.. 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_06_peano_curve.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_06_peano_curve.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_fractals_chaos_curves_plot_06_peano_curve.py:


Peano's space-filling curve
=================================

Peano's 1890 curve divides the square into a :math:`3 \times 3` grid,
visits the nine cells in a serpentine order, and repeats the pattern
inside each cell. The order-:math:`n` approximation visits every cell of
a :math:`3^n \times 3^n` grid exactly once, so in the limit the
continuous curve passes through every point of the square. Here the
approximations are generated by the standard L-system rewriting rules
for Peano's curve and drawn with turtle graphics.

.. GENERATED FROM PYTHON SOURCE LINES 15-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.fractals_chaos import lsystem, turtle_path

    rules = {"X": "XFYFX+F+YFXFY-F-XFYFX", "Y": "YFXFY-F-XFYFX+F+YFXFY"}








.. GENERATED FROM PYTHON SOURCE LINES 23-25

The first three approximations
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-43

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(12, 4.2))
    for order, ax in zip((1, 2, 3), axes):
        step = 1.0 / 3**order
        (path,) = turtle_path(lsystem("X", rules, order), angle=90.0, step=step, heading=90.0)
        path = path - path.min(axis=0) + step / 2
        ax.plot(path[:, 0], path[:, 1], lw=1.2 if order < 3 else 0.8)
        ax.set_xlim(0, 1)
        ax.set_ylim(0, 1)
        ax.set_aspect("equal")
        ax.set_xticks(np.linspace(0, 1, 4))
        ax.set_yticks(np.linspace(0, 1, 4))
        ax.grid(True, lw=0.4)
        ax.set_title(f"order {order}: {3**order}x{3**order} grid")

        cells = {tuple(v) for v in np.floor(path / step).astype(int)}
        print(f"order {order}: {len(path)} points, {len(cells)} distinct cells of {9**order}")
    fig.tight_layout()



.. image-sg:: /api/gallery/fractals_chaos/curves/images/sphx_glr_plot_06_peano_curve_001.png
   :alt: order 1: 3x3 grid, order 2: 9x9 grid, order 3: 27x27 grid
   :srcset: /api/gallery/fractals_chaos/curves/images/sphx_glr_plot_06_peano_curve_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    order 1: 9 points, 9 distinct cells of 9
    order 2: 81 points, 81 distinct cells of 81
    order 3: 729 points, 729 distinct cells of 729





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

   **Total running time of the script:** (0 minutes 0.050 seconds)


.. _sphx_glr_download_api_gallery_fractals_chaos_curves_plot_06_peano_curve.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_06_peano_curve.ipynb <plot_06_peano_curve.ipynb>`

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

      :download:`Download Python source code: plot_06_peano_curve.py <plot_06_peano_curve.py>`

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

      :download:`Download zipped: plot_06_peano_curve.zip <plot_06_peano_curve.zip>`


.. only:: html

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

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