Mandelbrot set: Difference between revisions

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ax.scatter(X, Y, s=S**2, c=D**0.1, cmap=plt.cm.twilight_shifted)

plt.savefig("Mandelbrot_plot.png", dpi=250)</lang>

A small change in the above code allows Mercator zooms of the Mandelbrot set (see David Madore: [http://www.madore.org/~david/math/mandelbrot.html ''Mandelbrot set images and videos'']).

A small change in the above code allows Mercator zooms of the Mandelbrot set (see David Madore: [http://www.madore.org/~david/math/mandelbrot.html ''Mandelbrot set images and videos''] and Anders Sandberg: [https://www.flickr.com/photos/arenamontanus/sets/72157615740829949 ''Mercator Mandelbrot Maps'']).

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The maximum magnification is exp(2*pi*h/d) = exp(2*pi*5.5) = 535.5^5.5 = 10^15, which is also the maximum for 64-bit arithmetic.

Compression is used as described by David Madore.

See also [https://www.flickr.com/photos/arenamontanus/sets/72157615740829949 ''Mercator Mandelbrot Maps''] by Anders Sandberg.

The largest magnification is exp(2*pi*h/d).

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In ~~the~~ ~~example it~~ is ~~equal~~ to exp(2*pi*5.5), ~~which~~ is ~~about~~ 10^15 ~~and~~ also the maximum for 64-bit arithmetic.

On some architectures, the precision can be extended a bit:

Try G = np.exp(2 * np.pi * (X * 1j - Y), dtype = np.clongdouble) if you are lucky.

Note that Anders Sandberg uses a different scaling.

He uses 10^(3*h/d) = 1000^(h/d) instead of exp(2*pi*h/d) = 535.5^(h/d), so his images appear somewhat compressed in comparison (but not much, because 1000^5 is ~~approximately~~ ~~equal to~~ 535.5^5.5).

He uses 10^(3*h/d) = 1000^(h/d) instead of exp(2*pi*h/d) = 535.5^(h/d), so his images appear somewhat compressed in comparison (but not much, because 1000^5 = 10^15 = 535.5^5.5).

With the same pixel density and the same maximum magnification, the difference in height between the maps is only about 10 percent.

<lang python>import numpy as np