.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crystal/crystal_systems/plot_02_steno_interfacial_angles.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_crystal_crystal_systems_plot_02_steno_interfacial_angles.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_crystal_systems_plot_02_steno_interfacial_angles.py:


Steno's law: constancy of interfacial angles
=============================================

Nicolas Steno (1669) found that quartz crystals, however unevenly grown,
always meet at the same angles between corresponding faces. The angle
between two faces depends only on their Miller indices -- not on how big
the faces are -- which
:func:`~chemistrykit.crystal.systems.crystal_systems.interplanar_angle_cubic`
computes for a cubic crystal from the face normals.

.. GENERATED FROM PYTHON SOURCE LINES 14-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.crystal.systems.crystal_systems import interplanar_angle_cubic

    pairs = [
        ((1, 0, 0), (0, 1, 0), "cube / cube"),
        ((1, 0, 0), (1, 1, 1), "cube / octahedron"),
        ((1, 1, 1), (1, 1, -1), "octahedron / octahedron"),
        ((1, 0, 0), (1, 1, 0), "cube / dodecahedron"),
        ((1, 1, 0), (1, 1, 1), "dodecahedron / octahedron"),
    ]
    for hkl_1, hkl_2, label in pairs:
        print(f"{str(hkl_1):12s} {str(hkl_2):12s} {label:28s} {interplanar_angle_cubic(hkl_1, hkl_2):7.2f} deg")





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

 .. code-block:: none

    (1, 0, 0)    (0, 1, 0)    cube / cube                    90.00 deg
    (1, 0, 0)    (1, 1, 1)    cube / octahedron              54.74 deg
    (1, 1, 1)    (1, 1, -1)   octahedron / octahedron        70.53 deg
    (1, 0, 0)    (1, 1, 0)    cube / dodecahedron            45.00 deg
    (1, 1, 0)    (1, 1, 1)    dodecahedron / octahedron      35.26 deg




.. GENERATED FROM PYTHON SOURCE LINES 30-34

Steno's point: growing one face bigger than another (a distorted crystal)
changes the face *sizes*, never the angles. Scaling the face normals --
the same faces pushed outward by different amounts -- leaves every
interfacial angle unchanged:

.. GENERATED FROM PYTHON SOURCE LINES 34-41

.. code-block:: Python

    rng = np.random.default_rng(0)
    for _ in range(3):
        s1, s2 = rng.uniform(0.5, 3.0, size=2)
        angle = np.degrees(np.arccos(np.dot(s1 * np.array([1, 0, 0]), s2 * np.array([1, 1, 1])) / (s1 * s2 * np.sqrt(3.0))))
        print(f"face distances {s1:.2f}, {s2:.2f}: cube/octahedron angle = {angle:.4f} deg")
        assert np.isclose(angle, interplanar_angle_cubic((1, 0, 0), (1, 1, 1)))





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

 .. code-block:: none

    face distances 2.09, 1.17: cube/octahedron angle = 54.7356 deg
    face distances 0.60, 0.54: cube/octahedron angle = 54.7356 deg
    face distances 2.53, 2.78: cube/octahedron angle = 54.7356 deg




.. GENERATED FROM PYTHON SOURCE LINES 42-44

A regular and a distorted cuboctahedron-like cross section (the {100}
and {110} faces cut in the (001) plane): same angles, different shapes.

.. GENERATED FROM PYTHON SOURCE LINES 44-62

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(9, 4.5))
    normals = [np.array([np.cos(t), np.sin(t)]) for t in np.radians(np.arange(0, 360, 45))]
    for ax, distances, title in (
        (axes[0], np.ones(8), "regular crystal"),
        (axes[1], np.array([1.0, 0.8, 1.6, 1.2, 1.3, 0.9, 1.1, 1.4]), "distorted crystal"),
    ):
        corners = []
        for i in range(8):
            n1, n2 = normals[i], normals[(i + 1) % 8]
            corners.append(np.linalg.solve(np.array([n1, n2]), np.array([distances[i], distances[(i + 1) % 8]])))
        corners = np.array(corners + [corners[0]])
        ax.plot(corners[:, 0], corners[:, 1], "k-")
        ax.fill(corners[:, 0], corners[:, 1], alpha=0.2)
        ax.set_aspect("equal")
        ax.set_title(f"{title}: every interior angle = 135 deg")
        ax.axis("off")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/crystal_systems/images/sphx_glr_plot_02_steno_interfacial_angles_001.png
   :alt: regular crystal: every interior angle = 135 deg, distorted crystal: every interior angle = 135 deg
   :srcset: /api/gallery/crystal/crystal_systems/images/sphx_glr_plot_02_steno_interfacial_angles_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.030 seconds)


.. _sphx_glr_download_api_gallery_crystal_crystal_systems_plot_02_steno_interfacial_angles.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_steno_interfacial_angles.ipynb <plot_02_steno_interfacial_angles.ipynb>`

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

      :download:`Download Python source code: plot_02_steno_interfacial_angles.py <plot_02_steno_interfacial_angles.py>`

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

      :download:`Download zipped: plot_02_steno_interfacial_angles.zip <plot_02_steno_interfacial_angles.zip>`


.. only:: html

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

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