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Mandelbrot set: Difference between revisions

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Mandelbrot set (view source)

Revision as of 15:04, 4 July 2021

381 bytes added ,  3 years ago
→‎Normalized Iteration Count, Distance Estimation and Mercator Maps: Added: e^(-|z|)-smoothing
(adding lambdatalk answer)
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===Normalized Iteration Count, Distance Estimation and Mercator Maps===
Actually the same, but without optimizations and therefore better suited for teaching.
The ''escape time'', ''e^(-|z|)-smoothing '', ''normalized iteration count'' and ''exterior distance estimation'' algorithms are used with NumPy and complex matrices (see WikipediaFrancisco Garcia, Angel Fernandez, Javier Barrallo, Luis Martin: [httpshttp://enmath.wikipediaunipa.orgit/wiki~grim/Plotting_algorithms_for_the_Mandelbrot_setJbarrallo.PDF ''PlottingColoring algorithmsDynamical forSystems in the MandelbrotComplex setPlane''] and SyntopiaMikael Hvidtfeldt Christensen: [http://blog.hvidtfeldts.net/index.php/2011/09/distance-estimated-3d-fractals-v-the-mandelbulb-different-de-approximations ''Distance Estimated 3D Fractals (V): The Mandelbulb & Different DE Approximations'']).
Finally, the Mandelbrot set is also printed with a scatter plot which will be misused later for a nice effect.
<lang python>import numpy as np
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Z, dZ = np.zeros_like(C), np.zeros_like(C)
T, S, D = np.zeros(C.shape), np.zeros(C.shape), np.zeros(C.shape)
 
for k in range(n):
M = abs(Z) < r
T[M], S[M] = T[M] + 1, S[M] + np.exp(-abs(Z[M]))
Z[M], dZ[M] = Z[M] ** 2 + C[M], 2 * Z[M] * dZ[M] + 1
T[M] = T[M] + 1
 
plt.imshow(T ** 0.1, cmap=plt.cm.twilight_shifted)
plt.savefig("Mandelbrot_set_1.png", dpi=250)
 
plt.imshow(S ** 0.1, cmap=plt.cm.twilight_shifted)
plt.savefig("Mandelbrot_set_2.png", dpi=250)
 
N = abs(Z) > r # normalized iteration count
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plt.imshow(T ** 0.1, cmap=plt.cm.twilight_shifted)
plt.savefig("Mandelbrot_set_2Mandelbrot_set_3.png", dpi=250)
 
N = abs(Z) > 2 # exterior distance estimation
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plt.imshow(D ** 0.1, cmap=plt.cm.twilight_shifted)
plt.savefig("Mandelbrot_set_3Mandelbrot_set_4.png", dpi=250)
 
X, Y = C.real, C.imag
SR = 150 * 2 / d # scaling depends on figsize
 
fig, ax = plt.subplots(figsize=(8, 6))
ax.scatter(X, Y, s=SR**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 allowsabove 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.
Note that Anders Sandberg uses a different scaling.
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Z, dZ = np.zeros_like(C), np.zeros_like(C)
T, S, D = np.zeros(C.shape), np.zeros(C.shape), np.zeros(C.shape)
 
for k in range(n):
M = abs(Z) < r
T[M], S[M] = T[M] + 1, S[M] + np.exp(-abs(Z[M]))
Z[M], dZ[M] = Z[M] ** 2 + C[M], 2 * Z[M] * dZ[M] + 1
T[M] = T[M] + 1
 
plt.imshow(T.T ** 0.1, cmap=plt.cm.nipy_spectral_r)
plt.savefig("Mercator_map_1.png", dpi=250)
 
plt.imshow(S.T ** 0.1, cmap=plt.cm.nipy_spectral_r)
plt.savefig("Mercator_map_2.png.png", dpi=250)
 
N = abs(Z) > r # normalized iteration count
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plt.imshow(T.T ** 0.1, cmap=plt.cm.nipy_spectral_r)
plt.savefig("Mercator_map_2Mercator_map_3.png", dpi=250)
 
N = abs(Z) > 2 # exterior distance estimation
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plt.imshow(D.T ** 0.1, cmap=plt.cm.nipy_spectral)
plt.savefig("Mercator_map_3Mercator_map_4.png", dpi=250)
 
X, Y = C.real, C.imag
SR = 150 * 2 / d * np.pi * np.exp(-B) # scaling depends on figsize
 
fig, ax = plt.subplots(2, 2, figsize=(16, 16))
ax[0, 0].scatter(X[0:300], Y[0:300], s=SR[0:300]**2, c=D[0:300]**0.5, cmap=plt.cm.nipy_spectral)
ax[0, 1].scatter(X[100:400], Y[100:400], s=SR[0:300]**2, c=D[100:400]**0.4, cmap=plt.cm.nipy_spectral)
ax[1, 0].scatter(X[200:500], Y[200:500], s=SR[0:300]**2, c=D[200:500]**0.3, cmap=plt.cm.nipy_spectral)
ax[1, 1].scatter(X[300:600], Y[300:600], s=SR[0:300]**2, c=D[300:600]**0.2, cmap=plt.cm.nipy_spectral)
plt.savefig("Mandelbrot_zoom.png", dpi=250)</lang>
 
Majow
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