Mandelbrot set: Difference between revisions
→Normalized Counting, Distance Estimation, Mercator Maps and Deep Zoom: Added comments and final adjustments
(→Normalized Counting, Distance Estimation, Mercator Maps and Perturbation Theory: Final changes and cuts) |
(→Normalized Counting, Distance Estimation, Mercator Maps and Deep Zoom: Added comments and final adjustments) |
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Line 8,890:
plt.imshow(D ** 0.1, cmap=plt.cm.twilight_shifted)
plt.savefig("Mandelbrot_set_3.png", dpi=200)</lang>
A small change in the code above creates Mercator maps 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'']).
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.
Line 8,896 ⟶ 8,897:
With the same pixel density and the same maximum magnification, the difference in height between the maps is only about 10 percent.
By misusing a scatter plot, it is possible to create zoom images of the Mandelbrot set.
<lang python>import numpy as np
import matplotlib.pyplot as plt
Line 8,936:
import matplotlib.pyplot as plt
import mpmath as mpm # complex floating-point arithmetic with arbitrary precision
mpm.mp.prec = 256 # set precision to 256 bits (extended)
d, h = 200, 150 # pixel density (= image width) and image height
Line 8,961:
C = 5.0e-55 * (A + B * 1j)
Z,
D = np.zeros(C.shape)
Line 8,978:
plt.savefig("Mandelbrot_deep_zoom.png", dpi=200)</lang>
Of course, deep Mercator maps can also be created. See also the image [https://www.flickr.com/photos/arenamontanus/3430921497/in/album-72157615740829949/ Deeper Mercator Mandelbrot] by Anders Sandberg. To reduce glitches, an additional reference sequence for the derivation (dS) is recorded with high precision. Compare the corresponding program examples for the Julia language for more details.
<lang python>import numpy as np
import matplotlib.pyplot as plt
import mpmath as mpm # complex floating-point arithmetic with arbitrary precision
mpm.mp.prec = 256 # set precision to 256 bits (extended)
d, h = 50, 1000 # pixel density (= image width) and image height
Line 9,007:
C = (- 4.0) * np.exp((A + B * 1j) * 1j)
Z,
D = np.zeros(C.shape)
def iteration(S,
return (2 * S + E) * E + C, 2 * ((S + E) * dE + E * dS)
for k in range(n):
M = Z.real ** 2 + Z.imag ** 2 < r ** 2
E[M], dE[M] = iteration(S[k],
Z[M], dZ[M] = S[k+1] + E[M], dS[k+1] + dE[M]
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