PMandelbrot.nlogo

globals [
  radius ; max distance that can be calculated
]

patches-own [
  creal ; initial value - real part
  cim ; initial value - imaginary part
  zreal ; current value - real
  zim ; current value - imaginary
  dist ; distance from the initial to the current
  done ; stop calculations for the agent - it is probably outside of Mandelbrot set
]

to setup
  clear-all
  ask patches [
    set creal center-x + ( pxcor / max-pxcor / zoom )
    set cim center-y + ( pycor / max-pycor / zoom )
    set zreal creal
    set zim cim
    set done false
  ]
  reset-ticks
end

to go
  tick
  ask patches with [ not done ] [
    carefully [
      let tzreal zreal
      let tzim zim
      set zreal ( tzreal ^ 2 - tzim ^ 2 + creal )
      set zim ( 2 * tzreal * tzim + cim )
      set dist sqrt ( ( zreal - creal ) ^ 2 + ( zim - cim ) ^ 2 )
    ]
    [
      set pcolor white
      set done true
    ]
  ]
  let notdone count patches with [ not done ]
  ifelse notdone > 0 [
    set radius max [ dist ] of patches with [ not done ]
    ; ask patches with [ not done ] [ set pcolor ( dist / radius ) * 10 ] - simple linear mapping
    ask patches with [ not done ] [ set pcolor 100 / ( 1 - ln ( ( dist + 1 ) / ( radius + 1 ) ) )] ; mapping to 0..100
  ]
  [ stop ]
end
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@#$#@#$#@
## WHAT IS IT?

This is NetLogo code that calculates and visualises Mandelbrot set using patches as pixel-agents.

## HOW IT WORKS

This is a simple implementation of [Mandelbrot set](https://en.wikipedia.org/wiki/Mandelbrot_set). Values of c are mapped to x,y coordinates of the world, and calculations are based on the patch coordinates.

## HOW TO USE IT

Slides set the center of visualisation ("center-x" - real, "center-y" - imaginary) and the "zoom" is scale - 1 is 1 in each direction while 50 is 1/50.
"Go" launches calculations of the model iterations.

## THINGS TO NOTICE

How quickly points outside of the Mandelbrot set run away! Explore agents by inspecting patches, and steppping through the iterations.

## THINGS TO TRY

Different values of center-x, center-y and zoom allow you to explore the fractal. For example, (-1.2,0.2) with 12x zoom.

## EXTENDING THE MODEL

One can easily imagine better controls over zoom and colouring. Dynamic zoom is a possibility. Dealing with floating point limitations is a challenging possibility.

## NETLOGO FEATURES

The model is ising "carefullly" block to detect overflow errors and mark affected agent as one outside of the set.

## RELATED MODELS

NetLogo standard library has alternative implementations of Mandelbrot set calculations.

## CREDITS AND REFERENCES

This is free as in free beer software, by Svyatoslav Pidgorny. It lives at https://github.com/SP3269/Agent-Based-Models
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NetLogo 6.0.4
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