the capacity dimension of cantor dust=1.26(approx) i know how this is arrived at mathematically...but can somebody give me an idea of its physical implication? like what does it mean physically to have fractional dimensions? for those who dunno cantor dust can be created by takin a line segment n removing the middle 3rd n continuing the process infinitely. moreover,i ve heard there r higher dimensions existing and most of them r curled up but space-time is flattened out which is why its dominant in the universe....i find this difficult to picture in my head...can sum1 pls clarify my concept of dimensions?
Mandelbrot, the discoverer of the Cantor set, spent some modest time at IBM. It was here that he learnt of problem of noise in transmission lines. Engineers at IBM knew to increase the current to make the signal stronger and the relative difference between it and the noise meant the signal 'drowned' out the noise. However, they found that no matter what they did to lessen the effect of noise they could never be rid of it. Engineers knew that transmission line noise was purely random, yet occurred in clusters. That is, for some time there would be no noise then all of a sudden there would be much noise. Mandelbrot took this as seeing periods of errorless communication followed by periods of errors. Mandelbrot investigated this and found that the more you look at the errors, the more complicated they become. In fact, Mandelbrot provided a 'error distribution' that predicted exactly a way of describing the noise however it was very confusing. As you probe deeper and deeper into periods of noise and anti-noise (excuse the term) you find more and more periods of noise and anti-noise within noise and anti-noise - strikingly familiar to fractals no? Or looking the other way, we can average out the noises to periods of say days where there is much noise and days where there is not much noise at all. The further you look at it the more SPARSE the errors become. Mandelbrot said that there is no time interval in which there is continuous noise. A big assumption because it was contrary to intuition. Within any burst of noise, no matter how short, there would always be further periods of anti-noise. But he went further. No matter what time interval (days, minutes, nanoseconds) the ratio between times of noise and anti-noise remained constant! The cantor set is a geometrical phenomenon that seems to duplicate the same error distribution for noise in a transmission line. The cantor set is fractal in nature and has an infinite number of points but has zero length. Obviously you know the mathematics behind fractal dimension. But to have a fractal dimensions means you have a degree of irregularity. For example, you pick up a rock. It looks 3D doesn't it? Well, in fact it isn't quite. You see, try measuring its circumference exactly. No matter how small you get there will be further and further twists and turns and you can never get the exact measurement. In fact, the degree of irregularity of an object corresponds to the effeciency of the object to take up space. A simple 1D line takes up no space at all. But were you twist and turn this line into, say, a Koch curve (easily done if you are a geometer) then all of a sudden, without adding any dimension to it, you have gone from 1D to 1.26D - which DOES take up space! Fractal dimension has many more applications in economics and statistics but I am not fluent in them so I leave it up to you to investigate.
hey that transmission line bit ws realy helpful;so far i had come across the use of fractals n chaos in climatic n population systems mostly. i think im beginning to understand,but i cant picture in graphically...like on an XYZ co-rdinate system how wud u depict an object with fractional dimensions? like the koch curve or cantor dust or other such fractals? ok temme if this is right: my surface of my desk looks like a 2-d object..a plane..bt if u look closer there r grooves n irregularities..so it actually isnt plane bt it really dusnt have a volume either...it is sort of like a 3-d object which is trying to reduce its 3rd dimension infinitisemally (approaching 2-d)..so its capacity dimension is a lil over 2 but not really 3....am i right or wrong?
Correct. To understand fractal dimension in terms of coordinate systems is a little complication. In fact, you don't generally use a coordinate system as such, rather a mapping of some sort. The Cantor Set is a fractal set and we can map it on some chosen interval. This way we can visualise what it looks like. But setting up and understanding how a fractal set evolves over time (or a number of iterations) requires a lot of preparation. A good place to start would be BIFURCATION THEORY. It basically lets you plot a function and then work out where there are stable points and nonstable points. You could try looking up "cobweb plots", "bifurcation theory", "strange attractors", "One and Two cycles", all of these topics build up to being able to plot fractal dimensions. In particular is the Mandelbrot set. It is governed by a simple complex recurrence relation z => z² + C. Mandelbrot discovered that he could create an image on the complex plane that would have fractal dimension. For the first time people could actually visualise in the real world what a pure fractal image looked like. In the early phases they where known as Julia Sets. But Mandelbrot used the complex recurrence relation above and ran the iterations over and over again. If you keep adding the result to itself repeatedly, then you either get an inifinte number the initial point is NOT in the Mandelbrot set. If the total loops or repeats then the initial point you used is in the Mandelbrot set. You record how many iterations it takes for points to reach either infinity or some finite repetition. The longer it takes depends on what colour you map it with on the plane. If it repeats then you make it black. That is why the Mandelbrot set looks so colourful. Just google search "Mandelbrot Set" and you will know what I mean.
yeah i ve tried these generation of fractals on graphs..especially equations regarding the popultation systems and verifying the stability of a system etc...there r sites available for computing the graphs for a certain number of iterations...n yeah i ve gone thru the bifurcation n cobweb stuff...real cool... um ok so can u tell me abt the higher dimensions curled/rolled up? i just wanna get my idea of dimensions totally clear? is there any particular book which wud be helpful?
