.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fractals_chaos/ifs/plot_01_sierpinski_triangle_and_carpet.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_ifs_plot_01_sierpinski_triangle_and_carpet.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_fractals_chaos_ifs_plot_01_sierpinski_triangle_and_carpet.py:


Sierpiński's triangle and carpet
======================================

Sierpiński's triangle (1915) removes the middle quarter of a filled
triangle and repeats on the three corner triangles; his carpet (1916)
removes the middle ninth of a square and repeats on the eight remaining
squares. Both limits have zero area and a dimension strictly between 1
and 2. Here both are drawn with the chaos game.

.. GENERATED FROM PYTHON SOURCE LINES 13-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.fractals_chaos import SierpinskiCarpet, SierpinskiTriangle
    from mathematicskit.fractals_chaos.visualizers.plots import plot_ifs_points








.. GENERATED FROM PYTHON SOURCE LINES 20-22

The two sets
------------

.. GENERATED FROM PYTHON SOURCE LINES 22-30

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(10, 5))
    plot_ifs_points(SierpinskiTriangle().generate(60000, seed=0), ax=axes[0], color="steelblue")
    axes[0].set_title("Sierpiński triangle")
    plot_ifs_points(SierpinskiCarpet().generate(80000, seed=0), ax=axes[1], color="firebrick")
    axes[1].set_title("Sierpiński carpet")
    fig.tight_layout()




.. image-sg:: /api/gallery/fractals_chaos/ifs/images/sphx_glr_plot_01_sierpinski_triangle_and_carpet_001.png
   :alt: Sierpiński triangle, Sierpiński carpet
   :srcset: /api/gallery/fractals_chaos/ifs/images/sphx_glr_plot_01_sierpinski_triangle_and_carpet_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 31-38

Area vanishes, dimension is fractional
--------------------------------------

After :math:`n` removals the triangle keeps :math:`(3/4)^n` of its
area and the carpet :math:`(8/9)^n`; both tend to zero. Three copies at
scale 1/2 and eight copies at scale 1/3 give dimensions
:math:`\log 3/\log 2` and :math:`\log 8/\log 3`.

.. GENERATED FROM PYTHON SOURCE LINES 38-42

.. code-block:: Python


    for n in (1, 5, 20, 50):
        print(f"n = {n:2d}: triangle area {(3 / 4) ** n:.2e}, carpet area {(8 / 9) ** n:.2e}")
    print(f"dimension of triangle: {np.log(3) / np.log(2):.4f}, carpet: {np.log(8) / np.log(3):.4f}")




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

 .. code-block:: none

    n =  1: triangle area 7.50e-01, carpet area 8.89e-01
    n =  5: triangle area 2.37e-01, carpet area 5.55e-01
    n = 20: triangle area 3.17e-03, carpet area 9.48e-02
    n = 50: triangle area 5.66e-07, carpet area 2.77e-03
    dimension of triangle: 1.5850, carpet: 1.8928





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

   **Total running time of the script:** (0 minutes 0.145 seconds)


.. _sphx_glr_download_api_gallery_fractals_chaos_ifs_plot_01_sierpinski_triangle_and_carpet.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_sierpinski_triangle_and_carpet.ipynb <plot_01_sierpinski_triangle_and_carpet.ipynb>`

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

      :download:`Download Python source code: plot_01_sierpinski_triangle_and_carpet.py <plot_01_sierpinski_triangle_and_carpet.py>`

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

      :download:`Download zipped: plot_01_sierpinski_triangle_and_carpet.zip <plot_01_sierpinski_triangle_and_carpet.zip>`


.. only:: html

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

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