Well, if I understand what you're asking, I've noticed in similar investigations that z = z*z z = z*z + c is equivalent to z = z^4 + c. In general, as becomes obvious when you look at it for awhile, z = z^m z = z^n + c is equivalent to z = z^(n+m) + c. Ciao, Russell --------- Original Message --------- DATE: Mon, 05 Apr 2004 04:04:45 From: "Raymond Filiatreault" <rayfil@hotmail.com> To: fractint@mailman.xmission.com Cc:
I did mention that the two formulas were FUNDAMENTALLY different. One uses the ACOS function of "z" while the other uses the ACOSH function of "z".
With real numbers, the acos function could only take arguments which have an absolute value less than or equal to 1.0, and produce results between 0 and 2PI. Similarly with real numbers, the acosh function could only take arguments which are positive and greater than or equal to 1.0, and produce results between +INFINITY and -INFINITY.
However, raising either of these functions of "z" (a complex number) to the 4th power and adding "c" have been producing the identical fractal, which also happens to be a quarter replica of the M-set. I wonder if some genius mathematician could make sense of this unexpected behaviour with complex numbers.
I got the same results with Fractint using the following formula and testing with the acos and acosh functions:
rayfil { c = z = pixel: z = fn1(z) z = z*z z = z*z + c |z| < 4 }
Raymond
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