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Recursion and Fractals

DocumentInQuestion.png It has been suggested that this page's contents be merged with the page Recursion. You can discuss this on the page's talk page.
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 itself. This can be used to create fractals. A fractal is pattern that produces a picture, which contains an infinite amount of copies of itself. Some well-known specimens are the Mandelbrot set, the Sierpinski Triangle (also, but less commonly known as the Sierpinski Gasket), 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 a piece of the 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

Implementation in Scratch

Basic Pen Path without Recursion

The triangle in the center is an equilateral triangle, therefore each of its 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 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

An animation of zooming in on the Mandelbrot set.

The Mandelbrot set is a mathematical fractal defined in the complex plane. It is completely self-similar, meaning that it repeats over and over as one zooms in. The Mandelbrot set was named after its discoverer, Benoit Mandelbrot, and has many close relationships to the Julia Sets.

Understanding the Definition

In its definition, the Mandelbrot uses complex numbers. A complex number is the sum of a real number and an imaginary one, with an imaginary number simply being the square root of a negative number. Since that can not be taken, the square root of -1 is assigned the value i. For example, 3+3i is a complex number.

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

Mandelbrot Equation.png

First, one starts with a z value of c (i.e. z1 = c). Then, when one puts that z back into the equation, it becomes z^2+c. That z is then taken and put back through the equation over and over. This is called iteration.

For example, let c=1. The value z then becomes 1. Once 1 is put back into the equation, the equation becomes 1^2+1, which equals 2. Once 2 is put back in, it becomes 2^2+1, or 5. Once 5 is put back in, it becomes 26. 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.

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

An animation of the Mandelbrot set under iteration.

In a basic Mandelbrot set, white is used for a c-values that escape to infinity and black is used for all c-values that do not. This would be the actual Mandelbrot set.

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. Color can also apply to the rate of the equation reaching infinity.

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

Coding

To start, a base of clones is needed to render a full screen due to the computing power needed to render the Mandelbrot set:

when gf clicked
hide//so that this sprite and its clones do not show
set y to (180)
clear//preparing the scene for the Mandelbrot set
repeat (12)
  create clone of [myself v]
  change y by (-1)//this makes it so that clones do not overlap
end

Next, it is important to give the clones a skeleton:

when I start as a clone
set pen size to (1.5)//any other size will appear transparent
set x to (-180)//the left hand side of the Mandelbrot set
repeat ((360) / (12))//12 clones and each gets 30 rows on the screen to render
  repeat (360)//360 pixels will be the width of the Mandelbrot set once rendered
    //this is where we'll be rendering each point
  end
  change y by (-12)//the clone finished a row and moves to another
  set x to (-180)
end

Although this is a functional script, it will take a while to render. For speed preferences, we'll be inserting a custom block:

when I start as a clone
set pen size to (1.5)
set x to (-180)
repeat (30)
  repeat (60)
    forced iteration::custom//renders 6 points without a screen refresh for speed benefits
  end
  change y by (-12)
  set x to (-180)
end

define forced iteration
//make sure this runs without screen refresh!
repeat (6)
//here is where we'll be rendering points now
end

As noted above, it is important that the custom block "forced iteration" runs without a screen refresh or else all the benefit of extra speed will be lost.

Anyway, now that that is coded, we need to create the skeleton for sampling a point and figuring out whether or not it is part of the Mandelbrot set:

define forced iteration
//make sure it runs without screen refresh
repeat (6)
  set [Real 1 v] to ((x position) / (90))//90 pixels to the right is the equivalent of 1
  set [Imaginary 1 v] to ((y position) / (90))//90 pixels upwards is the equivalent of i
  if <([sqrt v] of (((Real 1) * (Real 1)) + ((Imaginary 1) * (Imaginary 1)))) < (2.15)> then
    Test for Legibility at R: (Real 1) I: (Imaginary 1)::custom
    Set Pen Color::custom
    pen down//drawing the point
    pen up
  end
  change x by (1)//moving onto another point
end

define Test for Legibility at R: (Real) I: (Imaginary)//this is where we'll test if a point is part of the Mandelbrot set or not

define Set Pen Color//this is where we pick our color depending on the variable 'Best Fit'

