.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/surfaces/plot_02_klein_bottle_torsion.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_surfaces_plot_02_klein_bottle_torsion.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_surfaces_plot_02_klein_bottle_torsion.py:


Klein's bottle and torsion in homology (1882)
=============================================

Felix Klein's bottle glues a cylinder's ends with a reflection. It is a
closed surface with one side and no edge. It cannot sit in
:math:`\mathbb{R}^3` without crossing itself, as the figure-8 immersion
below does. Its first homology group is :math:`\mathbb{Z} \oplus
\mathbb{Z}/2`: a loop that is not a boundary but whose double is. The
:math:`\mathbb{Z}/2` torsion hides from rational Betti numbers and shows
up mod 2.

.. GENERATED FROM PYTHON SOURCE LINES 15-26

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.topology import betti_numbers, homology, is_orientable, klein_bottle, torus
    from mathematicskit.topology.visualizers import plot_complex

    K = klein_bottle(40, 24)
    T = torus(6, 4)
    for name, X in (("torus", T), ("Klein bottle", klein_bottle(6, 4))):
        print(f"{name:>12}: {homology(X)}; orientable = {is_orientable(X)}")
        print(f"{'':>12}  Betti over Q: {betti_numbers(X)}, over Z/2: {betti_numbers(X, field=2)}, over Z/3: {betti_numbers(X, field=3)}")





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

 .. code-block:: none

           torus: H_0 = Z, H_1 = Z^2, H_2 = Z; orientable = True
                  Betti over Q: (1, 2, 1), over Z/2: (1, 2, 1), over Z/3: (1, 2, 1)
    Klein bottle: H_0 = Z, H_1 = Z + Z/2, H_2 = 0; orientable = False
                  Betti over Q: (1, 1, 0), over Z/2: (1, 2, 1), over Z/3: (1, 1, 0)




.. GENERATED FROM PYTHON SOURCE LINES 27-29

The figure-8 immersion
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-37

.. code-block:: Python


    fig = plt.figure(figsize=(7, 6))
    ax = fig.add_subplot(projection="3d")
    heights = K.coordinates[:, 2]
    plot_complex(K, ax=ax, face_values=[heights[list(t)].mean() for t in K.simplices(2)], alpha=0.85, vertex_size=0)
    ax.view_init(elev=30, azim=-50)
    ax.set_axis_off()
    ax.set_title(r"Klein bottle: $H_1 = \mathbb{Z} \oplus \mathbb{Z}/2$")



.. image-sg:: /api/gallery/topology/surfaces/images/sphx_glr_plot_02_klein_bottle_torsion_001.png
   :alt: Klein bottle: $H_1 = \mathbb{Z} \oplus \mathbb{Z}/2$
   :srcset: /api/gallery/topology/surfaces/images/sphx_glr_plot_02_klein_bottle_torsion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Klein bottle: $H_1 = \\mathbb{Z} \\oplus \\mathbb{Z}/2$')




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

   **Total running time of the script:** (0 minutes 0.060 seconds)


.. _sphx_glr_download_api_gallery_topology_surfaces_plot_02_klein_bottle_torsion.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/surfaces/plot_02_klein_bottle_torsion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_klein_bottle_torsion.ipynb <plot_02_klein_bottle_torsion.ipynb>`

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

      :download:`Download Python source code: plot_02_klein_bottle_torsion.py <plot_02_klein_bottle_torsion.py>`

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

      :download:`Download zipped: plot_02_klein_bottle_torsion.zip <plot_02_klein_bottle_torsion.zip>`


.. only:: html

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

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