.. _sphx_glr_api_gallery_fractals_chaos_ifs:

Iterated function systems
---------------------------

The chaos game: Barnsley's fern and the Sierpinski triangle/carpet.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Sierpiński&#x27;s triangle (1915) removes the middle quarter of a filled triangle and repeats on the three corner triangles; his carpet (1916) removes the middle ninth of a square and repeats on the eight remaining squares. Both limits have zero area and a dimension strictly between 1 and 2. Here both are drawn with the chaos game.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/ifs/images/thumb/sphx_glr_plot_01_sierpinski_triangle_and_carpet_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/ifs/plot_01_sierpinski_triangle_and_carpet`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Sierpiński's triangle and carpet</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Barnsley&#x27;s fern is the attractor of just four contractive affine maps, drawn with the chaos game: start anywhere, and at each step apply one of the maps chosen at random with Barnsley&#x27;s probabilities. Hutchinson&#x27;s theorem says the fern is the unique set equal to the union of its four images under the maps, which the second figure shows directly.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/ifs/images/thumb/sphx_glr_plot_02_barnsley_fern_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/ifs/plot_02_barnsley_fern`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Barnsley's fern: an iterated function system</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/fractals_chaos/ifs/plot_01_sierpinski_triangle_and_carpet
   /api/gallery/fractals_chaos/ifs/plot_02_barnsley_fern