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<ul class="summary">
<li><a href="./"><h3>Chaos Game</h3></a></li>

<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Cover</a></li>
<li class="chapter" data-level="" data-path="how-to-read-this-book.html"><a href="how-to-read-this-book.html"><i class="fa fa-check"></i>How to read this book?</a></li>
<li class="chapter" data-level="1" data-path="hello-fractals.html"><a href="hello-fractals.html"><i class="fa fa-check"></i><b>1</b> Hello, fractals!</a>
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<li class="chapter" data-level="1.1" data-path="wacław-sierpinski.html"><a href="wacław-sierpinski.html"><i class="fa fa-check"></i><b>1.1</b> Wacław Sierpinski</a></li>
<li class="chapter" data-level="1.2" data-path="hello-fractals-1.html"><a href="hello-fractals-1.html"><i class="fa fa-check"></i><b>1.2</b> Hello, fractals!</a>
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<li class="chapter" data-level="1.2.1" data-path="hello-fractals-1.html"><a href="hello-fractals-1.html#cantor-dust"><i class="fa fa-check"></i><b>1.2.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.2.2" data-path="hello-fractals-1.html"><a href="hello-fractals-1.html#sierpinski-triangle"><i class="fa fa-check"></i><b>1.2.2</b> Sierpinski triangle</a></li>
</ul></li>
<li class="chapter" data-level="1.3" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html"><i class="fa fa-check"></i><b>1.3</b> But why does it work?</a>
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<li class="chapter" data-level="1.3.1" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html#topological-dimension"><i class="fa fa-check"></i><b>1.3.1</b> Topological dimension</a></li>
<li class="chapter" data-level="1.3.2" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html#minkowski-dimension-or-fractal-box-counting-dimension"><i class="fa fa-check"></i><b>1.3.2</b> Minkowski dimension, or fractal box-counting dimension</a></li>
<li class="chapter" data-level="1.3.3" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html#sierpinski-carpet"><i class="fa fa-check"></i><b>1.3.3</b> Sierpinski carpet</a></li>
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<li class="chapter" data-level="1.4" data-path="examples-in-python.html"><a href="examples-in-python.html"><i class="fa fa-check"></i><b>1.4</b> Examples in Python</a>
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<li class="chapter" data-level="1.4.1" data-path="examples-in-python.html"><a href="examples-in-python.html#cantor-dust-1"><i class="fa fa-check"></i><b>1.4.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.4.2" data-path="examples-in-python.html"><a href="examples-in-python.html#sierpinski-triangle-1"><i class="fa fa-check"></i><b>1.4.2</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="1.4.3" data-path="examples-in-python.html"><a href="examples-in-python.html#sierpinski-carpet-1"><i class="fa fa-check"></i><b>1.4.3</b> Sierpinski carpet</a></li>
</ul></li>
<li class="chapter" data-level="1.5" data-path="examples-in-r.html"><a href="examples-in-r.html"><i class="fa fa-check"></i><b>1.5</b> Examples in R</a>
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<li class="chapter" data-level="1.5.1" data-path="examples-in-r.html"><a href="examples-in-r.html#cantor-dust-2"><i class="fa fa-check"></i><b>1.5.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.5.2" data-path="examples-in-r.html"><a href="examples-in-r.html#sierpinski-triangle-2"><i class="fa fa-check"></i><b>1.5.2</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="1.5.3" data-path="examples-in-r.html"><a href="examples-in-r.html#sierpinski-carpet-2"><i class="fa fa-check"></i><b>1.5.3</b> Sierpinski carpet</a></li>
</ul></li>
<li class="chapter" data-level="1.6" data-path="examples-in-julia.html"><a href="examples-in-julia.html"><i class="fa fa-check"></i><b>1.6</b> Examples in Julia</a>
<ul>
<li class="chapter" data-level="1.6.1" data-path="examples-in-julia.html"><a href="examples-in-julia.html#cantor-dust-3"><i class="fa fa-check"></i><b>1.6.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.6.2" data-path="examples-in-julia.html"><a href="examples-in-julia.html#sierpinski-triangle-3"><i class="fa fa-check"></i><b>1.6.2</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="1.6.3" data-path="examples-in-julia.html"><a href="examples-in-julia.html#sierpinski-carpet-3"><i class="fa fa-check"></i><b>1.6.3</b> Sierpinski carpet</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="2" data-path="fractal-that-is-fixed-point.html"><a href="fractal-that-is-fixed-point.html"><i class="fa fa-check"></i><b>2</b> Fractal, that is fixed point</a>
<ul>
