# Mandelbrot Tutorial

First lets include the needed libraries

```#include <SDL2/SDL.h>
#include <numeric>
#include <complex>
```

Now we're going to need a function to decide if some complex number c is in the set.

We're going to define a function int is_in_set(std::complex<double> c) this function will decide if the complex number c is in the set and return to us 0 if it is or some number depending on how many iterations it took to decide if it was not.

```int is_in_set(std::complex<double> c)
{
std::complex<double> z(0,0);
for(int i = 0; i < 2500; i++)
{
z = std::pow(z,2) + c;
if(std::norm(z) > 10)
{
return i;
}
}
return 0;
}
```

This function is all you need to decide if some point c is in the set or not. We take a c initialize z to (0,0) then iterate 250 times to see if the squared magnitude of z ever exceeds 10 (an arbitrary number I made up, feel free to change it to whatever you want).

Next we're going to loop over x and y and run the is_in_set function for every x,y to see if the x,y combination lands in the set.

```int iters = 0;
for(double x = -2.0; x < 2.0; x+=0.001)
{
for(double y = -2.0; y < 2.0; y+=0.001)
{
iters = is_in_set(std::complex<double>(x,y);
// If iters == 0 paint this point black.
// Otherwise paint the point
// a color depending on the number of iters
}
}
```

doing this will give you something similar to this. The full source code is available below:

```#include <SDL2/SDL.h>
#include <numeric>
#include <complex>

int is_in_set(std::complex<double> c)
{
std::complex<double> z(0,0);
for(int i = 0; i < 2500; i++)
{
z = std::pow(z,2) + c;
if(std::norm(z) > 10)
{
return i;
}
}
return 0;
}

int main()
{
SDL_Init(SDL_INIT_EVERYTHING);
SDL_Window* window = nullptr;
SDL_Renderer* renderer = nullptr;
SDL_CreateWindowAndRenderer(1000*2,1000*2,0, &window, &renderer);
SDL_RenderSetScale(renderer,2,2);

for(double x = 0.0; x < 1.0; x+=0.001)
for(double y = 0.0; y < 1.0; y+=0.001)
{
double point_x = std::lerp(-2.0, 2.0,x);
double point_y = std::lerp(-2.0, 2.0,y);
int iters = is_in_set(std::complex<double>(point_x, point_y));
if(iters == 0)
{
SDL_SetRenderDrawColor(renderer,0,0,0,255);
SDL_RenderDrawPointF(renderer, static_cast<float>(x * 1000.0),  static_cast<float>(y * 1000.0));
}
else{
SDL_SetRenderDrawColor(
renderer,
static_cast<Uint8>(3 * iters % 255),
static_cast<Uint8>(3 * iters % 255),
static_cast<Uint8>(3 * iters % 255),
255);
SDL_RenderDrawPointF(renderer, static_cast<float>(x * 1000.0), static_cast<float>(y * 1000.0));
}
}

SDL_RenderPresent(renderer);
SDL_Delay(10000);
}
```

A video demo is available here: