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From: bet@ecsvax.UUCP (Bennett E. Todd III)
Newsgroups: net.graphics
Subject: Re: Mandelbrot set membership iteration limit question
Message-ID: <502@ecsvax.UUCP>
Date: Fri, 27-Sep-85 14:58:07 EDT
Article-I.D.: ecsvax.502
Posted: Fri Sep 27 14:58:07 1985
Date-Received: Sun, 29-Sep-85 06:36:23 EDT
References: <819@gitpyr.UUCP>
Reply-To: duccpc!bet@ecsvax.UUCP (Bennett E. Todd III)
Distribution: net
Organization: Duke University Computation Center
Lines: 45
In article <819@gitpyr.UUCP> roy@gitpyr.UUCP (Roy Mongiovi) writes:
>Ok. I know that for points close to the edge of the set, the numbers
>may grow without limit and yet still remain bounded and therefore "in"
>the set. Fine. The edge of the set is hard to see given a fixed
>length word size.
>
>But how does the maximum iteration count affect the perceived edge of
>the set? I would expect that to be a greater effect (especially for
>as few as 255 iterations).
I thought the Scientific American article indicated that the numbers
*can't* grow without bound -- if ever the magnitude of the iterated
value 'z' exceeds 2, you can stop, and know that the number lies outside
the Mandelbrot set. Indeed, the algorithms I have seen (and that I
wrote) color the region outside the Mandelbrot set based on the number
of iterations required for the magnitude to exceed 2. The only impact
word size has on resolution is in error propogation. The cumulative
error cannot be permitted to swamp the computed values; large
magnifications need a fair computational accuracy.
The iteration count does indeed have an important effect on resolution.
I am running a varient of the program I posted to the net, and when
computing even mild zooms (~90x) the boundary of the set is fuzzy with
iterations to 255. The boundary is quite sharp with iterations to 1024,
even with a zoom of ~2200x displaying 2808 pixels horizontally by 900
vertically (8 color printer). On the other hand, it takes the poor 8088
about 17 hours to compute this beastie!
I would be interested in hearing the details of the analysis that lead
one poster to this group to conclude that the Mandelbrot set is a
fiction, an artifact whose graphic representations are dominated by
computational errors. Does anybody have anything conclusive to state
about the error propogation characteristics of IEEE floating point as
implemented by an 8087? How about 32 bit scaled fixed point with
rounding? Both seem to produce identical pictures, nearly byte-for-byte.
The difference appears to be entirely attributable to slightly different
assignments of coordinates to pixels.
-Bennett
--
"Hypocrisy is the vaseline of social intercourse." (Who said that?)
Bennett Todd -- Duke Computation Center, Durham, NC 27706-7756; (919) 684-3695
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