diff --git a/fractal_project/README.md b/fractal_project/README.md index 2e44796c088bb2bf3d1ff7c1f8dc3c16d266d086..fba577fbece36dca87a7eabb7a06c187db0864b3 100644 --- a/fractal_project/README.md +++ b/fractal_project/README.md @@ -1,3 +1,29 @@ +# Mandelbrot set + +## What is the Mandelbrot set? + +## Calculation of the Mandelbrot set + +The **Mandelbrot set** is the set of **complex numbers** $c$ for which the function $f_{c}(z) = z^{2} + c$ does **not** diverge when iterated from $z_{0} = 0$. This means that starting with $z = 0$, if the iterations of the function $f_{c}(z) = z^{2} + c$ do not cause $\left\lvert z \right\rvert$ to grow beyond a certain threshold (usually $2$) after many iterations, then the point $c$ is part of the Mandelbrot set. + +### 1. Choose a complex constant $c$ + +> * Fot the Mandelbrot set, $c$ **is the parameter** that changes with each point we check in the complex plane. The set is defined by **which value of $c$ produce bounded iterates**. + +### 2. Define the complex plane + +### 3. Start with $z_{0} = 0$ + +### 4. Iterate the function $f_{c}(z) = z^{2} + c$ + +### 5. Check if $\left\lvert z \right\rvert$ (the modulus of $z$) escapes + +### 6. Repeat the iteration for a maximum number of iterations + +### 7. Color the point + +### 8. Repeat for all points in the complex plane + # Julia set ## What is a Julia set? @@ -36,7 +62,7 @@ Where: > * **Second iteration**: Now use $z_{1} = ... + ... \imath$ and apply the same function. > * **Finally**, you repeat this process for several iterations. -### 5. Check if $\left\lvert z \right\rvert$ (the module of $z$) escapes +### 5. Check if $\left\lvert z \right\rvert$ (the modulus of $z$) escapes > * **Escape condition**: If at any point during the iterations $\left\lvert z_{n} \right\rvert$ (the magnitude of $z_{n}$) exceeds a threshold (commonly $2$), we say the point $z$ escapes and is **not** in the Julia set we stop iterating.