.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/complexes/plot_01_barycentric_subdivision_invariance.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_complexes_plot_01_barycentric_subdivision_invariance.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_complexes_plot_01_barycentric_subdivision_invariance.py:


Brouwer, Alexander, and the invariance of homology under subdivision (1911-1915)
================================================================================

Is homology a property of the *space* or of the chosen triangulation?
Brouwer's simplicial approximation (1911) answers it. Any continuous
map is close to a simplicial one after subdividing finely enough.
Alexander (1915) used this to prove that homology is a topological
invariant. Barycentric subdivision replaces every simplex by the chains
of its faces. The simplex counts grow about sixfold per round for
surfaces, while the Betti numbers do not change.

.. GENERATED FROM PYTHON SOURCE LINES 15-26

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.topology import barycentric_subdivision, betti_numbers, projective_plane, simplex, torus
    from mathematicskit.topology.visualizers import plot_complex

    for name, K in (("torus", torus(3, 3)), ("projective plane", projective_plane())):
        for level in range(3):
            print(f"{name:>16}, sd^{level}: f = {K.f_vector!s:>18}  chi = {K.euler_characteristic}  Betti = {betti_numbers(K)}  mod 2: {betti_numbers(K, field=2)}")
            if level < 2:
                K = barycentric_subdivision(K)





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

 .. code-block:: none

               torus, sd^0: f =        (9, 27, 18)  chi = 0  Betti = (1, 2, 1)  mod 2: (1, 2, 1)
               torus, sd^1: f =     (54, 162, 108)  chi = 0  Betti = (1, 2, 1)  mod 2: (1, 2, 1)
               torus, sd^2: f =    (324, 972, 648)  chi = 0  Betti = (1, 2, 1)  mod 2: (1, 2, 1)
    projective plane, sd^0: f =        (6, 15, 10)  chi = 1  Betti = (1, 0, 0)  mod 2: (1, 1, 1)
    projective plane, sd^1: f =       (31, 90, 60)  chi = 1  Betti = (1, 0, 0)  mod 2: (1, 1, 1)
    projective plane, sd^2: f =    (181, 540, 360)  chi = 1  Betti = (1, 0, 0)  mod 2: (1, 1, 1)




.. GENERATED FROM PYTHON SOURCE LINES 27-29

Subdividing a triangle
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-40

.. code-block:: Python


    triangle = simplex(2)
    triangle.coordinates = [[0.0, 0.0], [1.0, 0.0], [0.5, 0.866]]
    fig, axes = plt.subplots(1, 3, figsize=(12, 4))
    K = triangle
    for level, ax in enumerate(axes):
        plot_complex(K, ax=ax, face_values=list(range(len(K.simplices(2)))), cmap="tab20", alpha=0.6, vertex_size=6)
        ax.set_title(f"sd^{level}: f = {K.f_vector}, χ = {K.euler_characteristic}")
        ax.axis("off")
        K = barycentric_subdivision(K)
    fig.suptitle("Barycentric subdivision changes the triangulation, not the space")



.. image-sg:: /api/gallery/topology/complexes/images/sphx_glr_plot_01_barycentric_subdivision_invariance_001.png
   :alt: Barycentric subdivision changes the triangulation, not the space, sd^0: f = (3, 3, 1), χ = 1, sd^1: f = (7, 12, 6), χ = 1, sd^2: f = (25, 60, 36), χ = 1
   :srcset: /api/gallery/topology/complexes/images/sphx_glr_plot_01_barycentric_subdivision_invariance_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 0.98, 'Barycentric subdivision changes the triangulation, not the space')




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

   **Total running time of the script:** (0 minutes 0.622 seconds)


.. _sphx_glr_download_api_gallery_topology_complexes_plot_01_barycentric_subdivision_invariance.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/complexes/plot_01_barycentric_subdivision_invariance.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_barycentric_subdivision_invariance.ipynb <plot_01_barycentric_subdivision_invariance.ipynb>`

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

      :download:`Download Python source code: plot_01_barycentric_subdivision_invariance.py <plot_01_barycentric_subdivision_invariance.py>`

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

      :download:`Download zipped: plot_01_barycentric_subdivision_invariance.zip <plot_01_barycentric_subdivision_invariance.zip>`


.. only:: html

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

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