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Author Topic: Imaginary numbers  (Read 4976 times)
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« on: October 19, 2006, 05:51:04 PM »

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,
Jules Ruis
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Posts: 209

Jules Ruis

« Reply #1 on: October 19, 2006, 06:34:00 PM »

I think you mean 'complex numbers'.

Complex numbers are used n the formular for the creation of the Mandelbrot Set and Julia sets  z'=z^2 + c

'c' is a complex number.

See: http://en.wikipedia.org/wiki/Complex_number

Jules J.C.M. Ruis
« Reply #2 on: October 19, 2006, 08:21:34 PM »

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?
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Posts: 238

« Reply #3 on: October 23, 2006, 06:54:10 PM »

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

Complex numbers have many applications, such as in fluid dynamics.  In most cases, they are not "needed," in that the same analyses could be conducted with real numbers.  However, using complex numbers often simplifies things, as one complex equation can be used instead of multiple real equations.

Complex numbers are not essential in creating fractals.  The Mandelbrot and Julia sets are defined on the complex plane, so complex numbers are part of the definition.  Working with complex numbers opens up the entire complex plane for the dynamics of the formula, instead of just the real axis, which can make things more interesting.  However, there are many other types of fractals that don't need complex numbers, such as the Lorenz butterfly and the Hilbert curve.

« Reply #4 on: October 24, 2006, 01:54:51 PM »

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
« Reply #5 on: October 24, 2006, 02:27:14 PM »

Thanks All,
When I see fractals based on real numbers I find a kind of chaos introduced in the Euclidean world, pushing it to its limits.
When I look at complex based fractals, I find a glimpse of order in a non Euclidean plane.
« Reply #6 on: November 17, 2006, 07:05:51 PM »

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. eiX 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 ejwt, 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

ejwt = Cos(wt) + jSin(wt)

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.


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Posts: 7044

formerly known as 'Trifox'

« Reply #7 on: November 26, 2006, 03:01:55 PM »

did someone mentioned that in complex numbers the square root of -1 exists ? ( 0+1i)

this makes writing formulas far more easy, beside of that, the formula z^2+c is a beautiful easy formula which generates the complex mandelbrot set ...
« Last Edit: November 26, 2006, 03:03:36 PM by Trifox » Logged


divide and conquer - iterate and rule - chaos is No random!
« Reply #8 on: November 27, 2006, 03:23:47 PM »

did someone mentioned that in complex numbers the square root of -1 exists ? ( 0+1i)

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.

« Reply #9 on: December 08, 2006, 06:34:52 AM »

Both the Mandelbrot set and Julia sets are created using iterative addition and multiplication of complex numbers. Intuitively the twisty, spiral-like appearance of these fractals can be related to a property of complex multiplication, which is that the geometric equivalent of multiplying two complex numbers together is adding together their angles from the origin. In other words: rotation. In simplified terms, it is this rotation that produces the spirals of the Mandelbrot and Julia sets.
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« Reply #10 on: March 03, 2007, 02:26:52 AM »

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/

* mcgoo.gif (2.55 KB, 223x398 - viewed 461 times.)

* inv_mbrot.gif (18.19 KB, 293x395 - viewed 476 times.)
« Last Edit: March 03, 2007, 02:30:00 AM by web2k » Logged
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« Reply #11 on: March 05, 2007, 01:51:38 AM »

John Bailey (web2k) wrote:
>    ....fractals that are plotted as the complex inverse of another fractal....
>    http://home.rochester.rr.com/jbxroads/interests/sci.fractals/Java_Fractals/

Thank you for sharing this link here.  I had almost forgotten about viewing these about 4 or 5 years ago, and thought they were very interesting.     smiley


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