We have seen a first program called “**DrawWorld**” we introduced the JAVA programming oriented graphics. Programming This module has helped us to see a first recursive fractal: The triangle Sierpisnki.

Let's change this basic program to generate a new basic recursive fractal: The **Kuch curve**.

(See and generating a recursive fractal)

It is a fractal that is built recursively from a straight line. Its sides are divided into three equal parts and the central segment is changed by two equal forming 60 degrees between the previous and if.

The

Koch curve, also known as snowflake is a fractal that can be obtained by different methods as so-calledIFSorFunction Systemsiterated(Deterministic or), rule-based systems, etc..The

recursive algorithmhas also the virtue of representing a concept closely associated with fractals: infinity. The essence of recursion to describe a very simple form of the curve itself. A universe that contains another and this in turn copy the pattern on a smaller scale (so contractive) an endlessly repeating sequence.Koch curve belongs to the

self-similar fractal[1], be the method of obtaining the deterministic.

## Fractal Dimension

The **dimension **an object is placed or topological concept that classifies objects **metric spaces**. The intuitive notion of whole dimensions spaces clashes with the so-called fractal dimensions, taking actual values.

The **Peano curve** is a curve capable of filling up. Do you have therefore two dimensions?, one wonders.

Is associated with a fractal dimension of roughness, or fragmentation, thereof, so that a greater dimension present a more rough or jagged. In any case gives information about its complexity characterizing.

Koch curve has a ratio **s = 1/3**, with **n = 4**, so its fractal dimension is:

**D = ln4/ln3 ~ 1.269**

If each of these new segments are again divided recursively curve obtained Hoch

.

If we use three lines, instead of one as an initiator, equilateral triangle shaped form appears classic snowflake, name with which this configuration is known fractal.

## Generator algorithm

We have defined a function “paintRecursivo” (I called from the method “paint”) spent the points of the line or lines of the triangle, and the recursion. The function computes the vertices of the new segments, paints the figure and calls itself again reducing the recursion.

Therefore, on each call to the function reduces the value of recursion, so that when it is zero finishes carrying recursion.

importjava.applet.Applet;importjava.awt.Graphics; /** * @ Author José Juan Aliaga */publicclassMainAppextendsApplet {doublexp1 = 300;doublep1 = 300;doublexp2 = 10;doublep2 = 300;doublesin60 = Math.sin(3.14/3.);intnivel_de_recursividad = 6;publicMainApp() { }publicstaticvoidmain(String[] args) { }public voidpaint(Graphics g){ paintRecursivo(g,nivel_de_recursividad,xp1, p1, xp2, p2); }private voidpaintRecursivo(Graphics g,inti,doubleXP12,doubleyp12,doubleXp22,doubleyp22 ){doubledx =(Xp22-XP12)/3.;doubledy =(yp22-yp12)/3.;doublexx = 3 * XP12 * sin60 dx/2.-dy;doubleyp12 3 * y = dy / 2. dx * sin60;if(i<= 0){g.drawLine((int)XP12,(int)yp12,(int)Xp22,(int)yp22);}else{paintRecursivo(g,i-1, XP12, yp12, XP12 dx,yp12 dy); paintRecursivo(g,i-1 dx XP12,yp12 dy,xx,yy); paintRecursivo(g,i-1, xx,yy,Xp22-dx,yp22-d); paintRecursivo(g,i-1, Xp22-dx,yp22-d,Xp22, yp22);}} }