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‎SUMMARY.md

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* [Plotting](contents/plotting/plotting.md)
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* [Domain Coloring](contents/domain_coloring/domain_coloring.md)
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* [Iterated Function Systems](contents/IFS/IFS.md)
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* [Fractal Flame](contents/fractal_flame/fractal_flame.md)
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* [Data Structures](contents/data_structures/data_structures.md)
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* [Stacks and Queues](contents/stacks_and_queues/stacks_and_queues.md)
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* [Mathematical Background](contents/mathematical_background/mathematical_background.md)
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* [Matrix Methods](contents/matrix_methods/matrix_methods.md)
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* [Gaussian Elimination](contents/gaussian_elimination/gaussian_elimination.md)
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* [Thomas Algorithm](contents/thomas_algorithm/thomas_algorithm.md)
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* [Convolutions](contents/convolutions/convolutions.md)
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* [Signal Processing](contents/signal_processing/signal_processing.md)
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* [Computational Geometry](contents/computational_geometry/computational_geometry.md)
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* [Gift Wrapping](contents/gift_wrapping/gift_wrapping.md)
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* [Jarvis March](contents/jarvis_march/jarvis_march.md)

‎contents/IFS/IFS.md

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##### Text
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The text of this chapter was written by [James Schloss](https://github.com/leio) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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[<p><img class="center" src="../cc/CC-BY-SA_icon.svg" /></p>](https://creativecommons.org/licenses/by-sa/4.0/)
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‎contents/computus/computus.md

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##### Text
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The text of this chapter was written by [James Schloss](https://github.com/leio) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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[<p><img class="center" src="../cc/CC-BY-SA_icon.svg" /></p>](https://creativecommons.org/licenses/by-sa/4.0/)
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‎contents/correlations/correlations.md

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# Correlations
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In the world of signal processing, there are plenty of situations where one might wish to determine how similar (or correlated) different signals might be.
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The most straightforward method to do this might be to multiply two signals together.
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For example, take two sine waves super-imposed on one another:
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ADD IMAGE
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Here, we show the two sine waves and their product.
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‎contents/flood_fill/flood_fill.md

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The text of this chapter was written by [James Schloss](https://github.com/leio) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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[<p><img class="center" src="../cc/CC-BY-SA_icon.svg" /></p>](https://creativecommons.org/licenses/by-sa/4.0/)
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‎contents/fractal_flame/fractal_flame.md

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# Fractal Flame
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To begin this chapter, please read through the chapter on [Iterated Function Systems](../IFS/IFS.md) where we discuss methods through which fractals and fractal-like patterns can be generated with an assortment of functions known as the Hutchinson operator:
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$$
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H(P) = \bigcup_{i=1}^Nf_i(P).
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$$
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Here, $$P$$ represents a point in some coordinate system, and $$\bigcup_{i=1}^Nf_i(P)$$ signifies a union of $$N$$ functions ($$f_i(P)$$) that act on that point to create new, unique patterns.
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The classical example is the Sierpinski triangle, where the Hutchinson operator encompasses three distinct functions:
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$$
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\begin{align}
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f_1(P) &= \frac{P + A}{2}\\
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f_2(P) &= \frac{P + B}{2}\\
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f_3(P) &= \frac{P + C}{2}\\
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\end{align}
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$$
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Here, $$A$$, $$B$$, and $$C$$ are three arbitrarily chosen points on a two-dimensional plane ($$\mathcal{R}^2$$).
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There are many ways to iterate through this function system, but the most common method is with a "chaos game."
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This process works by choosing a random point on the plane and and then randomly selecting any function it every iteration.
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This will chaotically produce the an image of a triangle of triangles shown below:
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This is the Sierpinsky triangle and might be one of the most famous fractals currently known.
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To make sure we are all on the same page, the code to generate this triangle can be seen here:
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The idea of a chaos game will be used throughout this chapter to recreate similar fractal-like structures; however, the functions have much, much more variation than those shown for the Sierpinski triangle.
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In addition, several other concepts (like [convolutions](../signal_processing/signal_processing.md)) are used to make sure the resulting image is visually ... .
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To start, let's discuss the breadth of available functions and use them in the chaos game before.
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After that, we will discuss several image manipulation strategies make the resulting images really *pop*!
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## Possible variations
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In the fractal flame method, each function is known as a *variation* and there are at least 48 outlined in the original paper {{ "draves2003fractal" | cite }}.
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For the purposes of this chapter, we will not be covering all 48, but will instead cover a subset that allows for the greatest understanding of all possible variations allowed by this method.
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## Image processing
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## Example Code
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{% method %}
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{% sample lang="jl" %}
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[import, lang:"julia"](code/julia/IFS.jl)
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{% endmethod %}
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### Bibliography
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{% references %} {% endreferences %}
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<script>
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MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
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</script>
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## License
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##### Code Examples
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The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)).
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##### Text
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The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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[<p><img class="center" src="../cc/CC-BY-SA_icon.svg" /></p>](https://creativecommons.org/licenses/by-sa/4.0/)
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#### Images/Graphics
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- The image "[IFS triangle 1](res/IFS_triangle_1.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).

‎contents/signal_processing/signal_processing.md

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# Signal Processing
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Signal processing is the art of taking a signal and processing it into something more easily understandable.
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Not only is this a field of study unto itself, but it is also the heart of many computational and experimental research.
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To be completely honest, this chapter is a bit of a stub.
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There is a lot of content to cover in this area, but I need some discussion of signal processing to cover other methods in the Archive right now, so I will put the necessary information here and restructure the chapter later when I can look into everything in more depth.
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For now, please excuse the fact that we have an entire section on signal processing, but only really cover two related topics: cross-correlations and convolutions.
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## Cross-Correlations
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## Convolutions
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## Example Code
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{% method %}
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{% sample lang="jl" %}
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[import, lang:"julia"](code/julia/IFS.jl)
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{% endmethod %}
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### Bibliography
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{% references %} {% endreferences %}
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<script>
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MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
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</script>
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## License
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##### Code Examples
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The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)).
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##### Text
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The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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[<p><img class="center" src="../cc/CC-BY-SA_icon.svg" /></p>](https://creativecommons.org/licenses/by-sa/4.0/)
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#### Images/Graphics
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- The image "[IFS triangle 1](res/IFS_triangle_1.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
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‎literature.bib

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publisher={Springer}
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}
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#------------------------------------------------------------------------------#
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# Fractal Flame
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#------------------------------------------------------------------------------#
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@article{draves2003fractal,
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title={The fractal flame algorithm},
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author={Draves, Scott and Reckase, Erik},
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journal={Forthcoming, available from http: I Itlam3. com/flame. pdf},
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year={2003},
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publisher={Citeseer}
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}
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