:orphan:

Examples
========

This gallery walks through every public feature of ``mathematicskit.fractals_chaos``:
Lyapunov exponent estimation, box-counting fractal dimension, Mandelbrot/
Julia set generation, iterated function systems, and cellular automata.

See also the narrative tutorial:

- :doc:`/tutorials/chaos_and_fractals`

Each script in this gallery is self-contained and can be run directly with
``python examples/fractals_chaos/<section>/<script>.py``.

Sections
--------

- **lyapunov** -- Lyapunov exponent estimation for 1D maps and flows.
- **box_counting** -- fractal (box-counting) dimension estimation.
- **mandelbrot_julia** -- Mandelbrot and Julia set escape-time generation.
- **ifs** -- iterated function systems: the Barnsley fern and Sierpinski
  triangle/carpet.
- **cellular_automata** -- elementary (Wolfram rule) cellular automata and
  Conway's Game of Life.


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Box-counting dimension
-----------------------

Estimating fractal dimension from a point cloud via box-counting.


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    <div class="sphx-glr-thumbcontainer" tooltip="Generates the Sierpinski triangle via the chaos game and estimates its box-counting dimension, comparing against the known closed-form value \log 3/\log 2 \approx 1.585.">

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  .. image:: /api/gallery/fractals_chaos/box_counting/images/thumb/sphx_glr_plot_01_sierpinski_dimension_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/box_counting/plot_01_sierpinski_dimension`

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      <div class="sphx-glr-thumbnail-title">Box-counting dimension of the Sierpinski triangle</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Starts from [0, 1] and repeatedly removes the open middle third of every remaining interval. After n steps there are 2^n intervals of total length (2/3)^n \to 0, yet the limit set is uncountable. Box counting recovers its fractional dimension \log 2/\log 3 \approx 0.631.">

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  .. image:: /api/gallery/fractals_chaos/box_counting/images/thumb/sphx_glr_plot_02_cantor_set_dimension_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/box_counting/plot_02_cantor_set_dimension`

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      <div class="sphx-glr-thumbnail-title">Cantor's middle-thirds set and its dimension</div>
    </div>


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    </div>

Cellular automata
-------------------

Elementary 1D cellular automata (Wolfram rule numbering) and Conway's
Game of Life.


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    <div class="sphx-glr-thumbcontainer" tooltip="Rule 30 (chaotic) and rule 90 (a discrete Sierpinski triangle) from a single seed cell, plus a glider gun-free Game of Life demonstration using the classic &quot;glider&quot; spaceship.">

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  .. image:: /api/gallery/fractals_chaos/cellular_automata/images/thumb/sphx_glr_plot_01_wolfram_rules_and_life_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/cellular_automata/plot_01_wolfram_rules_and_life`

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      <div class="sphx-glr-thumbnail-title">Elementary cellular automata and Conway's Game of Life</div>
    </div>


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    </div>

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

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


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    <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.">

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  .. 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`

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


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    <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.">

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  .. 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`

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


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    <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.">

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  .. 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`

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


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    <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.">

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  .. 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`

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


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    <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.">

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  .. 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`

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


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    <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.">

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  .. 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`

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


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Aggregation
-----------

Diffusion-limited aggregation: fractal clusters grown by random walkers.


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    <div class="sphx-glr-thumbcontainer" tooltip="Grows a cluster by releasing random walkers that stick on first contact, and measures its fractal dimension from the mass-radius relation N(r) ~ r^D.">

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  .. image:: /api/gallery/fractals_chaos/dla/images/thumb/sphx_glr_plot_01_dla_cluster_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/dla/plot_01_dla_cluster`

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      <div class="sphx-glr-thumbnail-title">Diffusion-limited aggregation</div>
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Iterated function systems
---------------------------

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


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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.">

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  .. 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`

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


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    <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.">

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  .. 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`

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


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    </div>

Lindenmayer systems
-------------------

Parallel string rewriting and turtle graphics for plants and fractal
curves.


