**Recursion** is the process of repeating items in a self-similar way. Recursion can be implemented in Scratch by making a block that uses itself. 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 set, 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 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.

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.

### Implementation 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°.

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

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 each time.

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 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 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

To add recursion, instead of drawing a line, a smaller Koch Curve can be drawn. When each iteration above one is drawn, it contains four smaller Koch Curves that are one iteration less than it self. For example, when drawing the second iteration you must draw four copies that are 1/3 the size of the first iteration. When the program gets to the first iteration it must draw straight lines. The following code will make the fifth iteration of the Koch Curve

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 the (5) iteration of the Koch Curve with a length of (480) pen up //put the pen up so movement afterwards is not recorded define make the (iteration) iteration of the Koch Curve with a length of (length) if <(iteration) = [1]> then //is it the first iteration? move ((length) / (3)) steps //draw a line segment else make the ((iteration) - (1)) iteration of the Koch Curve with a length of ((length) / (3)) //make a smaller Koch Curve end turn ccw (60) degrees //first turn if <(iteration) = [1]> then //is it the first iteration? move ((length) / (3)) steps //draw a line segment else make the ((iteration) - (1)) iteration of the Koch Curve with a length of ((length) / (3)) //make a smaller Koch Curve end turn cw (120) degrees //second turn if <(iteration) = [1]> then //is it the first iteration? move ((length) / (3)) steps //draw a line segment else make the ((iteration) - (1)) iteration of the Koch Curve with a length of ((length) / (3)) //make a smaller Koch Curve end turn ccw (60) degrees //third turn if <(iteration) = [1]> then //is it the first iteration? move ((length) / (3)) steps //draw a line segment else make the ((iteration) - (1)) iteration of the Koch Curve with a length of ((length) / (3)) //make a smaller Koch Curve end

## Creating the Mandelbrot Set

The Mandelbrot set is a mathematical fractal defined in the complex plane. It was named after its discoverer, Benoit Mandelbrot, and has many close relationships to the Julia Sets.

### Understanding the Definition

The Mandelbrot set is defined as all c values in the complex plain which are bounded under iteration in the following equation:

For example, letting c=1 yields the sequence 0, 1, 2, 5, 26, 677, ect... That sequence escapes to infinity and therefore c=1 is not part of the Mandelbrot set. Meanwhile, the sequence c=-1 gives 0, -1, 0, -1, 0, ect..., is bounded and so belongs to the Mandelbrot set.

To note, it has been proven that if any sequence contains a complex value that is outside a distance of 2 from the origin, it will escape to infinity.

#### Coloring

In a basic Mandelbrot set, white is used for a c-values that escapes to infinity and black is used for all c-values that do not.

In most Mandelbrot sets, though, colors are used to help depict the Mandelbrot set or make it more art-orientated. Colors are not defined through an equation, but rather through the last iteration before escaping a distance of two from the origin. The iteration is then assigned a color of the creator's preference.

### Implementation in Scratch

#### Creating the Variables

Due to the fact that Scratch does not directly support mathematics in the complex plain, a simple workaround has to be used. Each complex number will be defined as two variables, the real part, and the complex part. And since there will be operations based on the complex numbers, two complex numbers will be needed, or four variables. For the tutorial, these names will be used:

- Real 1
- Real 2
- Complex 1
- Complex 2

Along with that, a variable Best Fit will be used to figure out the color to be used when coloring a complex value:

- Best Fit