.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geometry/polyhedra/plot_01_euler_polyhedron_formula.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_geometry_polyhedra_plot_01_euler_polyhedron_formula.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_geometry_polyhedra_plot_01_euler_polyhedron_formula.py:


Euler's polyhedron formula: V - E + F = 2
===============================================

Counts vertices, edges, and faces of the five Platonic solids and of
random convex polyhedra. Every one satisfies V - E + F = 2.

.. GENERATED FROM PYTHON SOURCE LINES 10-18

.. code-block:: Python

    import itertools

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.spatial import ConvexHull

    from mathematicskit.geometry import polyhedron_counts








.. GENERATED FROM PYTHON SOURCE LINES 19-21

The Platonic solids
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-35

.. code-block:: Python


    phi = (1 + 5**0.5) / 2
    cube = list(itertools.product((-1, 1), repeat=3))
    solids = {
        "tetrahedron": [(1, 1, 1), (1, -1, -1), (-1, 1, -1), (-1, -1, 1)],
        "cube": cube,
        "octahedron": [tuple(s if k == i else 0 for k in range(3)) for i in range(3) for s in (-1, 1)],
        "dodecahedron": cube + [p for a, b in itertools.product((-1, 1), repeat=2) for p in ((0, a / phi, b * phi), (a / phi, b * phi, 0), (b * phi, 0, a / phi))],
        "icosahedron": [p for a, b in itertools.product((-1, 1), repeat=2) for p in ((0, a, b * phi), (a, b * phi, 0), (b * phi, 0, a))],
    }
    for name, pts in solids.items():
        r = polyhedron_counts(np.array(pts, dtype=float))
        print(f"{name:12s}: V = {r.vertices:2d}, E = {r.edges:2d}, F = {r.faces:2d}, V - E + F = {r.euler_characteristic}")





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

 .. code-block:: none

    tetrahedron : V =  4, E =  6, F =  4, V - E + F = 2
    cube        : V =  8, E = 12, F =  6, V - E + F = 2
    octahedron  : V =  6, E = 12, F =  8, V - E + F = 2
    dodecahedron: V = 20, E = 30, F = 12, V - E + F = 2
    icosahedron : V = 12, E = 30, F = 20, V - E + F = 2




.. GENERATED FROM PYTHON SOURCE LINES 36-38

Random convex polyhedra
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-50

.. code-block:: Python


    rng = np.random.default_rng(0)
    for n in (10, 50, 250):
        r = polyhedron_counts(rng.normal(size=(n, 3)))
        print(f"hull of {n:3d} random points: V = {r.vertices:3d}, E = {r.edges:3d}, F = {r.faces:3d}, V - E + F = {r.euler_characteristic}")

    pts = np.array(solids["dodecahedron"], dtype=float)
    hull = ConvexHull(pts)
    fig = plt.figure()
    ax = fig.add_subplot(projection="3d")
    ax.plot_trisurf(*pts.T, triangles=hull.simplices, alpha=0.5, edgecolor="k", linewidth=0.2)
    ax.set_title("dodecahedron: 20 - 30 + 12 = 2")



.. image-sg:: /api/gallery/geometry/polyhedra/images/sphx_glr_plot_01_euler_polyhedron_formula_001.png
   :alt: dodecahedron: 20 - 30 + 12 = 2
   :srcset: /api/gallery/geometry/polyhedra/images/sphx_glr_plot_01_euler_polyhedron_formula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    hull of  10 random points: V =  10, E =  24, F =  16, V - E + F = 2
    hull of  50 random points: V =  24, E =  66, F =  44, V - E + F = 2
    hull of 250 random points: V =  24, E =  66, F =  44, V - E + F = 2

    Text(0.5, 1.0, 'dodecahedron: 20 - 30 + 12 = 2')




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

   **Total running time of the script:** (0 minutes 0.034 seconds)


.. _sphx_glr_download_api_gallery_geometry_polyhedra_plot_01_euler_polyhedron_formula.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

      :download:`Download Jupyter notebook: plot_01_euler_polyhedron_formula.ipynb <plot_01_euler_polyhedron_formula.ipynb>`

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

      :download:`Download Python source code: plot_01_euler_polyhedron_formula.py <plot_01_euler_polyhedron_formula.py>`

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

      :download:`Download zipped: plot_01_euler_polyhedron_formula.zip <plot_01_euler_polyhedron_formula.zip>`


.. only:: html

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

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