globals [
  radius   ;; distance of the farthest green patch from the center
]

to setup
  clear-all
  set radius 0
  ask patch 0 0 0
    [ set pcolor green ]
  reset-ticks
end

to go
  ;; stop when we get near the edge of the screen
  if radius >= max-pxcor
    [ 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 (only happens
  ;; USE-WHOLE-WORLD? is false)
  while [count turtles < max-particles and
         count turtles < ticks]
    [ make-new-turtle ]
  ;; now move the turtles
  ask turtles
    [ wander
      if any? neighbors with [pcolor = green]
        [ set pcolor green
          ;; increase radius if appropriate
          if distancexyz 0 0 0 > radius
            [ set radius distancexyz 0 0 0 ]
          die ]
      ;; kill turtles that wander too far away from the center
      if not use-whole-world? and distancexyz 0 0 0 > radius + 3
        [ die ] ]
  ;; advance clock
  tick
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
 create-turtles 1
   [ set color red
     tilt-up asin (1.0 - random-float 2.0)
     ifelse use-whole-world?
       [ jump max-pxcor ]
       [ jump radius + 2 ]
     tilt-up 180
     ; set random roll, so when we start wiggling, there's no bias.
     roll-right random-float 360
]
end

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


; Copyright 2006 Uri Wilensky.
; See Info tab for full copyright and license.
@#$#@#$#@
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@#$#@#$#@
## WHAT IS IT?

This is a 3D version of the 2D model DLA. 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.

## HOW IT WORKS

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-WORLD? switch controls whether the red particles start at the edge of the screen, or from just outside a sphere 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 sphere 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 sphere 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 sphere, 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-WORLD? 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?

Can you find a way to modify the code so the resulting pattern spirals out instead of radiating straight out?

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 `neighbors6` 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 `hide-turtle` command).

## NETLOGO FEATURES

Note the use of the `neighbors` and `distancexyz` 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.

## HOW TO CITE

If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

* Wilensky, U. (2006).  NetLogo DLA 3D model.  http://ccl.northwestern.edu/netlogo/models/DLA3D.  Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

Please cite the NetLogo software as:

* Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

Copyright 2006 Uri Wilensky.

![CC BY-NC-SA 3.0](http://ccl.northwestern.edu/images/creativecommons/byncsa.png)

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

This is a 3D version of the 2D model DLA.

<!-- 2006 3D -->
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