.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fractals_chaos/mandelbrot_julia/plot_02_mandelbrot_set.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_fractals_chaos_mandelbrot_julia_plot_02_mandelbrot_set.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_fractals_chaos_mandelbrot_julia_plot_02_mandelbrot_set.py:


The Mandelbrot set
========================

Colours every parameter :math:`c` by how quickly the orbit of
:math:`z_0 = 0` under :math:`z \mapsto z^2 + c` escapes. The black
region, the Mandelbrot set, is the set of :math:`c` whose orbit stays
bounded -- equivalently, whose Julia set is connected. Zooming into its
boundary reveals ever finer detail, including small copies of the whole
set.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.fractals_chaos import mandelbrot_set
    from mathematicskit.fractals_chaos.visualizers.plots import plot_escape_time








.. GENERATED FROM PYTHON SOURCE LINES 20-22

The whole set
-------------

.. GENERATED FROM PYTHON SOURCE LINES 22-27

.. code-block:: Python


    mandelbrot = mandelbrot_set(resolution=400, max_iter=200)
    ax = plot_escape_time(mandelbrot)
    ax.set_title("Mandelbrot set")




.. image-sg:: /api/gallery/fractals_chaos/mandelbrot_julia/images/sphx_glr_plot_02_mandelbrot_set_001.png
   :alt: Mandelbrot set
   :srcset: /api/gallery/fractals_chaos/mandelbrot_julia/images/sphx_glr_plot_02_mandelbrot_set_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'Mandelbrot set')



.. GENERATED FROM PYTHON SOURCE LINES 28-33

Zooming into the boundary
-------------------------

Near :math:`c \approx -1.77` on the real axis, a small copy of the
Mandelbrot set sits inside the boundary's filaments.

.. GENERATED FROM PYTHON SOURCE LINES 33-43

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(10, 4.5))
    seahorse = mandelbrot_set(extent=(-0.80, -0.70, 0.05, 0.15), resolution=300, max_iter=400)
    plot_escape_time(seahorse, ax=axes[0], cmap="twilight")
    axes[0].set_title("Seahorse valley")
    minibrot = mandelbrot_set(extent=(-1.80, -1.74, -0.03, 0.03), resolution=300, max_iter=400)
    plot_escape_time(minibrot, ax=axes[1], cmap="twilight")
    axes[1].set_title("A small copy near c = -1.77")
    fig.tight_layout()




.. image-sg:: /api/gallery/fractals_chaos/mandelbrot_julia/images/sphx_glr_plot_02_mandelbrot_set_002.png
   :alt: Seahorse valley, A small copy near c = -1.77
   :srcset: /api/gallery/fractals_chaos/mandelbrot_julia/images/sphx_glr_plot_02_mandelbrot_set_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 44-49

Membership check
----------------

:math:`c=-1` and :math:`c=0.25` lie in the set (bounded orbits);
:math:`c = 0.3` and :math:`c = 1` do not.

.. GENERATED FROM PYTHON SOURCE LINES 49-55

.. code-block:: Python


    for c in (-1.0, 0.25, 0.3, 1.0):
        z, steps = 0.0, 0
        while abs(z) <= 2.0 and steps < 1000:
            z, steps = z * z + c, steps + 1
        print(f"c = {c:5.2f}: {'bounded (in the set)' if abs(z) <= 2.0 else f'escapes after {steps} steps'}")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    c = -1.00: bounded (in the set)
    c =  0.25: bounded (in the set)
    c =  0.30: escapes after 12 steps
    c =  1.00: escapes after 3 steps





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.227 seconds)


.. _sphx_glr_download_api_gallery_fractals_chaos_mandelbrot_julia_plot_02_mandelbrot_set.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_mandelbrot_set.ipynb <plot_02_mandelbrot_set.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_mandelbrot_set.py <plot_02_mandelbrot_set.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_mandelbrot_set.zip <plot_02_mandelbrot_set.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_