simple fractal | Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

Fractals are shapes that contain copies of themselves on smaller and smaller scales. There are many of them in nature: ferns, trees, frost-flowers, ice-floes, clouds and lungs, for example. Fractals are also easy to create on a computer, because you all need do is take a single rule and repeat it at smaller and smaller scales. One of the simplest fractals follows this rule:

1. Take a line of length l and find the midpoint.
2. Erect a new line of length l x lm on the midpoint at right angles.
3. Repeat with each of the four new lines (i.e., the two halves of the original line and the two sides of the line erected at right angles).

When lm = 1/3, the fractal looks like this:

(Please open image in a new window if it fails to animate)

When lm = 1/2, the fractal is less interesting:

But you can adjust rule 2 like this:

2. Erect a new line of length l x lm x lm1 on the midpoint at right angles.

When lm1 = 1, 0.99, 0.98, 0.97…, this is what happens:

The fractals resemble frost-flowers on a windowpane or ice-floes on a bay or lake. You can randomize the adjustments and angles to make the resemblance even stronger:

Ice floes (see Owen Kanzler)

Frost on window (see Kenneth G. Libbrecht)

To create a simple fractal, take an equilateral triangle and divide it into four more equilateral triangles. Remove the middle triangle. Repeat the process with each new triangle and go on repeating it. You’ll end up with a shape like this, which is known as the Sierpiński triangle, after the Polish mathematician Wacław Sierpiński (1882-1969):

But you can also create the Sierpiński triangle one pixel at a time. Choose any point inside an equilateral triangle. Pick a corner of the triangle at random and move half-way towards it. Mark this spot. Then pick a corner at random again and move half-way towards the corner. And repeat. The result looks like this:

A simple program to create the fractal looks like this:

initial()
repeat
  fractal()
  altervariables()
until false

function initial()
  v = 3 [v for vertex]
  r = 500
  lm = 0.5
endfunc

function fractal()
  th = 2 * pi / v
[the following loop creates the corners of the triangle]
  for l = 1 to v
    x[l]=xcenter + sin(l*th) * r
    y[l]=ycenter + cos(l*th) * r
  next l
  fx = xcenter
  fy = ycenter
  repeat
    rv = random(v)
    fx = fx + (x[rv]-fx) * lm
    fy = fy + (y[rv]-fy) * lm
    plot(fx,fy)
  until keypressed
endfunc

function altervariables()
[change v, lm, r etc]
endfunc

In this case, more is less. When v = 4 and the shape is a square, there is no fractal and plot(fx,fy) covers the entire square.

When v = 5 and the shape is a pentagon, this fractal appears:

But v = 4 produces a fractal if a simple change is made in the program. This time, a corner cannot be chosen twice in a row:

function initial()
  v = 4
  r = 500
  lm = 0.5
  ci = 1 [i.e, number of iterations since corner previously chosen]
endfunc

function fractal()
  th = 2 * pi / v
  for l = 1 to v
    x[l]=xcenter + sin(l*th) * r
    y[l]=ycenter + cos(l*th) * r
    chosen[l]=0
  next l
  fx = xcenter
  fy = ycenter
  repeat
    repeat
      rv = random(v)
    until chosen[rv]=0
    for l = 1 to v
      if chosen[l]>0 then chosen[l] = chosen[l]-1
    next l
    chosen[rv] = ci
    fx = fx + (x[rv]-fx) * lm
    fy = fy + (y[rv]-fy) * lm
    plot(fx,fy)
  until keypressed
endfunc

One can also disallow a corner if the corner next to it has been chosen previously, adjust the size of the movement towards the chosen corner, add a central point to the polygon, and so on. Here are more fractals created with such variations:

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