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 ---
-title: "HPC Mandelbrot, Julia, and Multibot Sets"
+title: "HPC Complex Fractal Generation"
 author:
 - "JP Appel"
 - "David Marrero"
 ---
 
-![Mandelbrot Set]()
-
 # Prerequisite Knowledge
 
 ## Complex Numbers
 
+$$i^2 = -1$$
+$$z = x + iy$$
+
+::: notes
+
+* complex numbers are an extension of real numbers, stemming from the square root of $-1$
+* a complex number is just a pair of two real numbers (x,y) with different ways to add and multiply
+* in computer science we model real numbers with a single float or double, so we will need 2 floats or doubles to model a complex number
+
+:::
+
+. . .
+
+### Addition
+
+$$z_1 + z_2 = (x_1 + x_2) + i (y_1 + y_2)$$
+
+::: notes
+
+* addition behaves as you expect
+* multiplication involves multiplying then distributing, and using the fact that $i^2 = -1$
+* so adding two complex numbers is 2 float additions
+* and multiplying them is 4 multiplications and 2 additions
+
+:::
+
+### Multiplication
+
+$$z_1z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)$$
+
 ## What is the Mandelbrot Set
 
+. . .
+
+$$z_n = z^2_{n-1} + z_0$$
+
+
+::: notes
+
+
+* this sequence is used to generate the mandelbrot set
+* if for some complex number $z_0$ the sequence remains bounded as it $n$ approaches infinity then $z_0$ lies within the mandelbrot set
+* there are many recursive sequences related to this, where you modify that happens with the recursive term
+
+:::
+
 ## Fractals
 
-## Handling Recursion
+::::: {.columns}
 
-# Implementations
+:::: {.column width=40%}
+* infinite self-similar geometric shape
+* have "fractional dimension"
+::::
 
-## Serial
+:::: {.column width=60%}
+![Sripenski Triangle](https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Sierpinski_triangle.svg/1920px-Sierpinski_triangle.svg.png){ width=80% }
+::::
+:::::
 
-## Serial Animation
+. . .
 
-## Shared
+The Mandelbrot set is a fractal in the complex plane
 
-## Shared Animation
+::: notes
+
+* fractals are a infinite self-similar geometric shape
+* so if you zoom in on any one part it will look like the entire object
+* the Sripenski triangle is an example of a fractal
+* the mandelbrot set forms a fractal in the complex plane
+
+:::
+
+## Fractal in Nature
+
+![Romanesco Cauliflower](https://www.rocketgardens.co.uk/wp-content/uploads/2016/02/Cauliflower20Romanesco.jpg){ width=50%}
+
+::: notes
+
+* fractals often show up in nature
 
-## GPU
+:::
 
-## GPU Animation
+
+## Escape Time Algorithm
+
+::: incremental
+
+* Inputs
+    * Maximum Number of iterations
+    * Upper bound
+1. Create a grid of points to sample
+2. For each point in the sample space
+    1. Compute the next term in the sequence
+    2. if greater than the upper bound return the number of iterations
+    3. else repeat until the maximum number of iterations and return
+
+:::
+
+::: notes
+
+* to compute the complex sets we used an escape time algorithm
+* in general it takes in a maximum number of iterations, an initial value, and an upper bound
+* the escape time algorithm is as follows (read of the slides)
+* note that each sampled point is completely independent of any other point
+    * this hints to us that the problem will parallelize well
+* for most of the sets we looked at, there is a proven bound
+    * ie if the sequence is ever larger than a value we know it diverges
+:::
+
+# Implementation
+
+## Program Structure
+
+![](diagram.png)
+
+::: notes
+
+* the translation units are roughly as pictured here
+* black ellipses are for serial code, color is for parallel
+* each version of the program fills in an array with the number of iterations it took the sequence to grow too large
+* that array along with some extra data is a grid object, which can be serialized and deserialized
+* the main fractals unit handles cli argument parsing for the sampling resolution, fractal type etc
+* a separate renderer program handles creating images from the `.grid` file
+* the `.grid` file format is really simple, it's a magic number, the grid dimensions, the maximum number of iterations, the lower left and upper right most points of the region, and then the data
+
+:::
+
+## Mandelbrot
+
+::: notes
+
+* when mandelbrot initially tried to have this printed, the printers kept removing the "dust" thinking it was an error in their printing process
+
+:::
+
+## Tricorn
+
+## Burning Ship
+
+## Multibrot
+
+## Multicorn
+
+## Julia
+
+## Serial Animation
+
+## Shared Animation
+
+## CUDA Animation
 
 # Analysis
 
-## Speedups
+##
+
+[Interactive Plots](../analysis/analysis.html)