.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/surfaces/plot_03_classification_of_surfaces.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_03_classification_of_surfaces.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_03_classification_of_surfaces.py:


The classification of surfaces (Dehn and Heegaard 1907, Brahana 1921)
=====================================================================

Every compact connected surface is a sphere with handles or with
cross-caps, minus some discs. Three numbers decide which: orientability,
the Euler characteristic, and the number of boundary circles. Here
:func:`~mathematicskit.topology.classify_surface` names nine
triangulated surfaces from those invariants alone.

.. GENERATED FROM PYTHON SOURCE LINES 13-46

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.topology import (
        SimplicialComplex,
        barycentric_subdivision,
        classify_surface,
        klein_bottle,
        mobius_strip,
        projective_plane,
        simplex,
        sphere,
        torus,
    )

    T = torus(6, 4)
    surfaces = {
        "boundary of a tetrahedron": sphere(2),
        "triangle": simplex(2),
        "ring of squares": T.induced_subcomplex([v for v in T.vertices if v // 4 in range(4)]),
        "doughnut grid": T,
        "torus minus a triangle": SimplicialComplex(T.simplices(2)[1:]),
        "twisted band": mobius_strip(),
        "6-vertex RP^2": projective_plane(),
        "subdivided RP^2": barycentric_subdivision(projective_plane()),
        "twisted grid": klein_bottle(),
    }

    rows = []
    for label, K in surfaces.items():
        c = classify_surface(K)
        rows.append([label, f"{K.f_vector}", str(c.euler_characteristic), "yes" if c.orientable else "no", str(c.boundary_components), c.name])
        print(f"{label:>26} -> {c.name}")





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

 .. code-block:: none

     boundary of a tetrahedron -> sphere
                      triangle -> disk
               ring of squares -> annulus
                 doughnut grid -> torus
        torus minus a triangle -> orientable surface of genus 1 with 1 boundary component
                  twisted band -> Möbius strip
                 6-vertex RP^2 -> projective plane
               subdivided RP^2 -> projective plane
                  twisted grid -> Klein bottle




.. GENERATED FROM PYTHON SOURCE LINES 47-49

The invariants side by side
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-58

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(12, 3.6))
    ax.axis("off")
    table = ax.table(cellText=rows, colLabels=["triangulation", "(V, E, F)", "χ", "orientable", "boundary", "surface"], loc="center", cellLoc="left")
    table.auto_set_font_size(False)
    table.set_fontsize(9)
    table.auto_set_column_width(range(6))
    table.scale(1, 1.4)
    ax.set_title("Orientability, χ and boundary circles determine the surface")



.. image-sg:: /api/gallery/topology/surfaces/images/sphx_glr_plot_03_classification_of_surfaces_001.png
   :alt: Orientability, χ and boundary circles determine the surface
   :srcset: /api/gallery/topology/surfaces/images/sphx_glr_plot_03_classification_of_surfaces_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Orientability, χ and boundary circles determine the surface')




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

   **Total running time of the script:** (0 minutes 0.091 seconds)


.. _sphx_glr_download_api_gallery_topology_surfaces_plot_03_classification_of_surfaces.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_03_classification_of_surfaces.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_classification_of_surfaces.ipynb <plot_03_classification_of_surfaces.ipynb>`

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

      :download:`Download Python source code: plot_03_classification_of_surfaces.py <plot_03_classification_of_surfaces.py>`

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

      :download:`Download zipped: plot_03_classification_of_surfaces.zip <plot_03_classification_of_surfaces.zip>`


.. only:: html

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

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