globals [
  clock    ;; how many clock ticks have passed
  radius   ;; distance of the farthest green patch from the center
]

to setup
  ca
  set clock 0
  set radius 0
  ask patch 0 0
    [ set pcolor green ]
end

to go
  ;; stop when we get near the edge of the screen
  if radius >= screen-edge-x - 3
    [ stop ]
  ;; make new turtles, up to a maximum controlled by the MAX-PARTICLES
  ;; slider; also check clock so we don't make too many turtles too
  ;; soon, otherwise we get a big green clump at the center
  while [count turtles < max-particles and
         count turtles < clock]
    [ make-new-turtle ]
  ;; now move the turtles
  ask turtles
    [ wander
      if any? neighbors with [pcolor = green]
        [ stamp green
          ;; increase radius if appropriate
          if distancexy 0 0 > radius
            [ set radius distancexy 0 0 ]
          die ]
      ;; kill turtles that wander too far away from the center
      if not use-whole-screen? and distancexy 0 0 > radius + 3
        [ die ] ]
  ;; advance clock
  set clock clock + 1
end

to make-new-turtle
  ;; each new turtle starts its random walk from a position
  ;; a bit outside the current radius and facing the center
  cct 1
    [ set color red
      setxy 0 0
      rt random-float 360
      ifelse use-whole-screen?
        [ jump screen-edge-x ]
        [ jump radius + 1.5 ]
      rt 180 ]
end

to wander   ;; turtle procedure
  ;; the WIGGLE-ANGLE slider makes our path straight or wiggly
  rt random-float wiggle-angle - random-float wiggle-angle
  fd 1
end


; *** NetLogo Model Copyright Notice ***
;
; This model was created as part of the project:
; PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN
; CLASSROOMS.  The project gratefully acknowledges the support of the
; National Science Foundation (REPP program) -- grant number REC #9814682.
;
; Copyright 2002 by Uri Wilensky.  Updated 2002.  All rights reserved.
;
; Permission to use, modify or redistribute this model is hereby granted,
; provided that both of the following requirements are followed:
; a) this copyright notice is included.
; b) this model will not be redistributed for profit without permission
;    from Uri Wilensky.
; Contact Uri Wilensky for appropriate licenses for redistribution for
; profit.
;
; To refer to this model in academic publications, please use:
; Wilensky, U. (2002).  NetLogo Diffusion Ltd Aggregation model.
; http://ccl.northwestern.edu/netlogo/models/DiffusionLtdAggregation.
; Center for Connected Learning and Computer-Based Modeling,
; Northwestern University, Evanston, IL.
;
; In other publications, please use:
; Copyright 1998 by Uri Wilensky.  All rights reserved.  See
; http://ccl.northwestern.edu/netlogo/models/DiffusionLtdAggregation
; for terms of use.
;
; *** End of NetLogo Model Copyright Notice ***
@#$#@#$#@
GRAPHICS-WINDOW
267
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75
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0
10
1
1
1

CC-WINDOW
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Command Center

SLIDER
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max-particles
max-particles
1
300
100
1
1
NIL

BUTTON
65
42
128
75
NIL
setup
NIL
1
T
OBSERVER
T

BUTTON
140
42
203
75
NIL
go
T
1
T
OBSERVER
NIL

SLIDER
8
88
261
121
wiggle-angle
wiggle-angle
0.0
360.0
60.0
1.0
1
degrees

SWITCH
54
185
219
218
use-whole-screen?
use-whole-screen?
0
1
-1000

@#$#@#$#@
WHAT IS IT?
-----------
This model demonstrates diffusion-limited aggregation, in which randomly moving (diffusing) particles stick together (aggregate) to form beautiful treelike branching fractal structures.  There are many patterns found in nature that resemble the patterns produced by this model: crystals, coral, fungi, lightning, and so on.

