Revision as of 15:14, 1 June 2013 by Bsteward (talk | contribs) (finished script)

Warning.png This article or section is currently undergoing major changes by Bsteward (talk | contribs). Please avoid largely modifying this page's contents until this template has been removed. (Date?)
The droste effect is an example of Recursion.

Recursion is the process of repeating items in a self-similar way. Recursion can be implemented in Scratch by making a block that uses it self. This can be used to create fractals. A fractal is pattern that produces a picture, which can be zoomed into infinity and will still produce the same picture. Some common examples of fractals are the mandelbrot, the Sierpinski Triangle, and the Koch Snowflake.

Creating the Koch Curve

The Koch Curve is a fractal that can be created relatively easily in Scratch. The Koch Curve is part of a larger fractal, the Koch Snowflake.

Understanding Recursion in the Koch Curve

The Koch Curve

The Koch Curve is made of four Koch Curves that are a third of the size of the original Koch Curve. They are they are arranged so that the first and fourth are flat and the middle two point up to make an equilateral that is triangle missing one side.

Recersion in Koch Curve.png

To make it easier to draw, the Koch Curve can be broken down into iterations, each one more complicated than the last. The first iteration is made up of four straight lines. The second iteration contains four copies of the first iteration. The third iteration contains four copies of the second iteration or sixteen copies of the first iteration. As iterations are added it gets more complicated and looks more and more like the real Koch Curve.

Iterations of Koch Curve.png

Implementing it in Scratch

Basic Pen Path without Recursion

The triangle in the center is an equilateral triangle, therefore each of it's angles have a measure of 60°.

Equilateral Triangle in Koch Curve.png

Using basic geometry the angles of the rotations the sprite must make can be found.

Sprite Turns in Koch Curve.png

Using this these angles, a script can be created that draws the first iteration of Koch Curve. Since each line segment is 1/3 the total length of the Koch Curve, the sprite should move 1/3 of the length given e

when gf clicked
point in direction (90) //make sure the sprite is pointed right
go to x: (-240) y: (-179) //put the sprite in the lower left corner
clear //clear graphics from previous runs
pen down //put the pen down for drawing
make of the first iteration of the Koch Curve with a length of (480)
pen up //put the pen up so movement afterwards is not recorded

define make of the first iteration of the Koch Curve with a length of (length)
move ((length) / (3)) steps //draw a line segment
turn ccw (60) degrees //first turn
move ((length) / (3)) steps //draw a line segment
turn cw (120) degrees //second turn
move ((length) / (3)) steps //draw a line segment
turn ccw (60) degrees //third turn
move ((length) / (3)) steps //draw a line segment


Adding Recursion

See Also