<li class="chapter" data-level="2.1" data-path="stefan-banach.html"><a href="stefan-banach.html"><i class="fa fa-check"></i><b>2.1</b> Stefan Banach</a></li>
<li class="chapter" data-level="2.2" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html"><i class="fa fa-check"></i><b>2.2</b> Fractal, that is fixed point</a>
<ul>
<li class="chapter" data-level="2.2.1" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#dragons-and-ferns"><i class="fa fa-check"></i><b>2.2.1</b> Dragons and ferns</a></li>
<li class="chapter" data-level="2.2.2" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#sierpinski-pentagon"><i class="fa fa-check"></i><b>2.2.2</b> Sierpinski pentagon</a></li>
<li class="chapter" data-level="2.2.3" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#pythagoras-tree"><i class="fa fa-check"></i><b>2.2.3</b> Pythagoras tree</a></li>
<li class="chapter" data-level="2.2.4" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#heighway-dragon"><i class="fa fa-check"></i><b>2.2.4</b> Heighway Dragon</a></li>
<li class="chapter" data-level="2.2.5" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#barnsley-fern"><i class="fa fa-check"></i><b>2.2.5</b> Barnsley fern</a></li>
</ul></li>
<li class="chapter" data-level="2.3" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html"><i class="fa fa-check"></i><b>2.3</b> But why does it work?</a>
<ul>
<li class="chapter" data-level="2.3.1" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#metric-space"><i class="fa fa-check"></i><b>2.3.1</b> Metric space</a></li>
<li class="chapter" data-level="2.3.2" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#hausdorff-distance"><i class="fa fa-check"></i><b>2.3.2</b> Hausdorff distance</a></li>
<li class="chapter" data-level="2.3.3" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#cauchy-sequence"><i class="fa fa-check"></i><b>2.3.3</b> Cauchy sequence</a></li>
<li class="chapter" data-level="2.3.4" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#complete-space"><i class="fa fa-check"></i><b>2.3.4</b> Complete space</a></li>
<li class="chapter" data-level="2.3.5" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#contraction"><i class="fa fa-check"></i><b>2.3.5</b> Contraction</a></li>
<li class="chapter" data-level="2.3.6" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#affine-transformation"><i class="fa fa-check"></i><b>2.3.6</b> Affine transformation</a></li>
<li class="chapter" data-level="2.3.7" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#hutchinsons-theorem"><i class="fa fa-check"></i><b>2.3.7</b> Hutchinson’s theorem</a></li>
<li class="chapter" data-level="2.3.8" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#banachs-fixed-point-theorem"><i class="fa fa-check"></i><b>2.3.8</b> Banach’s fixed point theorem</a></li>
</ul></li>
<li class="chapter" data-level="2.4" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html"><i class="fa fa-check"></i><b>2.4</b> Examples in Python</a>
<ul>
<li class="chapter" data-level="2.4.1" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#sierpinski-triangle-4"><i class="fa fa-check"></i><b>2.4.1</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="2.4.2" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#sierpinski-pentagon-1"><i class="fa fa-check"></i><b>2.4.2</b> Sierpinski pentagon</a></li>
<li class="chapter" data-level="2.4.3" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#heighways-dragon"><i class="fa fa-check"></i><b>2.4.3</b> Heighway’s Dragon</a></li>
<li class="chapter" data-level="2.4.4" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#symmetric-binary-tree-pythagoras-tree"><i class="fa fa-check"></i><b>2.4.4</b> Symmetric binary tree / Pythagoras’ tree</a></li>
</ul></li>
<li class="chapter" data-level="2.5" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html"><i class="fa fa-check"></i><b>2.5</b> Examples in R</a>
<ul>
<li class="chapter" data-level="2.5.1" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#sierpiński-triangle"><i class="fa fa-check"></i><b>2.5.1</b> Sierpiński triangle</a></li>
<li class="chapter" data-level="2.5.2" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#sierpiński-pentagon"><i class="fa fa-check"></i><b>2.5.2</b> Sierpiński pentagon</a></li>
<li class="chapter" data-level="2.5.3" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#heighways-dragon-1"><i class="fa fa-check"></i><b>2.5.3</b> Heighway’s Dragon</a></li>
<li class="chapter" data-level="2.5.4" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#symmetric-binary-tree-pythagoras-tree-1"><i class="fa fa-check"></i><b>2.5.4</b> Symmetric binary tree / Pythagoras’ tree</a></li>
</ul></li>
<li class="chapter" data-level="2.6" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html"><i class="fa fa-check"></i><b>2.6</b> Examples in Julia</a>
