Darren New <dne### [at] sanrrcom> wrote:
> Warp wrote:
> >   The farther away you are from the border of the set, the fewer iterations
> > it takes for the complex numbers to bail out (ie. to exit the circle of
> > radius 2, which is usually what's used for this). The closer you get to
> > the border of the set, the more iterations it takes for it to bail out.

> Ah! I see. Very cool.  Thanks for the insight!  That makes sense now.  I 
> hadn't connected it with "how close to the edge are you".

  Btw, that also explains why you need more mantissa bits to zoom deeper.
The complex numbers which start "on screen" don't stay there during the
iterative process. Instead, they orbit around and start escaping to
infinity (in practical implementations it's tested if they leave the
circle of radius 2, because it's easy to prove that if an orbit leaves
that circle it will inevitably escape to infinity). Thus you need to be
able to take some complex numbers which require a very high accuracy
(dozens and even hundreds of decimal places, depending on the zoom level),
and do calculations until they reach the circle-of-radius-2 limit.

  You can see the mantissa limit being hit when you zoom enough into the
set when you start getting rectangular artifacts. As stated, standard
double-precision floating point variables allow a zoom factor of something
around 2^50 before you hit the limit.

-- 
                                                          - Warp

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