.. 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_03_koch_snowflake.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_03_koch_snowflake.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_fractals_chaos_curves_plot_03_koch_snowflake.py:


The Koch snowflake: infinite perimeter, finite area
=========================================================

Builds the Koch snowflake. Each refinement multiplies the perimeter by
4/3, so it grows without bound, while the enclosed area converges to
8/5 of the starting triangle.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.fractals_chaos import box_counting_dimension, koch_curve, koch_snowflake, similarity_dimension








.. GENERATED FROM PYTHON SOURCE LINES 17-19

The first refinements
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-33

.. code-block:: Python


    fig, axes = plt.subplots(1, 4, figsize=(12, 3.5))
    triangle_area = np.sqrt(3) / 4
    for ax, order in zip(axes, range(4)):
        pts = koch_snowflake(order)
        perimeter = np.sum(np.linalg.norm(np.diff(pts, axis=0), axis=1))
        area = 0.5 * abs(np.dot(pts[:-1, 0], pts[1:, 1]) - np.dot(pts[1:, 0], pts[:-1, 1]))
        ax.fill(*pts.T, alpha=0.4)
        ax.plot(*pts.T, lw=0.8)
        ax.set_aspect("equal")
        ax.axis("off")
        ax.set_title(f"perimeter {perimeter:.2f}\narea / triangle {area / triangle_area:.3f}")
    print(f"limiting area ratio 8/5 = {8 / 5}")




.. image-sg:: /api/gallery/fractals_chaos/curves/images/sphx_glr_plot_03_koch_snowflake_001.png
   :alt: perimeter 3.00 area / triangle 1.000, perimeter 4.00 area / triangle 1.333, perimeter 5.33 area / triangle 1.481, perimeter 7.11 area / triangle 1.547
   :srcset: /api/gallery/fractals_chaos/curves/images/sphx_glr_plot_03_koch_snowflake_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    limiting area ratio 8/5 = 1.6




.. GENERATED FROM PYTHON SOURCE LINES 34-36

Dimension of the Koch curve
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-39

.. code-block:: Python


    print(f"similarity dimension log 4 / log 3 = {similarity_dimension([1 / 3] * 4):.4f}")
    print(f"box-counting estimate on order 7: {box_counting_dimension(koch_curve(7)).dimension:.4f}")




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

 .. code-block:: none

    similarity dimension log 4 / log 3 = 1.2619
    box-counting estimate on order 7: 1.2771





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

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_fractals_chaos_curves_plot_03_koch_snowflake.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_koch_snowflake.ipynb <plot_03_koch_snowflake.ipynb>`

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

      :download:`Download Python source code: plot_03_koch_snowflake.py <plot_03_koch_snowflake.py>`

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

      :download:`Download zipped: plot_03_koch_snowflake.zip <plot_03_koch_snowflake.zip>`


.. only:: html

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

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