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Shaded & power normalized renderingΒΆ

The Mandelbrot set rendering can be improved by using a normalized recount associated with a power normalized colormap (gamma=0.3). Rendering can be further enhanced thanks to shading.

The maxiter gives the precision of the computation. maxiter=200 should take a few seconds on most modern laptops.

import numpy as np

def mandelbrot_set(xmin, xmax, ymin, ymax, xn, yn, maxiter, horizon=2.0):
    X = np.linspace(xmin, xmax, xn).astype(np.float32)
    Y = np.linspace(ymin, ymax, yn).astype(np.float32)
    C = X + Y[:, None] * 1j
    N = np.zeros_like(C, dtype=int)
    Z = np.zeros_like(C)
    for n in range(maxiter):
        I = np.less(abs(Z), horizon)
        N[I] = n
        Z[I] = Z[I]**2 + C[I]
    N[N == maxiter-1] = 0
    return Z, N

if __name__ == '__main__':
    import time
    import matplotlib
    from matplotlib import colors
    import matplotlib.pyplot as plt

    xmin, xmax, xn = -2.25, +0.75, 3000/2
    ymin, ymax, yn = -1.25, +1.25, 2500/2
    maxiter = 200
    horizon = 2.0 ** 40
    log_horizon = np.log(np.log(horizon))/np.log(2)
    Z, N = mandelbrot_set(xmin, xmax, ymin, ymax, xn, yn, maxiter, horizon)

    # Normalized recount as explained in:

    # This line will generate warnings for null values but it is faster to
    # process them afterwards using the nan_to_num
    with np.errstate(invalid='ignore'):
        M = np.nan_to_num(N + 1 -
                          np.log(np.log(abs(Z)))/np.log(2) +

    dpi = 72
    width = 10
    height = 10*yn/xn
    fig = plt.figure(figsize=(width, height), dpi=dpi)
    ax = fig.add_axes([0.0, 0.0, 1.0, 1.0], frameon=False, aspect=1)

    # Shaded rendering
    light = colors.LightSource(azdeg=315, altdeg=10)
    M = light.shade(M,, vert_exag=1.5,
                    norm=colors.PowerNorm(0.3), blend_mode='hsv')
    plt.imshow(M, extent=[xmin, xmax, ymin, ymax], interpolation="bicubic")

    # Some advertisement for matplotlib
    year = time.strftime("%Y")
    major, minor, micro = matplotlib.__version__.split('.', 2)
    text = ("The Mandelbrot fractal set\n"
            "Rendered with matplotlib %s.%s, %s -"
            % (major, minor, year))
    ax.text(xmin+.025, ymin+.025, text, color="white", fontsize=12, alpha=0.5)

Total running time of the script: ( 0 minutes 5.221 seconds)

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