L-Systems and the Fibonacci word fractal

open System
open System.Drawing
open System.Drawing.Imaging

let right = Math.PI / 2.0
let left  = 3.0 * Math.PI / 2.0

type Alphabet = 
    |K of float
    |Q of float
    |R of float
    |L of float
    |T of float

let fibonacci_word = function
    |L(x) -> [T(left) ; R(x); T(right); L(x); K(x); L(x); T(right); R(x); T(left) ]
    |R(x) -> [T(right); L(x); T(left) ; R(x); Q(x); R(x); T(left) ; L(x); T(right)]
    |K(x) -> [L(x)]
    |Q(x) -> [R(x)]
    |T(x) -> [T(x)]
    
let rec internal applyN f n x = 
    if n = 0 then x else f (applyN f (n-1) x)
    
type ITurtle =
    abstract draw_forward : float -> unit
    abstract move_forward : float -> unit
    abstract turn         : float -> unit

let interpreter (turtle: ITurtle) = function 
    |L(x) -> turtle.draw_forward(x)
    |R(x) -> turtle.draw_forward(x)
    |K(x) -> turtle.move_forward(x)
    |Q(x) -> turtle.move_forward(x)
    |T(x) -> turtle.turn(x)

let rotate (x,y) theta = 
    let x' = x * cos theta - y * sin theta
    let y' = x * sin theta + y * cos theta
    (x',y')

let Turtle(screen:Graphics) =  
    let x = ref 1.0 
    let y = ref 500.0  
    let dx = ref 1.0 
    let dy = ref 0.0 
    { new ITurtle with 
        member t.draw_forward(scale) = screen.DrawLine( new Pen(Color.Red),
                                                PointF(float32 !x,float32  !y),
                                                PointF(float32(!x + (!dx*scale)), 
                                                       float32( !y + (!dy*scale))))
                                       t.move_forward(scale)
        member t.move_forward(scale) = x := !x + (!dx*scale)
                                       y := !y + (!dy*scale)
        member t.turn(theta)    = let dx',dy' = rotate (!dx,!dy) (theta)
                                  dx := dx'
                                  dy := dy'
        
    };;

let bitmap = new Bitmap(1000,1000)    

let turtle = Turtle(Graphics.FromImage(bitmap))

List.iter (interpreter turtle) (applyN (List.collect fibonacci_word) 8 [L(0.5)])

bitmap.Save("fibonacci_word.bmp")