Koch snowflake

English: The Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described.

Deutsch: Wenn man den Ersetzungsprozess der Koch-Kurve nicht mit einer Strecke, sondern mit einem gleichseitigen Dreieck durchführt, erhält man die kochsche Schneeflocke.

Français : Le flocon de Koch est une construction de courbes planes et l'une des premières fractales à avoir été décrite.

See also: Koch curve

Illustrations

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Iteration

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  Snowflake, 0th iteration

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  Snowflake, 1st iteration

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  Snowflake, 2nd iteration

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  Snowflake, 3rd iteration

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  Snowflake, 4th iteration

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  Snowflake, 5th iteration

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  Snowflake, 6th iteration

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  Snowflake, 7th iteration

Animations

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Scripts

Programs that produce representations of the fractal in various languages.

Logo

Below is a recursive implementation in Logo. It can be tried out with most implementations of Logo, or online with the Java implementation XLogo.

Try start, call rt 30 koch 100.

to koch :x 
  repeat 3 [triline :x rt 120]
end
to triline :x 
  if :x < 1 [fd :x] [triline :x/3 lt 60 triline :x/3 rt 120 triline :x/3 lt 60 triline :x/3]
end

Web Turtle

Here follows a sample implementation of the Koch curve for a Turtle robot written in a Logo-like language. It can be tried out online with Web Turtle. Change the value of A in the first line to any number from 1 to 5 to see the different levels of complexity.

LET A 5
; calculate adjusted side-length
LET B 243
REPEAT A
  LET B B/3
NEXT
; place pointer
POINT 150
MOVE 140
POINT 0
; start
GO SIDE
RIGHT 120
GO SIDE
RIGHT 120
GO SIDE
; finished.
END

; main loop
# SIDE
 GO F
 LEFT 60
 GO F
 RIGHT 120
 GO F
 LEFT 60
 GO F
RETURN

; forward
# F
 IF A > 1
   ; go deeper depending on level
   LET A A-1
   GO SIDE
   LET A A+1
 ELSE
   ; or just do a single line
   DRAW B
 ENDIF
RETURN

Python

Here is the Koch curve in Python 3.1.

import turtle
 
koch_set = "F"
iterations = 5
 
for i in range(iterations):
    koch_set = koch_set.replace("F","FLFRFLF")
 
turtle.down()
 
for move in koch_set:
    if move == "F":
        turtle.forward(100.0 / (3 ** (iterations - 1)))
    elif move == "L":
        turtle.left(60)
    elif move == "R":
        turtle.right(120)

The program can be easily modified to show the entire snowflake:

import turtle
 
koch_flake = "FRFRF"
iterations = 5
 
for i in range(iterations):
    koch_flake = koch_flake.replace("F","FLFRFLF")
 
turtle.down()
 
for move in koch_flake:
    if move == "F":
        turtle.forward(100.0 / (3 ** (iterations - 1)))
    elif move == "L":
        turtle.left(60)
    elif move == "R":
        turtle.right(120)

Flash ActionScript 3

The source for a Koch Snowflake in Flash AS3:

var sideLength:Number = 400
var down:Number = (sideLength/2) * Math.tan(Math.PI/6)
var up:Number = (sideLength/2) * Math.tan(Math.PI/3) - down
var p1:Point = new Point(-sideLength/2,down);
var p2:Point = new Point(sideLength/2,down);
var p3:Point = new Point(0,-up);
var p4:Point = new Point();
var p5:Point = new Point();
var rot:Number = 0
var lines:Array = [p1,p2,p3]
var tempLines:Array = []
 
for(var iterations:uint = 1; iterations <= 9;iterations++){
        sideLength /= 3
        for(var loop:uint = 0; loop <= lines.length - 1; loop++){
                p1 = lines[loop]
                p2 = lines[loop+1]
                if(loop == lines.length-1){p2 = lines[0]}
                rot = Math.atan2(p2.y - p1.y, p2.x - p1.x)
                p3 = p1.add(Point.polar(sideLength,rot))
                rot += Math.PI/3
                p4 = p3.add(Point.polar(sideLength,rot))
                rot -= 2*Math.PI/3
                p5 = p4.add(Point.polar(sideLength,rot))
                tempLines.push(p1)
                tempLines.push(p3)
                tempLines.push(p4)
                tempLines.push(p5)
        }
        lines = tempLines
        tempLines = []
}
lines.push(p2)
var line:Sprite = new Sprite()
addChild(line)
line.graphics.lineStyle(0,0x000000)
line.graphics.moveTo(lines[0].x,lines[0].y)
trace(lines.length)
for(var a:uint = 0; a<lines.length;a++){
        line.graphics.lineTo(lines[a].x,lines[a].y)
}
stage.addEventListener(MouseEvent.MOUSE_DOWN, mouseDown) 
function mouseDown(event:MouseEvent):void {line.startDrag()}
stage.addEventListener(MouseEvent.MOUSE_UP, mouseReleased);
function mouseReleased(event:MouseEvent):void {line.stopDrag()}