At 02:46 26/07/2002, Jim Muth wrote:
While toying with the Lyapunov Mandelbrot formula recently posted by Morgan Owens, I noticed that the formula produces images virtually identical to those drawn by the Fractint type=hypercomplex Mandelbrot formula. I have included two parameter files, one of which uses the Lyapunov Mandelbrot formula, while the other uses the hypercomplex Mandelbrot formula, yet the two files draw identical images.
Perhaps one of the math experts on the list can explain what is happening here. . . .
It comes about because hypercomplex numbers can be modelled as pairs of complex numbers; by converting them into this "dual" form, one can use functions designed to operate on complex numbers on hypercomplex numbers as well, hence their implementation in Fractint: Hypercomplex numbers have an elegant generalization of any unary complex valued function defined on the complex numbers. First, note that hypercomplex numbers can be represented as a pair of complex numbers in the following way. Let h = x + yi + zj + wk. a = (x-w) + i(y+z) b = (x+w) + i(y-z) The numbers a and b are complex numbers. We can represent h as the pair of complex numbers (a,b). Conversely, if we have a hypercomplex number given to us in the form (a,b), we can solve for x, y, z, and w. The solution to c = (x-w) + i(y+z) d = (x+w) + i(y-z) is x = (real(c) + real(d))/2 y = (imag(c) + imag(d))/2 z = (imag(c) - imag(d))/2 w = (real(d) - real(c))/2 We can now, for example, compute sin(h). First compute the two complex numbers a and b as above, then set c = sin(a) and d = sin(b) where sin() is the complex version of the sin function. Now use the equations above to solve for x, y, z, and w in terms of c and d. The hypercomplex number (x,y,z,w) thus obtained is sin(h). The beauty of this is that it really doesn't make any difference what function we use. Instead of sin, we could have used cos, sinh, ln, or z^2. Using this technique, Fractint can create 3-D fractals using the formula h' = fn(h) + c, where "fn" is any of the built-in functions. Where fn is sqr(), this is the famous mandelbrot formula, generalized to four dimensions. Fractint Version 20.0, pp 178-179 What the LyapunovMandelbrot formula amounts to (basically by accident) is to make this duality between hypercomplex numbers and pairs of complex numbers explicit (after taking into account the different approaches to parameterisation). Morgan L. Owens "Morse's Theorem says that the typical smooth function we can expect to encounter is locally quadratic near a critical point. Discuss this genericity result within the framework of the quadratic maps leading to the Mandelbrot set and its many relatives."