linguist/samples/NetLogo/Star Fractal.nlogo

turtles-own [
  moves         ;; how many sides the turtles has drawn so far
  star-side     ;; The length of the side that the turtle is drawing
  generation    ;; How many generations of turtles have come before it.  So,
]               ;; if the turtle was hatched from a turtle that was hatched
                ;; by another turtle, it's generation is 2

to setup
  clear-all
  crt 1 [
    set size 8                    ;; so turtles are easy to see
    set ycor max-pycor * 0.9      ;; place near top of world
    set star-side ycor * 1.97538  ;; length of the sides of the star it will draw
    pen-down
    set heading 180 + 18
    recolor
  ]
  reset-ticks
end

to go
  if not any? turtles [ stop ]
  ask turtles [
    fd star-side
    rt 180 + 36
    if generation < fractal-level [      ;; If it's not drawing at the deepest level of the figure,
      hatch 1 [                          ;;  hatch a new turtle to draw at a lower level
        set generation generation + 1
        set star-side star-side / 2
        recolor
        set moves 0
        rt 18
      ]
    ]
    set moves moves + 1
    if moves > 5 [ die ]        ;; When the turtle has completed its star, it's done
  ]
  tick
end

to recolor  ;; turtle procedure
  set color scale-color red (fractal-level - generation) -1 (fractal-level + 1)
end
@#$#@#$#@
GRAPHICS-WINDOW
274
10
686
443
100
100
2.0
1
10
1
1
1
0
0
0
1
-100
100
-100
100
1
1
1
ticks
30.0

BUTTON
146
42
211
75
go
go
T
1
T
OBSERVER
NIL
NIL
NIL
NIL
1

SLIDER
32
88
248
121
fractal-level
fractal-level
0
4
3
1
1
NIL
HORIZONTAL

BUTTON
69
42
133
75
setup
setup
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1

MONITOR
105
128
190
173
turtles
count turtles
3
1
11

PLOT
2
179
272
394
Turtles vs. Time
Time
Num Turtles
0.0
25.0
0.0
180.0
true
false
"" ""
PENS
"default" 1.0 0 -16777216 true "" "plot count turtles"

@#$#@#$#@
## WHAT IS IT?

This model creates a fractal-like structure consisting of stars drawn at various scales, the number of which is specified by the user.  The structure is drawn by generations of turtles that draw stars at a given scale and hatch successive generations to draw new ones at smaller scales.

## HOW TO USE IT

The FRACTAL-LEVEL slider controls the number of levels the structure is drawn in.  For instance, a fractal level of 0 results in a single star, while a fractal level of 1 draws a star with smaller stars that begin at each of its 5 corners.  This pattern is repeated as the level increases.

Click the SETUP button to initialize the structure and GO to begin drawing it.

## THINGS TO NOTICE

The figure drawn embodies the most important fractal quality: self-similar structure at smaller scales.  This means that no matter how far you "zoom in" to it, you will still see the same shape (namely, a star).  The fractal level determines to how small a scale this property holds, so the greater the level the more you can "zoom in" and still see stars.

Notice that at a fractal level greater than 1, the model begins with one turtle, which hatches new turtles at each corner, and these turtles similarly hatch new ones at each of their corners as well. This behavior is made possible by a powerful mathematical technique called recursion.  A process is called recursive when its workings involve "performing itself".  In this case, the process performed by each turtle at each step is to draw a new side of its star and then create new turtles to perform the very same process.  A helpful property of recursive processes is that the instructions are often short, because each performer is executing the same instructions. Hence the brevity of Star Fractal's procedures.

The ideas behind fractal scaling (the property of self-similar structures at different scales) and recursion are essentially the same.  This is suggested by the fact that a recursive process is able to generate a fractal-like structure, as in this model.  One way to think about it is that recursion applies these ideas to processes (like NetLogo models), and fractal scaling applies them to physical or mathematical structures (like the figure drawn by Star Fractal).