Now here is where the mathematics of the Mandelbrot set comes into play. In the Test for Legibility custom block, we'll have to take a complex number, apply the equation which defines the Mandelbrot set, and repeat if it is still within a distance of 2 from the origin:

define Test for Legibility at R: (Real) I: (Imaginary)
set [Best Fit v] to (-1)
repeat until <<(Best Fit) = (20)> or <([sqrt v] of (((Real 1) * (Real 1)) + ((Imaginary 1) * (Imaginary 1)))) > (2)>>
  change [Best Fit v] by (1)//the complex number has survived one iteration
  set [Real 2 v] to ((((Real 1) * (Real 1)) - ((Imaginary 1) * (Imaginary 1))) + (Real))
  set [Imaginary 2 v] to (((2) * ((Real 1) * (Imaginary 1))) + (Imaginary))
  set [Real 1 v] to (Real 2)//setting the scene for another iteration
  set [Imaginary 1 v] to (Imaginary 2)
end

Up above, it may be noticed that the repeat continues until the complex number is found not to be part of the Mandelbrot set, or until it has iterated 20 times. That 20 can be changed to whatever one wants, with higher numbers producing higher quality; though the higher the number, the more lag will be caused.

To complete the Mandelbrot set, colors need to be implemented:

define Set Pen Color
if <(Best Fit) = (-1)> then
  set pen color to [#000]//this point is already outside a distance of 2 from the origin
else
  if <(Best Fit) = (20)> then
    set pen color to [#FFF]//this point is a solution to the Mandelbrot set
  else 
    set pen color to (50)//this complex number is not a solution, but survives several iterations
    set pen shade to (((0) - (Best Fit)) * (5))
  end
end

Those colors may be changed to one's wishes.

Final Product

In the end, this is all the code used to draw the Mandelbrot set (scroll to see all of the code):

when gf clicked
hide
set y to (180)
clear
repeat (12)
  create clone of [myself v]
  change y by (-1)//this makes it so that clones do not overlap
end

when I start as a clone
set pen size to (1.5)
set x to (-180)
repeat (30)
  repeat (60)
    forced iteration::custom
  end
  change y by (-12)
  set x to (-180)
end

define forced iteration
//remember, run this without screen refresh!
repeat (6)
  set [Real 1 v] to ((x position) / (90))
  set [Imaginary 1 v] to ((y position) / (90))
  if <([sqrt v] of (((Real 1) * (Real 1)) + ((Imaginary 1) * (Imaginary 1)))) < (2.15)> then
    Test for Legibility at R: (Real 1) I: (Imaginary 1)
    Set Pen Color
    pen down//drawing the point
    pen up
  end
  change x by (1)//moving onto another point
end

define Test for Legibility at R: (Real) I: (Imaginary)
set [Best Fit v] to (-1)
repeat until <<(Best Fit) = (20)> or <([sqrt v] of (((Real 1) * (Real 1)) + ((Imaginary 1) * (Imaginary 1)))) > (2)>>
  change [Best Fit v] by (1)//the complex number has survived one iteration
  set [Real 2 v] to ((((Real 1) * (Real 1)) - ((Imaginary 1) * (Imaginary 1))) + (Real))
  set [Imaginary 2 v] to (((2) * ((Real 1) * (Imaginary 1))) + (Imaginary))
  set [Real 1 v] to (Real 2)
  set [Imaginary 1 v] to (Imaginary 2)
end

define Set Pen Color
if <(Best Fit) = (-1)> then
  set pen color to [#000]//this point is already outside a distance of 2 from the origin
else
  if <(Best Fit) = (20)> then
    set pen color to [#FFF]//this point is a solution to the Mandelbrot set
  else 
    set pen color to (50)//this complex number is not a solution, but survives several iterations
    set pen shade to (((0) - (Best Fit)) * (5))
  end
end

Julia Set

The Julia set is a series of equations that are mathematical fractals, and that is defined very similarly to the Mandelbrot set. The official equation is:

Julia Set Equation.png

The only difference in its definition from the Mandelbrot set is that c is no longer a point in the complex plane, but rather a complex parameter, which is consistent whichever point you pick. Also, z1 is defined as being the point you pick in the complex plane. The technique above for rendering the Mandelbrot set may be used here again, with the required changes.

Here is a gallery of images on the Julia set:

See Also

  • This page was last modified on 22 June 2017, at 02:23.