<ul>
<li class="chapter" data-level="2.6.1" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#sierpinski-triangle-5"><i class="fa fa-check"></i><b>2.6.1</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="2.6.2" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#sierpinski-pentagon-2"><i class="fa fa-check"></i><b>2.6.2</b> Sierpinski pentagon</a></li>
<li class="chapter" data-level="2.6.3" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#heighways-dragon-2"><i class="fa fa-check"></i><b>2.6.3</b> Heighway’s Dragon</a></li>
<li class="chapter" data-level="2.6.4" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#symmetric-binary-tree-pythagoras-tree-2"><i class="fa fa-check"></i><b>2.6.4</b> Symmetric binary tree / Pythagoras’ tree</a></li>
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<li class="chapter" data-level="3" data-path="chaos-game.html"><a href="chaos-game.html"><i class="fa fa-check"></i><b>3</b> Chaos Game</a>
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<li class="chapter" data-level="3.1" data-path="hugo-steinhaus.html"><a href="hugo-steinhaus.html"><i class="fa fa-check"></i><b>3.1</b> Hugo Steinhaus</a></li>
<li class="chapter" data-level="3.2" data-path="chaos-game-1.html"><a href="chaos-game-1.html"><i class="fa fa-check"></i><b>3.2</b> Chaos Game</a>
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<li class="chapter" data-level="3.2.1" data-path="chaos-game-1.html"><a href="chaos-game-1.html#chaos-game-and-triangles"><i class="fa fa-check"></i><b>3.2.1</b> Chaos game and triangles</a></li>
<li class="chapter" data-level="3.2.2" data-path="chaos-game-1.html"><a href="chaos-game-1.html#whats-going-on-here"><i class="fa fa-check"></i><b>3.2.2</b> What’s going on here?</a></li>
<li class="chapter" data-level="3.2.3" data-path="chaos-game-1.html"><a href="chaos-game-1.html#chaos-game-and-fractals"><i class="fa fa-check"></i><b>3.2.3</b> Chaos game and fractals</a></li>
<li class="chapter" data-level="3.2.4" data-path="chaos-game-1.html"><a href="chaos-game-1.html#affine-transformations"><i class="fa fa-check"></i><b>3.2.4</b> Affine transformations</a></li>
<li class="chapter" data-level="3.2.5" data-path="chaos-game-1.html"><a href="chaos-game-1.html#ferns-and-chaos"><i class="fa fa-check"></i><b>3.2.5</b> Ferns and chaos</a></li>
<li class="chapter" data-level="3.2.6" data-path="chaos-game-1.html"><a href="chaos-game-1.html#more-formally"><i class="fa fa-check"></i><b>3.2.6</b> More formally</a></li>
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<li class="chapter" data-level="3.3" data-path="applications.html"><a href="applications.html"><i class="fa fa-check"></i><b>3.3</b> Applications</a>
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<li class="chapter" data-level="3.3.1" data-path="applications.html"><a href="applications.html#chaos-game-and-genetics"><i class="fa fa-check"></i><b>3.3.1</b> Chaos game and genetics</a></li>
<li class="chapter" data-level="3.3.2" data-path="applications.html"><a href="applications.html#chaos-game-and-music"><i class="fa fa-check"></i><b>3.3.2</b> Chaos game and music</a></li>
<li class="chapter" data-level="3.3.3" data-path="applications.html"><a href="applications.html#the-fractal-dimension-of-the-brain"><i class="fa fa-check"></i><b>3.3.3</b> The fractal dimension of the brain</a></li>
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<li class="chapter" data-level="3.4" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html"><i class="fa fa-check"></i><b>3.4</b> Examples in Python</a>
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<li class="chapter" data-level="3.4.1" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#the-sierpinski-triangle-once-again"><i class="fa fa-check"></i><b>3.4.1</b> The Sierpinski triangle once again</a></li>
<li class="chapter" data-level="3.4.2" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#barnsley-fern-1"><i class="fa fa-check"></i><b>3.4.2</b> Barnsley fern</a></li>
<li class="chapter" data-level="3.4.3" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#maple-leaf"><i class="fa fa-check"></i><b>3.4.3</b> Maple leaf</a></li>
<li class="chapter" data-level="3.4.4" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#spiral"><i class="fa fa-check"></i><b>3.4.4</b> Spiral</a></li>
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<li class="chapter" data-level="3.5" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html"><i class="fa fa-check"></i><b>3.5</b> Examples in R</a>
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<li class="chapter" data-level="3.5.1" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#the-sierpinski-triangle-once-again-1"><i class="fa fa-check"></i><b>3.5.1</b> The Sierpinski triangle once again</a></li>
