.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/homology/plot_02_betti_numbers.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_02_betti_numbers.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_02_betti_numbers.py:


Betti's connectivity numbers (1871)
===================================

Enrico Betti counted the independent closed curves, surfaces, and so
on, that do not bound in a space. In Poincaré's later algebra these are
the ranks :math:`\beta_k = \dim\ker\partial_k - \operatorname{rk}\partial_{k+1}`:
cycles minus boundaries. :math:`\beta_0` counts pieces, :math:`\beta_1`
independent loops, and :math:`\beta_2` enclosed cavities. The bars show
the bookkeeping for each space: chains :math:`f_k`, cycles, and
boundaries.

.. GENERATED FROM PYTHON SOURCE LINES 15-48



.. image-sg:: /api/gallery/topology/homology/images/sphx_glr_plot_02_betti_numbers_001.png
   :alt: Betti numbers: cycles that are not boundaries, two points: β = (2,), circle: β = (1, 1), theta graph: β = (1, 2), sphere: β = (1, 0, 1), torus: β = (1, 2, 1), sphere + circle: β = (2, 1, 1)
   :srcset: /api/gallery/topology/homology/images/sphx_glr_plot_02_betti_numbers_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

          two points: f = (2,), Betti numbers = (2,)
              circle: f = (6, 6), Betti numbers = (1, 1)
         theta graph: f = (5, 6), Betti numbers = (1, 2)
              sphere: f = (4, 6, 4), Betti numbers = (1, 0, 1)
               torus: f = (24, 72, 48), Betti numbers = (1, 2, 1)
     sphere + circle: f = (8, 10, 4), Betti numbers = (2, 1, 1)

    Text(0.5, 0.98, 'Betti numbers: cycles that are not boundaries')





|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import SimplicialComplex, betti_numbers, circle, sphere, torus

    theta = SimplicialComplex([[0, 1], [1, 2], [2, 3], [3, 0], [0, 4], [4, 2]])
    spaces = {
        "two points": SimplicialComplex([[0], [1]]),
        "circle": circle(6),
        "theta graph": theta,
        "sphere": sphere(2),
        "torus": torus(6, 4),
        "sphere + circle": sphere(2) | SimplicialComplex([[i + 10, (i + 1) % 4 + 10] for i in range(4)]),
    }

    fig, axes = plt.subplots(2, 3, figsize=(12, 6.5))
    for ax, (name, K) in zip(axes.ravel(), spaces.items(), strict=True):
        ranks = [np.linalg.matrix_rank(K.boundary_matrix(k)) if K.boundary_matrix(k).size else 0 for k in range(K.dimension + 2)]
        f = np.array(K.f_vector)
        cycles = f - np.array(ranks[: len(f)])
        boundaries = np.array(ranks[1 : len(f) + 1])
        betti = betti_numbers(K)
        print(f"{name:>16}: f = {K.f_vector}, Betti numbers = {betti}")
        k = np.arange(len(f))
        ax.bar(k - 0.25, cycles, 0.25, label="cycles: dim ker ∂_k")
        ax.bar(k, boundaries, 0.25, label="boundaries: rk ∂_(k+1)")
        ax.bar(k + 0.25, betti, 0.25, label="β_k", color="k")
        for i, b in enumerate(betti):
            ax.annotate(str(b), (i + 0.25, b), ha="center", va="bottom", fontweight="bold")
        ax.set_xticks(k, [f"k={i}" for i in k])
        ax.set_title(f"{name}: β = {betti}")
    axes[0, 0].legend(fontsize=8)
    fig.suptitle("Betti numbers: cycles that are not boundaries")


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

   **Total running time of the script:** (0 minutes 0.102 seconds)


.. _sphx_glr_download_api_gallery_topology_homology_plot_02_betti_numbers.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_02_betti_numbers.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_betti_numbers.ipynb <plot_02_betti_numbers.ipynb>`

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

      :download:`Download Python source code: plot_02_betti_numbers.py <plot_02_betti_numbers.py>`

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

      :download:`Download zipped: plot_02_betti_numbers.zip <plot_02_betti_numbers.zip>`


.. only:: html

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

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