.. _sphx_glr_api_gallery_topology_surfaces:

Surfaces
--------

One-sided surfaces, the Klein bottle, the classification theorem, and critical points.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="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 \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.">

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  .. image:: /api/gallery/topology/surfaces/images/thumb/sphx_glr_plot_01_mobius_one_sided_surface_thumb.png
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  :doc:`/api/gallery/topology/surfaces/plot_01_mobius_one_sided_surface`

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      <div class="sphx-glr-thumbnail-title">Möbius and Listing: the one-sided surface (1858-1865)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Felix Klein&#x27;s bottle glues a cylinder&#x27;s ends with a reflection. It is a closed surface with one side and no edge. It cannot sit in \mathbb{R}^3 without crossing itself, as the figure-8 immersion below does. Its first homology group is \mathbb{Z} \oplus \mathbb{Z}/2: a loop that is not a boundary but whose double is. The \mathbb{Z}/2 torsion hides from rational Betti numbers and shows up mod 2.">

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  .. image:: /api/gallery/topology/surfaces/images/thumb/sphx_glr_plot_02_klein_bottle_torsion_thumb.png
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  :doc:`/api/gallery/topology/surfaces/plot_02_klein_bottle_torsion`

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      <div class="sphx-glr-thumbnail-title">Klein's bottle and torsion in homology (1882)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="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 classify_surface names nine triangulated surfaces from those invariants alone.">

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  .. image:: /api/gallery/topology/surfaces/images/thumb/sphx_glr_plot_03_classification_of_surfaces_thumb.png
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  :doc:`/api/gallery/topology/surfaces/plot_03_classification_of_surfaces`

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      <div class="sphx-glr-thumbnail-title">The classification of surfaces (Dehn and Heegaard 1907, Brahana 1921)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Morse related the critical points of a function on a manifold to its topology: the alternating count of minima, saddles and maxima is the Euler characteristic. Banchoff found the polyhedral version. Charge each simplex to its highest vertex, and a vertex&#x27;s index is the alternating count of what it receives. On an upright torus the height has one minimum, two saddles and one maximum (1 - 2 + 1 = 0). A wobblier height function has more critical points, but the alternating sum stays at 0.">

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  .. image:: /api/gallery/topology/surfaces/images/thumb/sphx_glr_plot_04_morse_banchoff_critical_points_thumb.png
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  :doc:`/api/gallery/topology/surfaces/plot_04_morse_banchoff_critical_points`

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      <div class="sphx-glr-thumbnail-title">Morse theory and Banchoff's critical points (1925-1967)</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/topology/surfaces/plot_01_mobius_one_sided_surface
   /api/gallery/topology/surfaces/plot_02_klein_bottle_torsion
   /api/gallery/topology/surfaces/plot_03_classification_of_surfaces
   /api/gallery/topology/surfaces/plot_04_morse_banchoff_critical_points