Love Can Build A Bridge - Cher/Chrissie
Hyde/Neneh Cherry/Eric Clapton, 1997-Mama/Who do You Think You Are
- Spice Girls, 1999 - When The Going Gets Tough - Boyzone, 2001
- Uptown Girl - Westlife and this year 2003 - Spirit In The Sky
- Gareth Gates.
in fact, is covering the old Norman Greenbaum song that topped the
UK chart in 1970. In 1986 Doctor And Medics also took the song to
the top of the chart.
debuted at No:2 with her new song All I Have featuring the senior
rap star LL Cool J. The song entered the chart just as her new film
Maid In Manhatten was released. Jennifer has also received a lot
of attention in the UK print media including an almost pictorial
biography from infancy todate which appeared in a recent issue of
the Hello magazine.
All I Have
is a follow up to Jenny From The Block which peaked at No.3.
LL Cool J was
in the Top 20 on his own with the track 'Paradise' which peaked
at No:18 in February.
singer hit the UK chart this week. Delta Goodrem was playing her
role in the TV Series Neighbours and the next thing we know, she
follows Kylie Minogue, Jason Donovan, Stefan Dennis, Craig McLachlan,
Natalie Imbruglia and Holly Valance to move across the seas into
the UK chart debuting at No. 3 with Born To Try. Her single is a
power ballad and even if she was not a part of Neighbours, it is
believed the song would have in any case become a hit.
checked into the chart at No.11 with Shape. It's the lowest entry
point or peaking point for the all girl act that recently won the
Brit Award for best dance act.
Making a cameo
appearance on the track is former Police frontman Sting. The Sugababes
had two No.1 songs. Their previous single Freak Like Me peaked at
A new Irish
act from Dublin entered the chart at No.18. Thrills debuted in the
top 20 with One Horse Town. The song is laced with pianos, Hammond
organs, guitars and Beach boy-like harmonies. Stil it failed to
lift the spirit of the market and gain higher ground. Maybe Thrills
can do it with the next single.
singer Cheryl Tweedy is due to appear before the South West Surrey
magistrate on March 25 on assault charges.
Tweedy got into a fight with a toilet cleaner, Sophie Amogbokpa
at a night club in Guildford, Surrey in January. Amogbokpa suffered
a swollen eye after as she claims Tweedy hit her in the face.
said she will fight to prove she is not a racist and in fact innocent
of the charges brought against her.
Mary J. Blige
will work with P.Daddy as executive producer of her new album Love
And Life. P. Daddy had produced two earlier albums for her What's
The 411 and My Life.
The new album
will feature a track produced by Dr. Dre too. She will also collaborate
with 50 cents and Method Man. The singer who couples hip hop beats
with sultry vocals is expected to release Love And Life this summer.