.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/surfaces/plot_01_mobius_one_sided_surface.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_01_mobius_one_sided_surface.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_01_mobius_one_sided_surface.py:


Möbius and Listing: the one-sided surface (1858-1865)
=====================================================

Glue the ends of a strip with a half twist and the surface has only one
side. In triangle terms, its triangles cannot all be oriented so that
neighbours traverse their shared edge in opposite directions. Propagating
orientations around the strip comes back reversed. The untwisted band
(an annulus) has no such obstruction. Both have :math:`\chi = 0`, so
orientability is what tells them apart, along with the number of
boundary circles: one for the Möbius strip, two for the annulus.

.. GENERATED FROM PYTHON SOURCE LINES 15-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import SimplicialComplex, boundary_components, is_orientable, mobius_strip, torus
    from mathematicskit.topology.visualizers import plot_complex

    M = mobius_strip(n=24, m=3, width=0.8)
    T = torus(24, 4)
    annulus = T.induced_subcomplex([v for v in T.vertices if v % 4 in (0, 1, 2)])
    annulus = SimplicialComplex(annulus.simplices(), T.coordinates)
    for name, K in (("Möbius strip", M), ("annulus", annulus)):
        print(f"{name:>13}: chi = {K.euler_characteristic}, orientable = {is_orientable(K)}, boundary circles = {boundary_components(K)}")





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

 .. code-block:: none

     Möbius strip: chi = 0, orientable = False, boundary circles = 1
          annulus: chi = 0, orientable = True, boundary circles = 2




.. GENERATED FROM PYTHON SOURCE LINES 29-34

A walk along the centre line comes back upside down
---------------------------------------------------

Carrying the normal vector of the strip once around the centre line
returns it pointing the other way; for the annulus it returns unchanged.

.. GENERATED FROM PYTHON SOURCE LINES 34-52

.. code-block:: Python


    u = np.linspace(0, 2 * np.pi, 13)
    fig = plt.figure(figsize=(12, 5))
    for i, (name, K, twist) in enumerate((("Möbius strip", M, 0.5), ("annulus", annulus, 0.0))):
        ax = fig.add_subplot(1, 2, i + 1, projection="3d", computed_zorder=False)
        plot_complex(K, ax=ax, alpha=0.3, vertex_size=0)
        if twist:
            center = np.column_stack([np.cos(u), np.sin(u), 0 * u])
            normal = np.column_stack([-np.sin(twist * u) * np.cos(u), -np.sin(twist * u) * np.sin(u), np.cos(twist * u)])
        else:
            center = np.column_stack([2.0 * np.cos(u), 2.0 * np.sin(u), 1.0 + 0 * u])
            normal = np.column_stack([0 * u, 0 * u, 1 + 0 * u])
        colors = plt.cm.coolwarm(np.linspace(0, 1, len(u)))
        ax.quiver(*center.T, *(0.5 * normal).T, colors=colors, lw=2)
        ax.set_title(f"{name}: orientable = {is_orientable(K)}")
        ax.view_init(elev=35, azim=-60)
        ax.set_axis_off()
    fig.suptitle("Möbius (1865): a surface with one side cannot be oriented")



.. image-sg:: /api/gallery/topology/surfaces/images/sphx_glr_plot_01_mobius_one_sided_surface_001.png
   :alt: Möbius (1865): a surface with one side cannot be oriented, Möbius strip: orientable = False, annulus: orientable = True
   :srcset: /api/gallery/topology/surfaces/images/sphx_glr_plot_01_mobius_one_sided_surface_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 0.98, 'Möbius (1865): a surface with one side cannot be oriented')




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

   **Total running time of the script:** (0 minutes 0.037 seconds)


.. _sphx_glr_download_api_gallery_topology_surfaces_plot_01_mobius_one_sided_surface.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_01_mobius_one_sided_surface.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_mobius_one_sided_surface.ipynb <plot_01_mobius_one_sided_surface.ipynb>`

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

      :download:`Download Python source code: plot_01_mobius_one_sided_surface.py <plot_01_mobius_one_sided_surface.py>`

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

      :download:`Download zipped: plot_01_mobius_one_sided_surface.zip <plot_01_mobius_one_sided_surface.zip>`


.. only:: html

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

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