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Hello,
Until now I just understood that the two pillars of Fractals are self similarity and imaginary numbers. Hope it's right!

I remember a bit about complex numbers from school. I heard that solving many equations are far easier when imaginary numbers are used, even if the initial problem is all real, and so is the final answer.

Before understanding fractals, I wonder why imaginary numbers make things easier! Any concrete example? I'm ready to get into it!

I have many other questions...
Thanks to whomever may have the knowledge,
A.Real

Complex numbers are indeed a real number + an imaginary number (a+bi, or a+bj in electronics).

In electronics, we need complex numbers for capacitance, because no real current passes from one side of a capacitor to another, so we use "imaginary current", we use complex numbers.


The question is still : why we need complex numbers for creating fractals?
Thanks

the type of fractals that use complex numbers are the mandlebrot\julia, ultrafractal\fractint\fractal explorer sort
there are meany types though, most of which don't need complex numbers, but nearly all of which contain self simetery

In the world of electronics, i is taken - it is used for current. So electronics engineers use j. I like j. The descender makes it very conspicuous in a formula.

I have shown elsewhere on the Web that "Nature Will Not Imagine". Imaginary numbers were dreamt up by Rafael Bombelli, whose greatest supporter was Girolamo Cardano. They did not gain acceptance until Leonhard Euler produced his famous equation that links the antilogarithm of an imaginary number with the cosine and sine of the angle.

I showed that the Euler equation is a spiral in complex space:


There is another use for the term "Euler Spiral", so I had to specify that this is my Euler Spiral. On the back-plane, there is a red circle of radius 1. e^iX causes the plotting-point to rotate in the i-Y complex plane whilst X moves smoothly along the real (X) axis. The projection of a smooth rotation onto a smooth linear translation produces a spiral.

This is useful for repetitive cyclic functions, like a train of sine-waves in electronics. Here we use e^jwt, where e is 2.718281828459045 approximately - the natural exponential constant, j is the imaginary prefix in electronics, w is omega - the angular frequency - and t is time. With repetitive functions, one cycle (for example) is much like another, so it does not matter whether complex numbers are natural or man-made. The important point is that they are useful.

The shadow on the real XY wall is a cosine (green) The shadow on the i-X complex plane is a sine. The orange dot on each shows that a sine and a cosine differ only in phase - the value taken at zero degrees. Otherwise the shapes are the same.

So the electronics version of the Euler equation is


De Moivre showed that if this is so, when one multiplies two complex numbers together one must add the azimuth angles and multiply the distances from the origin (the ranges) by each other:


So one can consider the position on the plane not just in terms of the des Cartes (Descartes) Cartesian co-ordinates, but also in terms of polar co-ordinates, which up to now are easier to understand. For example, with Julia and Mandelbrot sets one simply doubles the angle and squares the range - because it is a complex squaring. Here we double the angle:



and here we square the range:



Whilst studying these things, and possibly using such graphical methods rather than arithmetic (mathematicians are always seeking the easiest way), Professor Gaston Julia added a tumble, along a fixed vector:



He then reiterated the process, and counted the number of times it was possible:



This tumble turned the crude de Moivre image:



into the first fractals like this:



Without research into complex numbers, such things would never have arrived.

Charles

That is the whole point of complex numbers. i1 (or 1i) is imaginary. Real numbers, when squared, give positive. Imaginaries squared give negative. Real together with imaginary is defined as complex. So 0+1i is complex (zero real, unit imaginary).

However, there are TWO square roots of -1. These are +i1 and -i1.

There are THREE cube roots of -1. These are -1,    0.5 + i(0.75)^0.5    and    0.5 - i(0.75)^0.5.

There are FOUR fourth roots &c. &c.

Charles

Another perspective on imaginary numbers is gained by examining fractals that are plotted as the complex inverse of another fractal.
The web pages at  http://home.rochester.rr.com/jbxroads/interests/sci.fractals/Java_Fractals/ contain several examples of this variation. For example, the Mandelbrot sets complex inverse is shown.  Each point plotted is found by taking the point that would be plotted, expressed as a complex number, and finding the point that is its inverse.
The formula for the complex inverse of a number a + ib is a/(a^2+b^2) - ib/(a^2 + b^2)

I have found this variation useful in a number of cases--especially when a novel fractal tends to behave badly.  Mr Magoo, was visually improved (somewhat) by this technique. http://home.rochester.rr.com/jbxroads/interests/sci.fractals/Java_Fractals/inv_i_ft/