.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fractals_chaos/curves/plot_04_coastline_paradox.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_curves_plot_04_coastline_paradox.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_fractals_chaos_curves_plot_04_coastline_paradox.py:


How long is the coast of Britain? The divider method
==========================================================

Walks a pair of dividers along a smooth circle and along the Koch curve
with ever smaller openings. The circle's measured length converges, but
the Koch curve's grows as a power of the ruler, the scaling Richardson
observed for real coastlines and Mandelbrot interpreted as a fractal
dimension.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.fractals_chaos import divider_length, koch_curve








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Measured length against ruler size
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-40

.. code-block:: Python


    t = np.linspace(0, 2 * np.pi, 20001)
    circle = np.column_stack([np.cos(t), np.sin(t)]) / (2 * np.pi)  # circumference 1
    koch = koch_curve(8)
    rulers = 3.0 ** -np.arange(1, 7)
    circle_lengths = [divider_length(circle, r) for r in rulers]
    koch_lengths = [divider_length(koch, r) for r in rulers]

    slope = np.polyfit(np.log(rulers), np.log(koch_lengths), 1)[0]
    print(f"Koch curve: L ~ ruler^{slope:.4f}, so D = 1 - slope = {1 - slope:.4f} (log 4 / log 3 = {np.log(4) / np.log(3):.4f})")

    fig, ax = plt.subplots()
    ax.loglog(rulers, circle_lengths, "o-", label="circle (smooth)")
    ax.loglog(rulers, koch_lengths, "s-", label="Koch curve (fractal)")
    ax.set_xlabel("ruler length")
    ax.set_ylabel("measured length")
    ax.invert_xaxis()
    ax.legend()
    ax.set_title("Richardson plot")



.. image-sg:: /api/gallery/fractals_chaos/curves/images/sphx_glr_plot_04_coastline_paradox_001.png
   :alt: Richardson plot
   :srcset: /api/gallery/fractals_chaos/curves/images/sphx_glr_plot_04_coastline_paradox_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Koch curve: L ~ ruler^-0.2619, so D = 1 - slope = 1.2619 (log 4 / log 3 = 1.2619)

    Text(0.5, 1.0, 'Richardson plot')




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

   **Total running time of the script:** (0 minutes 1.773 seconds)


.. _sphx_glr_download_api_gallery_fractals_chaos_curves_plot_04_coastline_paradox.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_coastline_paradox.ipynb <plot_04_coastline_paradox.ipynb>`

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

      :download:`Download Python source code: plot_04_coastline_paradox.py <plot_04_coastline_paradox.py>`

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

      :download:`Download zipped: plot_04_coastline_paradox.zip <plot_04_coastline_paradox.zip>`


.. only:: html

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

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