The model begins with an initial green "seed" in the center of the screen.  Red particles move around the screen randomly.  When a red particle hits a green square, it "sticks" and turns green (and a new red particle is created to keep the process going).


HOW TO USE IT
-------------
Press SETUP to make the initial seed, then press GO to run the model.

The WIGGLE-ANGLE slider controls how wiggly the paths the particles follow are.  If WIGGLE-ANGLE is 0, they move in straight lines.  If WIGGLE-ANGLE is 360, they move in a totally random direction at each time step.

The MAX-PARTICLES slider controls how many red particles can exist at the same time.

Both settings may be altered in the middle of a model run.

The USE-WHOLE-SCREEN? switch controls whether the red particles start at the edge of the screen, or from just outside a circle enclosing the green area.  If the switch is on, it's easier to see what's going on, but the model runs slower, particularly when WIGGLE-ANGLE is high.


THINGS TO NOTICE
----------------
Note that the resulting structure has a branching structure, like a tree.  Why does this happen?

What other phenomena in the world do the shapes remind you of?  Is this aggregation process a plausible model of how those phenomena occur?

When the enclosing circle gets too near to the edge of the screen, the model stops, since allowing the particles to wrap around the edges of the screen would distort the shape of the aggregate.

New red particles are created not at the edge of the screen, but at the edge of a circle enclosing the current size of the green aggregate, instead of traveling from the edge of the screen.  Also, if a red particle wanders too far outside the circle, it disappears and a new one is created.  Neither of these behaviors is essential to the model -- it is done this way just to the model runs fast.


THINGS TO TRY
-------------
Try different settings for WIGGLE-ANGLE.  What is the effect on the appearance of the resulting aggregate?  Why?

Does the MAX-PARTICLES slider make any difference?  Why or why not?

Do you think the USE-WHOLE-SCREEN? setting has an effect on the appearance of the resulting aggregate?  Why or why not?  Experiment and find out.  If you initially thought differently from what you found, why do you think you thought otherwise?  Can you explain why it does happen the way you found?


EXTENDING THE MODEL
-------------------
What happens if you start with more than one "seed" patch?  What happens if the seed is a line instead of a point?

The rule used in this model is that a particle "sticks" if any of the eight patches surrounding it are green.  What do the resulting structures look like if you use a different rule (for example, only testing the single patch ahead, or using NEIGHBORS4 instead of NEIGHBORS)?

Can you compute the fractal dimension of the aggregate?

If instead of using green, you gradually vary the color of deposited particles over time, you can see more vividly the accretion of "layers" over time.  (The effect is also visually pleasing.)

The model will run faster if the turtles are invisible, so you may want to add a switch that hides them (using the HT command).


NETLOGO FEATURES
----------------
Note the use of the NEIGHBORS and DISTANCEXY primitives.


RELATED MODELS
--------------
The various models in the "Fractals" subsection of the "Mathematics" section of the Models Library demonstrate some other ways of "growing" fractal structures.

The "Percolation" model in the "Earth Science" section produces patterns resembling the patterns in this model.


CREDITS AND REFERENCES
----------------------
The concept of diffusion limited aggregation was invented by T.A. Witten and L.M. Sander in 1981.  Tamas Viczek's book "Fractal Growth Phenomena" contains a discussion, as do many other books about fractals.

To refer to this model in academic publications, please use: Wilensky, U. (1998).  NetLogo Diffusion Ltd Aggregation model. http://ccl.northwestern.edu/netlogo/models/DiffusionLtdAggregation.  Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

In other publications, please use: Copyright 1998 by Uri Wilensky.  All rights reserved.  See http://ccl.northwestern.edu/netlogo/models/DiffusionLtdAggregation for terms of use.
@#$#@#$#@
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@#$#@#$#@
NetLogo 2.0beta5
@#$#@#$#@
set use-whole-screen? false
setup
repeat 450 [ go ]
@#$#@#$#@
@#$#@#$#@