使用Python组合九重尖点multibrot和enneagram

Question

我正在开发Python和LaTeX (最好是矢量图形TikZ/渐近图/PGF/Metapost/GeoGebra)代码，它们通过在终端上运行简单的代码来生成这个动画。

这是一个thread on Tex.SE，其中讨论了几种绘制Mandelbrot集的方法，但我不能像在LaTeX中那样简单地修改数学方程。因此，我转而使用Python，并生成尽可能接近矢量图形(~2,000 dpi)的高分辨率输出。我想把multibrots画得尽可能的笼统。例如，当d为负数时。

Enneabrot

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下面的代码取自GitHub上的this Mandelbrot (dimension=2) project，我需要通过更改定义Mandelbrot集的递归方程中的指数来创建这个分形项目的高维版本，例如Triabrot (dimension=4)，Pentabrot (dimension=6)，Heptabrot (dimension=8)和Enneabrot (dimension=10)。换句话说，代替曼德尔布洛特分形的z_{n+1} = z_n * z_n + z_0，我们定义了一个维数变量d，然后d维Multibrot的方程将是z_{n+1} = z_n^d + z_0，每个这样的方程都将在其稳定区图中产生d-1个尖点。要了解有关此主题的更多信息，请观看以下两个YouTube视频：Times Tables by Mathologer和Mandelbrot Set by Numberphile。

有必要改变每一块输出的轴；这是因为分形在复杂的平面上移动，我们需要沿着镜头移动才能通过Python观察它们。我们还可以使用哪些公式？也许axisFix = ((d^2-4)/d)/10？在这个建议的公式中，d^2-4在那里，因为我希望在没有任何位移的情况下，将曼德尔布洛特分形(d=2)打印在中心。因此，轴平移的值将为零。这很有趣，因为对于d=-2，它也是零，这意味着我们还必须尝试查看像z_{n+1}=z_n^{-2.0} + z_0这样的方程。目标是找到最平滑的函数f(d)，使得f(2)=0，并通过以小尺寸(0.01-0.1)的增量增加d的值并将这里定义的mandelbrot(阈值、密度、尺寸)函数的输出打印为动画来制作动画。函数越平滑，演示文稿中幻灯片动画中的幻灯片过渡就越平滑。

import numpy as np
import matplotlib.pyplot as plt
# counts the number of iterations until the function diverges or
# returns the iteration threshold that we check until
def countIterationsUntilDivergent(c, threshold, d):
    z = complex(0, 0)
    for iteration in range(threshold):
# here is the recurrence relation z_{n+1} = z_n^d + z_0, used for 
# drawing d-1-dimensional Mandelbrot creatures (growing fractals)
        z = z**d + c
        if abs(z) > 4:
            break
            pass
        pass
    return iteration

# takes the iteration limit before declaring function as convergent and
# takes the density of the atlas
# create atlas, plot mandelbrot set, display set
def mandelbrot(threshold, density, d):
# it is necessary to change the axis for every patch of outputs;
# this happens because the fractals move in the complex plane
# and we need to move along our lens to watch them through Python
## what other formulas could we use? Maybe axisFix = ((d^2-4)/d)/10?
## d^2-4 is there because I want the Mandelbrot fractal (d=2)
## to be right there were it is printed without any replacement
## so the value of the axis translation would be zero. This is
## funny because it is also zero for d=-2 which means that
## we must also be trying to look at equations like
### z_{n+1}=z_n^{-2.0} + z_0
### the goal is to find the smoothest function
### f(d) such that f(2)=0 and make an animation 
### by increasing the value of d by increments of small size (0.01-0.1)
### and printing the output of the mandelbrot function defined here
### as an animation. The smoother the function, the smoother
### the transition of slides in the animation
    axisFix = d/10 
    # location and size of the atlas rectangle
    realAxis = np.linspace(-2.25+axisFix, 0.75+axisFix, density)
    imaginaryAxis = np.linspace(-1.5, 1.5, density)
    # realAxis = np.linspace(-0.22, -0.219, 1000)
    # imaginaryAxis = np.linspace(-0.70, -0.699, 1000)
    realAxisLen = len(realAxis)
    imaginaryAxisLen = len(imaginaryAxis)

    # 2-D array to represent mandelbrot atlas
    atlas = np.empty((realAxisLen, imaginaryAxisLen))

    # color each point in the atlas depending on the iteration count
    for ix in range(realAxisLen):
        for iy in range(imaginaryAxisLen):
            cx = realAxis[ix]
            cy = imaginaryAxis[iy]
            c = complex(cx, cy)
            atlas[ix, iy] = countIterationsUntilDivergent(c, threshold, d)
            pass
        pass

    # plot and display mandelbrot set
    fig1 = plt.gcf()
    plt.axis('off')
    # plt.savefig('mandel.eps', format='eps')
    plt.imshow(atlas.T, interpolation="nearest")
    # plt.show()
    output_name = str(d)+'.pdf'
    fig1.savefig(output_name, format='pdf', bbox_inches='tight', dpi=2000)

# time to party!!
dimensions = np.arange(10, 100) / 10
# for d in dimensions:
#     mandelbrot(120, 1000, d)

# Enneabrot
mandelbrot(120, 1000, 10)
# Heptabrot
mandelbrot(120, 1000, 8)
# Pentabrot
mandelbrot(120, 1000, 6)
# Triabrot
mandelbrot(120, 1000, 4)
# Mandelbrot
mandelbrot(120, 1000, 2)

Answer 1

以下是结果，感谢雅利安·赫马蒂通过将灵魂的Enneagram与十维Mandelbrot分形合并来编辑最终的照片。

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我需要一个Python和LaTeX (最好是矢量图形TikZ/渐近/PGF/Metapost/GeoGebra，按那个顺序)，它通过在您的终端上运行简单的代码来生成此动画。我们可以很容易地改变参数来制作一个七叶树(d=8)，五叶树(d=6)，甚至三叶树(d=4)。我附上了使用Python绘制Enneabrot (d=10)的代码，我已经由Danyaal Rangwala修改了this code，并在计算解的精确平衡区的递归方程中定义了一个新的变量d(维度)，这最终显示Enneabrot是我尝试生成这种分形的最后一个数字(从d=1.0开始，增量为0.1，直到d=10.0)。

在这里，我将发布曼德尔布洛特分形(d=2)的输出，以及我在上面定义的分形: Enneabrot (d=10)，Heptabrot (d=8)，五叉树(d=6)，甚至三叉树(d=4)。在此answer的下一个版本中将遵循的其他图表。

曼德尔布洛特(d=2)

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三叉树(d=4)

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五角星(d=6)

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七叶树(d=8)

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Enneabrot (d=10)

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