<li class="chapter" data-level="3.5.2" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#barnsley-fern-2"><i class="fa fa-check"></i><b>3.5.2</b> Barnsley fern</a></li>
<li class="chapter" data-level="3.5.3" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#maple-leaf-1"><i class="fa fa-check"></i><b>3.5.3</b> Maple leaf</a></li>
<li class="chapter" data-level="3.5.4" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#spiral-1"><i class="fa fa-check"></i><b>3.5.4</b> Spiral</a></li>
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<li class="chapter" data-level="3.6" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html"><i class="fa fa-check"></i><b>3.6</b> Examples in Julia</a>
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<li class="chapter" data-level="3.6.1" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#the-sierpinski-triangle-once-again-2"><i class="fa fa-check"></i><b>3.6.1</b> The Sierpinski triangle once again</a></li>
<li class="chapter" data-level="3.6.2" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#barnsley-fern-3"><i class="fa fa-check"></i><b>3.6.2</b> Barnsley fern</a></li>
<li class="chapter" data-level="3.6.3" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#maple-leaf-2"><i class="fa fa-check"></i><b>3.6.3</b> Maple leaf</a></li>
<li class="chapter" data-level="3.6.4" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#spiral-2"><i class="fa fa-check"></i><b>3.6.4</b> Spiral</a></li>
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<li class="chapter" data-level="4" data-path="pocket-atlas-of-fractals.html"><a href="pocket-atlas-of-fractals.html"><i class="fa fa-check"></i><b>4</b> Pocket atlas of fractals</a>
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<li class="chapter" data-level="4.1" data-path="barnsley-fern-4.html"><a href="barnsley-fern-4.html"><i class="fa fa-check"></i><b>4.1</b> Barnsley fern</a></li>
<li class="chapter" data-level="4.2" data-path="maple-leaf-3.html"><a href="maple-leaf-3.html"><i class="fa fa-check"></i><b>4.2</b> Maple leaf</a></li>
<li class="chapter" data-level="4.3" data-path="mcworters-pentigree.html"><a href="mcworters-pentigree.html"><i class="fa fa-check"></i><b>4.3</b> McWorter’s pentigree</a></li>
<li class="chapter" data-level="4.4" data-path="trees.html"><a href="trees.html"><i class="fa fa-check"></i><b>4.4</b> Trees</a></li>
<li class="chapter" data-level="4.5" data-path="spiral-3.html"><a href="spiral-3.html"><i class="fa fa-check"></i><b>4.5</b> Spiral</a></li>
<li class="chapter" data-level="4.6" data-path="sierpinski-polygons.html"><a href="sierpinski-polygons.html"><i class="fa fa-check"></i><b>4.6</b> Sierpinski polygons</a></li>
<li class="chapter" data-level="4.7" data-path="dragons.html"><a href="dragons.html"><i class="fa fa-check"></i><b>4.7</b> Dragons</a></li>
<li class="chapter" data-level="4.8" data-path="sky-is-the-limit.html"><a href="sky-is-the-limit.html"><i class="fa fa-check"></i><b>4.8</b> Sky is the limit</a></li>
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<li class="chapter" data-level="" data-path="bibliography.html"><a href="bibliography.html"><i class="fa fa-check"></i>Bibliography</a></li>
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<li><a href="https://www.mi2.ai/beta-bit.html" target="_blank">Other books in the Beta and Bit series</a></li>

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<h2><span class="header-section-number">4.7</span> Dragons<a href="dragons.html#dragons" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p><img src="images/ifs_smoki_25.png" /></p>
<p><img src="images/ifs_smoki_75.png" /></p>
<p>In the second chapter we learned about the Heighway dragon and the Pythagorean tree. It turns out that both of these fractals belong to a richer family that can be parametrized by a single number. Such a pair can be defined for each <span class="math inline">\(T \in [0, 1]\)</span>.</p>
<p><span class="math display">\[
f_1(x, y) =   \frac{1}{\sqrt{2}}
\begin{bmatrix}
\cos(\pi/4) &amp; -\sin(\pi/4) \\
\sin(\pi/4) &amp;  \cos(\pi/4)
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}
\]</span>
<span class="math display">\[
f_2(x, y) =   \frac{1}{\sqrt{2}}
\begin{bmatrix}
\cos\left(\frac 34 \pi -  \frac{T}2 \pi\right) &amp; -\sin\left(\frac 34 \pi -  \frac{T}2 \pi\right) \\
\sin\left(\frac 34 \pi -  \frac{T}2 \pi\right) &amp;  \cos\left(\frac 34 \pi -  \frac{T}2 \pi\right)
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}  +
\begin{bmatrix}
1 - T/2 \\
T/2
\end{bmatrix}
\]</span></p>
<p>Six fractals from this family are shown below.</p>
<p><img src="images/fraktale_smoki.png" /></p>
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