Mirror Magazine


Techno Page
By Harendra Alwis
E-mail: technopage_lk@yahoo.com

More about fractals
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!

The second 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.

Later, Benoit 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 in.

Fractal dimensions
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 true.

Uses of fractals
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 fractal equations.

Fractals also 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.

Antenna arrays, 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.

Motorola has 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.
Source: JRAcademy.com
Sent in by:
Buthsanda Wettewe

Attention, all gamers!
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.

MindHead, the 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).

Please contact 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 computerclub@stcmount.sch.lk or visit our website at http://www.stcmount.org/gaming.htm for more details or confirmation.

Buddhima Kurukulasuriya

J.Lo 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 Crow.

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.

Fleetwood Mac 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.

Gareth Gates 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.

Also receiving 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.

Eleven songs 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.

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.

Jennifer Lopez 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.

Another Australian 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.

The Sugababes 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 No. 7.

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.

Girls Aloud singer Cheryl Tweedy is due to appear before the South West Surrey magistrate on March 25 on assault charges.

The 19-year-old 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.

Tweedy recently 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.

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