.. _sphx_glr_api_gallery_fractals_chaos_curves:

Fractal curves
--------------

Nowhere-differentiable and space-filling curves, coastline measurement,
and the similarity dimension.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Plots the Weierstrass function sum a^n cos(b^n pi x) at several zoom levels. The graph looks equally rough at every scale, difference quotients grow without bound as the step shrinks, and the graph&#x27;s measured length keeps growing as it is sampled more finely.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/curves/images/thumb/sphx_glr_plot_01_weierstrass_function_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/curves/plot_01_weierstrass_function`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Weierstrass function: continuous, nowhere differentiable</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Draws the first few Hilbert curves, each visiting every cell of a finer grid. In the limit the curve passes through every point of the square while keeping nearby points of the curve close in the plane.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/curves/images/thumb/sphx_glr_plot_02_hilbert_curve_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/curves/plot_02_hilbert_curve`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Hilbert's space-filling curve</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Builds the Koch snowflake. Each refinement multiplies the perimeter by 4/3, so it grows without bound, while the enclosed area converges to 8/5 of the starting triangle.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/curves/images/thumb/sphx_glr_plot_03_koch_snowflake_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/curves/plot_03_koch_snowflake`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Koch snowflake: infinite perimeter, finite area</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Walks a pair of dividers along a smooth circle and along the Koch curve with ever smaller openings. The circle&#x27;s measured length converges, but the Koch curve&#x27;s grows as a power of the ruler, the scaling Richardson observed for real coastlines and Mandelbrot interpreted as a fractal dimension.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/curves/images/thumb/sphx_glr_plot_04_coastline_paradox_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/curves/plot_04_coastline_paradox`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">How long is the coast of Britain? The divider method</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Solves Moran&#x27;s equation sum r_i^D = 1 for several self-similar sets and compares the answers with box-counting estimates computed from the sets themselves.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/curves/images/thumb/sphx_glr_plot_05_similarity_dimension_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/curves/plot_05_similarity_dimension`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Hausdorff and Moran: the similarity dimension</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Peano&#x27;s 1890 curve divides the square into a 3 \times 3 grid, visits the nine cells in a serpentine order, and repeats the pattern inside each cell. The order-n approximation visits every cell of a 3^n \times 3^n grid exactly once, so in the limit the continuous curve passes through every point of the square. Here the approximations are generated by the standard L-system rewriting rules for Peano&#x27;s curve and drawn with turtle graphics.">

.. only:: html

  .. image:: /api/gallery/fractals_chaos/curves/images/thumb/sphx_glr_plot_06_peano_curve_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/curves/plot_06_peano_curve`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Peano's space-filling curve</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/fractals_chaos/curves/plot_01_weierstrass_function
   /api/gallery/fractals_chaos/curves/plot_02_hilbert_curve
   /api/gallery/fractals_chaos/curves/plot_03_koch_snowflake
   /api/gallery/fractals_chaos/curves/plot_04_coastline_paradox
   /api/gallery/fractals_chaos/curves/plot_05_similarity_dimension
   /api/gallery/fractals_chaos/curves/plot_06_peano_curve