.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/homology/plot_03_poincare_homology_euler_formula.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_homology_plot_03_poincare_homology_euler_formula.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_homology_plot_03_poincare_homology_euler_formula.py:


Poincaré's Analysis Situs: homology and the Euler-Poincaré formula (1895-1900)
==============================================================================

Poincaré made Betti's numbers a theory. Chains of simplices, their
boundaries, and the homology groups :math:`H_k = \ker\partial_k / \operatorname{im}\partial_{k+1}`,
together with torsion coefficients. One consequence generalizes Euler:
the alternating count of simplices equals the alternating sum of Betti
numbers,

.. math:: \sum_k (-1)^k f_k = \sum_k (-1)^k \beta_k ,

so :math:`\chi` depends only on the space, not on the triangulation. Below,
many triangulations of five spaces with wildly different simplex counts
land on the same :math:`\chi`.

.. GENERATED FROM PYTHON SOURCE LINES 19-44



.. image-sg:: /api/gallery/topology/homology/images/sphx_glr_plot_03_poincare_homology_euler_formula_001.png
   :alt: The Euler characteristic is a topological invariant
   :srcset: /api/gallery/topology/homology/images/sphx_glr_plot_03_poincare_homology_euler_formula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

              sphere: H_0 = Z, H_1 = 0, H_2 = Z  (chi = 2)
              circle: H_0 = Z, H_1 = Z  (chi = 0)
               torus: H_0 = Z, H_1 = Z^2, H_2 = Z  (chi = 0)
        Klein bottle: H_0 = Z, H_1 = Z + Z/2, H_2 = 0  (chi = 0)
    projective plane: H_0 = Z, H_1 = Z/2, H_2 = 0  (chi = 1)

    Text(0.5, 1.0, 'The Euler characteristic is a topological invariant')





|

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.topology import circle, homology, klein_bottle, projective_plane, sphere, torus

    families = {
        "sphere": [sphere(2), sphere(2, "cross_polytope")],
        "circle": [circle(n) for n in (3, 5, 9, 17)],
        "torus": [torus(n, m) for n, m in ((3, 3), (5, 4), (8, 6), (12, 9))],
        "Klein bottle": [klein_bottle(n, m) for n, m in ((3, 3), (6, 4), (10, 7))],
        "projective plane": [projective_plane()],
    }

    fig, ax = plt.subplots(figsize=(9, 5))
    for i, (name, complexes) in enumerate(families.items()):
        for K in complexes:
            H = homology(K)
            assert H.euler_characteristic == K.euler_characteristic
            ax.scatter(len(K), K.euler_characteristic, s=50, color=f"C{i}", label=name if K is complexes[0] else None)
        print(f"{name:>16}: {H}  (chi = {H.euler_characteristic})")
    ax.set_xscale("log")
    ax.set_xlabel("number of simplices in the triangulation")
    ax.set_ylabel(r"$\chi = \sum (-1)^k f_k = \sum (-1)^k \beta_k$")
    ax.set_yticks([0, 1, 2])
    ax.legend()
    ax.set_title("The Euler characteristic is a topological invariant")


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

   **Total running time of the script:** (0 minutes 0.539 seconds)


.. _sphx_glr_download_api_gallery_topology_homology_plot_03_poincare_homology_euler_formula.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/homology/plot_03_poincare_homology_euler_formula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_poincare_homology_euler_formula.ipynb <plot_03_poincare_homology_euler_formula.ipynb>`

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

      :download:`Download Python source code: plot_03_poincare_homology_euler_formula.py <plot_03_poincare_homology_euler_formula.py>`

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

      :download:`Download zipped: plot_03_poincare_homology_euler_formula.zip <plot_03_poincare_homology_euler_formula.zip>`


.. only:: html

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

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