I believe that I can clarify the first part of this statement (and Osher Doctorow can correct me if it's wrong): What is being said here is that the Mset can be generated as easily from real numbers as from complex. E.g., in the form of an frm: C4 { x=real(pixel), y=imag(pixel) a=b=0: a1 = a^2-b^2 b1 = 2*a*b a=a1+x, b=b1+y z = sqrt(a^2 + b^2) z < 4 } Recall that y=imag(pixel) is a real number (u da man Gerald) and so imaginary numbers don't come into play in this formula. One virtue of this approach is that it leads to interesting generalizations. A somewhat trivial example is offered in the formula below; C4G {; p1 is (plus or minus) 1 or i x=real(pixel), y=imag(pixel)*p1 a=b=0: a1 = a^2-p1^2*b^2 b1 = 2*a*b a=a1+x, b=b1+y z = sqrt(a^2 + b^2) z < 4 } -- On Fri, 31 Jan 2003 02:07:12 Guy Marson wrote:

At 12:20 29/01/03 -0800, you wrote:

...

From Osher Doctorow (user in mdoctorow@comcast.net network).

Let's examine the expression z^2 + c (where z^2 means z-squared) for z, c complex (c constant), that is to say z = x + iy, c = c1 + ic2, i = square root of -1. For the Mandelbrot set (M for short), we want /c/ < 2 where /c/ = sqrt(c1^2 + c2^2), sqrt( ) meaning square root of whatever's inside ( ). We'll rewrite this in terms of x and y as follows.

1) z^2 = (x + iy)(x + iy) = x^2 + iyx + ixy + (iy)^2 = [(x^2) - y^2] + 2ixy

2) z^2 + c = [x^2 - y^2 + c1] + i[2xy + c2]

In Rare Event Theory (RET), the expression u + iv changes to real variables as follows.

3) u + iv --> u - v

where the arrow indicates the result of the change or transformation. So from (2) bove, z^ + c changes as follows.

4) z^2 + c --> [x^2 - y^2 + c1] - [2xy + c2]

Now we get a strange result. J. C. Sprott of Dept. Physics U. Wisconsin Madison, in "Mandelbrot Set Chaos," July 3, 1997, http://sprott.physics.wisc.edu/chaos/manchaos.htm computes the Mandelbrot set orbits in the c1-c2 plane, which he calls the a-b plane, respectively by the following where xn+1 means x with subscript n+1, etc.

5) xn+1 = xn^2 - yn^2 + a 6) yn+1 = 2xnyn + b

These are precisely the right hand side expressions in the respective brackets in (4) with a = c1 and b = c2.

Now, p1 is optimized at p1(x, y) = 1, where p1(x, y) = 1 + y - x. How do we write the right hand side of (4) in terms of p1? The right hand side of (4) has the form y - x for some y, x, so it is p1 - 1 for some p1. In fact, we have:

7) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1) - 1

Let's incorporate the -1 into c1 and call the result c1', so we can write (7) as follows.

8) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1')

Now we can set p1 = 1, which says the following.

9) x^2 - y^2 -(2xy + c2) + (c1' + 1) = 1

and therefore we have the following.

10) x^2 - y^2 -(2xy + c2) + c1 = 0

or, expanding:

11) x^2 - y^2 - 2xy + (c1 - c2) = 0

Except for possibly degenerate cases, it is proved in analytic geometry that this is part of a hyperbola (by proper rotation of axes and translations).

We therefore can study the Mandelbrot set M as a part of a hyperbola in Rare Event Theory, which simplifies it considerably. We have c1^2 + c2^2 < = 4 from before.

Osher Doctorow

ooooh, big stuffff..... please put it into a .frm (for 'normal people'), so we pass it through the evolver and will see what happens..

thanks,

Guy

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