.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/fixed_points/plot_02_brouwer_fixed_point.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_topology_fixed_points_plot_02_brouwer_fixed_point.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_topology_fixed_points_plot_02_brouwer_fixed_point.py:


Brouwer's fixed-point theorem (1911)
====================================

Every continuous map of a disc (or triangle) to itself has a fixed
point. The proof here is constructive. Label each vertex of a fine
triangulation with a coordinate the map does not increase. That
labelling obeys Sperner's rule, so some small triangle carries all three
labels, and on it the map nearly fixes every coordinate. Refining the
grid closes in on a fixed point, which matches a Newton-type solve with
:func:`scipy.optimize.fsolve`.

.. GENERATED FROM PYTHON SOURCE LINES 15-39

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import fsolve

    from mathematicskit.topology import brouwer_fixed_point, triangle_grid


    def f(x):
        """A nonlinear swirl of the triangle onto itself (barycentric coordinates in and out)."""
        a, b, c = x
        y = np.array([0.15 + 0.6 * b + 0.2 * a * c, 0.1 + 0.7 * c**2 + 0.1 * a, 0.0])
        y = np.clip(y, 0, None)
        y[2] = max(0.0, 1 - y[0] - y[1])
        return y / y.sum()


    exact = fsolve(lambda z: f(np.array([z[0], z[1], 1 - z[0] - z[1]]))[:2] - z, [0.3, 0.3])
    exact = np.array([*exact, 1 - exact.sum()])
    print("fixed point by fsolve:", np.round(exact, 6))
    for n in (4, 16, 64, 256):
        result = brouwer_fixed_point(f, n)
        error = np.max(np.abs(result.point - exact))
        print(f"n = {n:3d}: Sperner triangle centroid {np.round(result.point, 4)}, error {error:.2e}, residual {result.residual:.2e}")





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

 .. code-block:: none

    fixed point by fsolve: [0.329915 0.254114 0.415971]
    n =   4: Sperner triangle centroid [0.3333 0.3333 0.3333], error 8.26e-02, residual 1.22e-01
    n =  16: Sperner triangle centroid [0.3333 0.2708 0.3958], error 2.01e-02, residual 2.78e-02
    n =  64: Sperner triangle centroid [0.3177 0.2552 0.4271], error 1.22e-02, residual 1.68e-02
    n = 256: Sperner triangle centroid [0.3294 0.2552 0.4154], error 1.09e-03, residual 1.50e-03




.. GENERATED FROM PYTHON SOURCE LINES 40-42

The labels and the fully labelled triangle
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-59

.. code-block:: Python


    n = 24
    grid = triangle_grid(n)
    x = grid.points / n
    labels = np.argmax((np.array([f(p) for p in x]) <= x + 1e-15) & (x > 0), axis=1)
    result = brouwer_fixed_point(f, n)
    corners = np.array([[0.0, 0.0], [1.0, 0.0], [0.5, np.sqrt(3) / 2]])
    fig, ax = plt.subplots(figsize=(7, 6.5))
    ax.triplot(*grid.cartesian.T, grid.triangles, color="0.8", lw=0.5)
    for k, color in enumerate(("tab:red", "tab:green", "tab:blue")):
        ax.scatter(*grid.cartesian[labels == k].T, s=14, color=color, label=f"label {k}: f decreases coordinate {k}")
    ax.add_patch(plt.Polygon(result.triangle @ corners, color="gold", ec="k", zorder=3))
    ax.scatter(*(exact @ corners), marker="*", s=200, color="k", zorder=4, label="true fixed point")
    ax.set_aspect("equal")
    ax.axis("off")
    ax.legend(loc="upper right", fontsize=8)
    ax.set_title("Brouwer: a fully labelled triangle traps the fixed point")



.. image-sg:: /api/gallery/topology/fixed_points/images/sphx_glr_plot_02_brouwer_fixed_point_001.png
   :alt: Brouwer: a fully labelled triangle traps the fixed point
   :srcset: /api/gallery/topology/fixed_points/images/sphx_glr_plot_02_brouwer_fixed_point_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Brouwer: a fully labelled triangle traps the fixed point')




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

   **Total running time of the script:** (0 minutes 0.177 seconds)


.. _sphx_glr_download_api_gallery_topology_fixed_points_plot_02_brouwer_fixed_point.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/topology/fixed_points/plot_02_brouwer_fixed_point.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_brouwer_fixed_point.ipynb <plot_02_brouwer_fixed_point.ipynb>`

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

      :download:`Download Python source code: plot_02_brouwer_fixed_point.py <plot_02_brouwer_fixed_point.py>`

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

      :download:`Download zipped: plot_02_brouwer_fixed_point.zip <plot_02_brouwer_fixed_point.zip>`


.. only:: html

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

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