## Fractals in Clojure - Sierpinski Triangle

This fractal is called the Sierpinski triangle, after the Polish mathematician Waclaw Sierpinski.

In order to generate it we'll use the following algorithm,

- We pick a triangle on the surface.
- We shrink the triangle to half the height and width, make three copies, and position the three triangles so that each triangle touches the two other triangles at a corner.
- We repeat the above step until required depth is reached.

(ns sierpinski (:import (java.awt Dimension Polygon Color) (javax.swing JFrame JLabel) (java.awt.image BufferedImage))) (defstruct point :x :y) (defstruct triangle :left :rigth :bottom)

We need two structures for representing a point on the canvas and the triangle we are drawing.

(defn midpoints [trig] (struct triangle (struct point (/ (+ (:x (:bottom trig)) (:x (:left trig))) 2) (/ (+ (:y (:bottom trig)) (:y (:left trig))) 2)) (struct point (/ (+ (:x (:rigth trig)) (:x (:bottom trig))) 2) (/ (+ (:y (:rigth trig)) (:y (:bottom trig))) 2)) (struct point (/ (+ (:x (:left trig)) (:x (:rigth trig))) 2) (/ (+ (:y (:left trig)) (:y (:rigth trig))) 2))))

Given a triangle this function will return a new triangle thats half the original triangle.

(defn paint-triangle [g trig] (let [polygon (new Polygon)] (doto polygon (.addPoint (:x (:left trig)) (:y (:left trig))) (.addPoint (:x (:rigth trig)) (:y (:rigth trig))) (.addPoint (:x (:bottom trig)) (:y (:bottom trig)))) (.drawPolygon g polygon))) (defn create-triangles [trig step depth g] (paint-triangle g trig) (let [points (midpoints trig) left (struct triangle (:left trig) (:left points) (:bottom points)) rigth (struct triangle (:rigth trig) (:rigth points) (:bottom points)) top (struct triangle (:bottom trig) (:rigth points) (:left points)) ] (if (< step depth ) (do (create-triangles left (inc step) depth g) (create-triangles rigth (inc step) depth g) (create-triangles top (inc step) depth g)))))

create-triangles is a recursive function that will call it self until required depth is reached. It first draws the given triangle then calculate 3 smaller triangles that will be placed inside the parent triangle.

(defn draw [depth width height] (let [image (BufferedImage. width height BufferedImage/TYPE_INT_RGB) canvas (proxy [JLabel] [] (paint [g] (.drawImage g image 0 0 this))) graphics (.createGraphics image)] (.setColor graphics Color/white) (create-triangles (struct triangle (struct point 0 height) (struct point width height) (struct point (/ width 2) 0)) 0 depth graphics) (doto (JFrame.) (.add canvas) (.setSize (Dimension. width height)) (.show))))

(draw 4 400 400)