## THINGS TO TRY

Edit GO so that it's not a forever button, and/or use the speed slider to slow the model down, and note what happens during each tick.  Notice that what begins as a single drawing multiplies into many drawings, each of which is at a smaller scale but proceeds exactly like the larger one that spawned it.  Thus, recursion begets self-sameness at different scales.

Try the model initially with small fractal levels and work your way up to higher ones.  Try to figure out how many more stars are incorporated into the figure each time the level increases by one.  What kind of general rule you can come up with? That is, given a generation of turtles g, how many turtles will be in the generation g + 1?  Start from g = 0.  (This is a recursive rule.)

## EXTENDING THE MODEL

In increasing order of difficulty:

Change the color scaling so that the smaller stars are brighter than the larger ones.

Change the model so that it draws shapes other than stars.  Will any shape work?

The structure that this model draws is only "fractal-like", because even when you "zoom in" and look at smaller scales, the smaller stars aren't actually part of the larger ones; they're just connected to them at a vertex. An important property of fractals is that the larger structures are composed of the smaller ones, something which clearly doesn't hold here.  For a tougher exercise, try changing the model so that the smaller stars (or whatever shapes are being drawn) are actually part of the larger ones, for instance by drawing them in the middle of their sides.  Such a figure will be a lot less "noisy" than this one, in terms of how difficult it is to discern individual stars, and it will have different properties when you zoom in.

## NETLOGO FEATURES

This model makes use of an important NetLogo command: `hatch`.  `hatch` allows one turtle to create a new one on the same patch and give it some initial instructions.  In this case, the parent turtle sets the length of the star its offspring will draw, its initial heading, and several other variables.  Notice that the new turtle uses the values of its parent turtle's variables in the body of the hatch procedure.

Another useful primitive this model uses is `scale-color`.  It is used to make the brightness of the color the turtles are drawing inversely proportional to the turtle's generation, so that larger stars are brighter than smaller ones (this was done to make the larger ones more visible).  This code:

    scale-color red (fractal-level - generation) -1 (fractal-level + 1)

works as follows: Subtract the value of `generation` from that of `fractal-level`.  The higher that resulting value is on the scale from -1 to `(fractal-level + 1)`, the darker the shade of red the turtle will draw with.

## CREDITS AND REFERENCES
@#$#@#$#@
default
true
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airplane
true
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arrow
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box
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bug
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butterfly
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car
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circle
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circle 2
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cow
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cylinder
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dot
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face happy
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face neutral
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face sad
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fish
false
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Circle -16777216 true false 215 106 30

flag
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Polygon -7500403 true true 90 150 270 90 90 30
Line -7500403 true 75 135 90 135
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flower
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Circle -16777216 true false 113 68 74
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house
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Polygon -7500403 true true 15 120 150 15 285 120
Line -16777216 false 30 120 270 120

leaf
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line
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line half
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pentagon
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person
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Polygon -7500403 true true 105 90 60 150 75 180 135 105

plant
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Polygon -7500403 true true 135 105 90 60 45 45 75 105 135 135
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square
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square 2
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star
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target
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Circle -7500403 true true 60 60 180
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Circle -7500403 true true 120 120 60

tree
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Rectangle -6459832 true false 120 195 180 300
Circle -7500403 true true 65 21 108
Circle -7500403 true true 116 41 127
Circle -7500403 true true 45 90 120
Circle -7500403 true true 104 74 152

triangle
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triangle 2
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truck
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Circle -16777216 true false 234 174 42
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turtle
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wheel
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Line -7500403 true 40 84 269 221
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x
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Polygon -7500403 true true 270 75 225 30 30 225 75 270
Polygon -7500403 true true 30 75 75 30 270 225 225 270

@#$#@#$#@
NetLogo 5.0.5
@#$#@#$#@
setup
while [any? turtles] [ go ]
@#$#@#$#@
@#$#@#$#@
@#$#@#$#@
@#$#@#$#@
default
0.0
-0.2 0 0.0 1.0
0.0 1 1.0 0.0
0.2 0 0.0 1.0
link direction
true
0
Line -7500403 true 150 150 90 180
Line -7500403 true 150 150 210 180

@#$#@#$#@
0
@#$#@#$#@