z=(cosh(z))^2+c
Status: Beta
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jmmcg
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What's so good about this one? Well - it is unconnected and
infinite in area. It has (perfect?) copies of itself up and down
the imaginary axis. So it is unbounded, unlike the Mandelbrot
set that never exceeds to circle or radius 2 in the complex
plain, this one extends infinitely. It was inspired by an article I
read in Scientific American many years ago about fractal
tessellations (about 1993 - *real* shame I can' t remember
the reference), which got me thinking: Can I make fractal that
uniquely different to the Mandelbot set? (The purists argue:
The Mandelbrot set contains all the features other fractals
contain, so why look at others?) Well, it isn't infinite in area,
so this has a property uniquely different to the Mandelbrot set
(infinite area), thus it is worthy of study, event to the purists!
Go and roll it up in the "Custom Fractal" type and see for
yourself? My challenge: Can you make a fractal that is
unbounded along the real axis? What about *both* axes?
Then the mother of all challenges: An inexact copy on the
real, imaginary or both axes? If you get this far, can you prove
that there does not exist another exact copy anywhere in the
plane? These are a fractal artists' nightmare: Not only is a
single set infinitely complex, but each set would be
*guaranteed* to be different to all others (thus potentially
worthy of individual study), and there would be an infinite
number of them! Could the answer to self similarity be related
to the number of real numbes, the transfinite number: Aleph
1. (You guessed the name now? ;)
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And just for a laugh - it has an infinite number of
*unconnected* Mandelbrot sets within each repeat! Tee hee!
Infinity is a *very* big number!
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The equation 'z=(sinh(z))^2+c exhibits very similar properties.