May 28, 1997: Nowhere in a Hurry | May 27 | May 29 | 1997 | FOTD Home |
Fractal
visionaries:
No other fractal is quite like the Mandelbrot set. No other
fractal gives so great a variety from so simple a formula.
But the Mandelbrot set is but one in an endless series of similar
fractals. There is no reason one must stop at Z^2+C; the
formula
works just as well with Z^3+C, Z^4+C, or Z^n+C.
However the higher order mandeloids soon degenerate into a monotonous
circle with n-1 identical bays. Indeed, when n is greater
than
12, the sets are barely distinguishable from one another, and the
midgets, which grow in size and number until they nearly obliterate all
else, soon become uninteresting off-center circles. The only
really interesting higher order
Mandeloids are the Z^3, the Z^4, and to some extent the Z^5.
Today's fractal, all_nine, is a symmetrical order 3 midget that I
picked in honor of the Z^3 mandeloid, which gets far less attention
than it deserves. The midget is located on the negative
x-axis of
its parent fractal. I realize that the
order 3 mandeloid doesn't normally have midgets on its negative tail,
in fact it doesn't have a negative tail at all. But I cheated
a bit by writing a formula that gives it one, complete with midgets.
The little two-headed midget sits surrounded by its pattern, glowing
like a vision of the holy grail, with its arms radiating around it in
ascending powers of three, rather than in powers of two. Thus
instead of 4, 8, 16, we have 9, 27, 81.
BTW, this same increasing-powers-of-the-exponent sequence continues
into the higher orders, though the features soon become too tiny to be
distinguished. The sequence also prevails in the
fractional order mandeloids, which is why those figures must be filled
with such annoying discontinuities.
Jim Muth
jamth@mindspring.com
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