.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/complexes/plot_02_kunneth_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_topology_complexes_plot_02_kunneth_formula.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_complexes_plot_02_kunneth_formula.py:


The Künneth formula for products (1923)
=======================================

Hermann Künneth showed that the Betti numbers of a product are a
convolution:
:math:`\beta_k(X \times Y) = \sum_{i+j=k} \beta_i(X)\,\beta_j(Y)`,
the coefficients of the product of the two Poincaré polynomials. The
product of two simplicial complexes is triangulated by staircase paths
through each product of simplices. Circle times circle is the torus,
:math:`(1 + t)^2 = 1 + 2t + t^2`, and circle times sphere gives
:math:`(1 + t)(1 + t^2)`.

.. GENERATED FROM PYTHON SOURCE LINES 16-38

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import betti_numbers, circle, simplex, simplicial_product, sphere


    def poincare_polynomial(betti):
        return " + ".join(f"{b}t^{k}" if k else f"{b}" for k, b in enumerate(betti) if b) or "0"


    pairs = {
        "circle x circle": (circle(3), circle(4)),
        "circle x interval": (circle(4), simplex(1)),
        "circle x sphere": (circle(3), sphere(2)),
        "sphere x sphere": (sphere(2), sphere(2)),
        "torus x circle": (simplicial_product(circle(3), circle(3)), circle(3)),
    }
    for name, (X, Y) in pairs.items():
        product = simplicial_product(X, Y)
        predicted = poincare_polynomial(np.convolve(betti_numbers(X), betti_numbers(Y)))
        print(f"{name:>17}: predicted {predicted:>22}, computed {poincare_polynomial(betti_numbers(product)):>22}, f = {product.f_vector}")





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

 .. code-block:: none

      circle x circle: predicted        1 + 2t^1 + 1t^2, computed        1 + 2t^1 + 1t^2, f = (12, 36, 24)
    circle x interval: predicted               1 + 1t^1, computed               1 + 1t^1, f = (8, 16, 8)
      circle x sphere: predicted 1 + 1t^1 + 1t^2 + 1t^3, computed 1 + 1t^1 + 1t^2 + 1t^3, f = (12, 48, 72, 36)
      sphere x sphere: predicted        1 + 2t^2 + 1t^4, computed        1 + 2t^2 + 1t^4, f = (16, 84, 216, 240, 96)
       torus x circle: predicted 1 + 3t^1 + 3t^2 + 1t^3, computed 1 + 3t^1 + 3t^2 + 1t^3, f = (27, 189, 324, 162)




.. GENERATED FROM PYTHON SOURCE LINES 39-45

A square times a square, drawn as a flat torus
----------------------------------------------

Each square of the grid is a product of two edges, cut into two
triangles along the staircase diagonal. The squares that wrap around
(glued to the opposite side) are left out of the drawing.

.. GENERATED FROM PYTHON SOURCE LINES 45-57

.. code-block:: Python


    T = simplicial_product(circle(4), circle(4))
    grid = np.array([(i, j) for i in range(4) for j in range(4)], dtype=float)
    fig, ax = plt.subplots(figsize=(5, 5))
    for tri in T.simplices(2):
        pts = grid[list(tri)]
        if np.ptp(pts[:, 0]) <= 1 and np.ptp(pts[:, 1]) <= 1:
            ax.add_patch(plt.Polygon(pts, alpha=0.4, ec="k"))
    ax.set_xlim(-0.2, 3.2)
    ax.set_ylim(-0.2, 3.2)
    ax.set_aspect("equal")
    ax.set_title(f"$S^1 \\times S^1$: f = {T.f_vector}, β = {betti_numbers(T)}")



.. image-sg:: /api/gallery/topology/complexes/images/sphx_glr_plot_02_kunneth_formula_001.png
   :alt: $S^1 \times S^1$: f = (16, 48, 32), β = (1, 2, 1)
   :srcset: /api/gallery/topology/complexes/images/sphx_glr_plot_02_kunneth_formula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, '$S^1 \\times S^1$: f = (16, 48, 32), β = (1, 2, 1)')




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

   **Total running time of the script:** (0 minutes 0.113 seconds)


.. _sphx_glr_download_api_gallery_topology_complexes_plot_02_kunneth_formula.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/complexes/plot_02_kunneth_formula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_kunneth_formula.ipynb <plot_02_kunneth_formula.ipynb>`

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

      :download:`Download Python source code: plot_02_kunneth_formula.py <plot_02_kunneth_formula.py>`

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

      :download:`Download zipped: plot_02_kunneth_formula.zip <plot_02_kunneth_formula.zip>`


.. only:: html

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

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