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    <div class="sphx-glr-thumbcontainer" tooltip="Runs Lindenmayer&#x27;s 1968 algae system, whose string lengths are Fibonacci numbers, and draws a Koch curve and a branching plant with turtle graphics.">

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  .. image:: /api/gallery/fractals_chaos/lsystems/images/thumb/sphx_glr_plot_01_lindenmayer_plants_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/lsystems/plot_01_lindenmayer_plants`

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      <div class="sphx-glr-thumbnail-title">Lindenmayer systems: from algae to plants</div>
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Lyapunov exponents
-------------------

Estimating the Lyapunov exponent of 1D maps and simple flows.


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    <div class="sphx-glr-thumbcontainer" tooltip="Sweeps the logistic map&#x27;s growth-rate parameter r and estimates the Lyapunov exponent at each value: negative in the periodic windows, crossing zero exactly at each bifurcation, and mostly positive beyond the onset of chaos at r ~ 3.56995.">

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  .. image:: /api/gallery/fractals_chaos/lyapunov/images/thumb/sphx_glr_plot_01_logistic_map_route_to_chaos_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/lyapunov/plot_01_logistic_map_route_to_chaos`

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      <div class="sphx-glr-thumbnail-title">Lyapunov exponent across the logistic map's route to chaos</div>
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Mandelbrot and Julia sets
---------------------------

Escape-time generation of the Mandelbrot set and Julia sets.


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    <div class="sphx-glr-thumbcontainer" tooltip="Iterates the quadratic map z \mapsto z^2 + c from every starting point z_0 of a grid. Points whose orbits stay bounded form the filled Julia set; its boundary, the Julia set, separates the Fatou set, where nearby orbits behave alike, from the region where orbits escape to infinity. Changing c turns the Julia set from a circle into a fractal curve or a disconnected dust.">

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  .. image:: /api/gallery/fractals_chaos/mandelbrot_julia/images/thumb/sphx_glr_plot_01_julia_sets_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/mandelbrot_julia/plot_01_julia_sets`

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      <div class="sphx-glr-thumbnail-title">Fatou and Julia: Julia sets of z -> z^2 + c</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Colours every parameter c by how quickly the orbit of z_0 = 0 under z \mapsto z^2 + c escapes. The black region, the Mandelbrot set, is the set of c whose orbit stays bounded -- equivalently, whose Julia set is connected. Zooming into its boundary reveals ever finer detail, including small copies of the whole set.">

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  .. image:: /api/gallery/fractals_chaos/mandelbrot_julia/images/thumb/sphx_glr_plot_02_mandelbrot_set_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/mandelbrot_julia/plot_02_mandelbrot_set`

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      <div class="sphx-glr-thumbnail-title">The Mandelbrot set</div>
    </div>


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    </div>

Discrete maps
-------------

Two-dimensional maps with strange attractors.


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    <div class="sphx-glr-thumbcontainer" tooltip="Iterates Hénon&#x27;s map (x, y) -&gt; (1 - 1.4 x^2 + y, 0.3 x), zooms in on its strange attractor to reveal repeated layers, and estimates its fractal and Lyapunov characteristics.">

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  .. image:: /api/gallery/fractals_chaos/maps/images/thumb/sphx_glr_plot_01_henon_attractor_thumb.png
    :alt:

  :doc:`/api/gallery/fractals_chaos/maps/plot_01_henon_attractor`

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      <div class="sphx-glr-thumbnail-title">The Hénon attractor</div>
    </div>


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    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/fractals_chaos/box_counting/index.rst
   /api/gallery/fractals_chaos/cellular_automata/index.rst
   /api/gallery/fractals_chaos/curves/index.rst
   /api/gallery/fractals_chaos/dla/index.rst
   /api/gallery/fractals_chaos/ifs/index.rst
   /api/gallery/fractals_chaos/lsystems/index.rst
   /api/gallery/fractals_chaos/lyapunov/index.rst
   /api/gallery/fractals_chaos/mandelbrot_julia/index.rst
   /api/gallery/fractals_chaos/maps/index.rst



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