.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/exhaustion/plot_01_archimedes_pi.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_calculus_exhaustion_plot_01_archimedes_pi.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_exhaustion_plot_01_archimedes_pi.py:


Archimedes' method of exhaustion: squeezing pi
====================================================

Doubles the sides of inscribed and circumscribed regular polygons from
6 to 96, reproducing Archimedes' bounds 3 10/71 < pi < 3 1/7, and shows
that the gap shrinks by a factor of four with each doubling.

.. GENERATED FROM PYTHON SOURCE LINES 11-17

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt

    from mathematicskit.calculus import archimedes_pi_bounds








.. GENERATED FROM PYTHON SOURCE LINES 18-20

The 96-gon bounds
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-26

.. code-block:: Python


    result = archimedes_pi_bounds(4)
    for n, lo, hi in zip(result.sides, result.lower, result.upper):
        print(f"{n:3d} sides: {lo:.6f} < pi < {hi:.6f}")
    print(f"Archimedes: {3 + 10 / 71:.6f} < pi < {3 + 1 / 7:.6f}")





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

 .. code-block:: none

      6 sides: 3.000000 < pi < 3.464102
     12 sides: 3.105829 < pi < 3.215390
     24 sides: 3.132629 < pi < 3.159660
     48 sides: 3.139350 < pi < 3.146086
     96 sides: 3.141032 < pi < 3.142715
    Archimedes: 3.140845 < pi < 3.142857




.. GENERATED FROM PYTHON SOURCE LINES 27-29

Convergence of the gap
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-40

.. code-block:: Python


    result = archimedes_pi_bounds(15)
    gaps = [hi - lo for lo, hi in zip(result.lower, result.upper)]
    fig, ax = plt.subplots()
    ax.loglog(result.sides, gaps, "o-", label="upper - lower")
    ax.loglog(result.sides, [gaps[0] * (6 / n) ** 2 for n in result.sides], "--", label=r"$\propto n^{-2}$")
    ax.axhline(abs(math.pi - 22 / 7), color="0.6", lw=0.8)
    ax.set_xlabel("polygon sides n")
    ax.set_ylabel("width of the bracket on pi")
    ax.legend()
    ax.set_title("Each doubling cuts the gap by 4")



.. image-sg:: /api/gallery/calculus/exhaustion/images/sphx_glr_plot_01_archimedes_pi_001.png
   :alt: Each doubling cuts the gap by 4
   :srcset: /api/gallery/calculus/exhaustion/images/sphx_glr_plot_01_archimedes_pi_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Each doubling cuts the gap by 4')




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

   **Total running time of the script:** (0 minutes 0.047 seconds)


.. _sphx_glr_download_api_gallery_calculus_exhaustion_plot_01_archimedes_pi.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_archimedes_pi.ipynb <plot_01_archimedes_pi.ipynb>`

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

      :download:`Download Python source code: plot_01_archimedes_pi.py <plot_01_archimedes_pi.py>`

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

      :download:`Download zipped: plot_01_archimedes_pi.zip <plot_01_archimedes_pi.zip>`


.. only:: html

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

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