globals [ balls y-upper-bound counters ]
breeds [ pegs falling-balls piling-balls piled-balls ]
patches-own [ shade ]

to setup
  ca
  set-default-shape turtles "circle"
  set balls 0
  ; need y-upper-bound to be even
  set y-upper-bound screen-edge-y - 2 + (screen-edge-y mod 2)
  setup-pegs
end

to go
  locals [stop-running?]
  set stop-running? false
  if balls < number-of-balls
    [ new-ball ]
  balls-fall
  ask counters
    [ if any? falling-balls-at 0 1 
        [ ask falling-balls-at 0 1 [ set breed piling-balls ]
          set plabel plabel + 1 
        ]
      ; stop running when the pile of balls reaches the green patches
      if any? piled-balls-at 0 -1 [ set stop-running? true ]   
    ]
  ifelse pile-up? ; if not pile-up, just kill balls at bucket
     [ pile-up-balls ]
     [ ask piling-balls [ die ] ] 
  if stop-running? or (not any? piling-balls and not any? falling-balls)
  [
    while [any? piling-balls]
    [ pile-up-balls ]
    stop
  ]
  set-peg-angle
  do-trials
end

to setup-pegs
  locals [bucket-xcor]
  set-default-shape pegs "default"
  ask patches with [valid-pycor? pycor and valid-pxcor? pxcor pycor]
    [ sprout 1 [ set color yellow set breed pegs ] ] 
  set-peg-angle
  set counters patches with [valid-counter-patch? pycor]
  ask counters
    [ set plabel 0
      set pcolor scale-color green 40 50 0
      ; set the serial labels at the bottom of the graphics screen
      set plabel-of (patch pxcor (- screen-edge-y)) round ((number-of-rows + pxcor) / 2)
    ]
end

to new-ball
  create-custom-falling-balls 1 
    [ set color red
      set heading 180
      set xcor 0
      set ycor screen-edge-y - 1 + screen-edge-y mod 2
    ]
  set balls balls + 1
end

to balls-fall
  ask falling-balls [
    fd 1
    shade-patch
    fd 1
    shade-patch
  ]
end

to do-trials
  ask falling-balls
    [ ifelse random-float 100 < chance-of-bouncing-right
        [ lt 90 ] ; actually turning right
        [ rt 90 ] ; actually turning left
      fd 1
      shade-patch
      set heading 180
    ]
end

;; pile up balls in columns
to pile-up-balls
  ask piling-balls 
    [ pu
      repeat 2
      [
        ifelse ycor != (1 - screen-edge-y) and not any? piled-balls-at 0 -1
          ; have to make sure piling-balls aren't on top of each other
          [ if not any? piling-balls-at 0 -1 [ fd 1 ] ]
          [ set breed piled-balls ]
      ]
    ]
end

to set-peg-angle
  ask pegs
    [ ifelse chance-of-bouncing-right >= 50
        [ set heading 360 - (chance-of-bouncing-right - 50) * 0.9 ]
        [ set heading 45 - (chance-of-bouncing-right * 0.9) ]
    ]
end

;; shades the path of the balls, but only to a certain darkness
;; you can adjust the darkness by changing the value of darkness
to shade-patch 
  locals [darkness]
  set darkness 1.5
  if shade-path?
    [ ask patch-here
        [ if shade < number-of-balls / darkness
          [ set shade shade + 2
            set pcolor scale-color blue shade number-of-balls 0 
          ]
        ]
    ]
end

;;; predicate like reporters that help with graphic drawing
;;; valid coordinates depend on the number of rows in the triangle
;; a valid y coordinate is even coordinate, below the y-upper-bound
to-report valid-pycor? [y]
  report y >= y-upper-bound - (2 * number-of-rows) and y <= y-upper-bound and y mod 2 = 0
end

;; a valid x coordinate depends on a patches y coordinate
;; vertically, patches with valid x coordinates alter from having
;; odd to even pxcor
to-report valid-pxcor? [x y]
  if x <= pxcor-bounds y and x >= (- (pxcor-bounds y))
    [ if (pxcor-bounds y) mod 2 = 0 and x mod 2 = 1 and x != 0
        [ report true ]
      if (pxcor-bounds pycor) mod 2 = 1 and pxcor mod 2 = 0
        [ report true ]
    ]
  report false
end

;; below last row and inline with a peg
to-report valid-counter-patch? [y]
  report y = y-upper-bound - 2 * (number-of-rows + 1) and (any? pegs-at 1 2 or any? pegs-at -1 2)  
end

;; helps restrict pxcor depending on the size of y.
;; lower y ~ larger pxcor range
to-report pxcor-bounds [y]
  report (y-upper-bound - y) / 2
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 Galton Box model.
; http://ccl.northwestern.edu/netlogo/models/GaltonBox.
; 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/GaltonBox
; for terms of use.
;
; *** End of NetLogo Model Copyright Notice ***
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@#$#@#$#@
WHAT IS IT?
-----------
A Galton Box is a triangular board that contains several rows of staggered but equally spaced pegs.  Balls are dropped from the top, bounce off the pegs and stack up at the bottom of the triangle.  In this model, the user is able to adjust the probability of the balls bouncing right or left when it hits a peg.  

The model enables you to observe how nature produces the binomial coefficients from Pascal's Triangle and their relation to a Gaussian bell-shaped normal curve. The model can also simulate coin tossing experiments with biased coins which result in skewed distributions

There are many applications for the concepts encompassed in a Galton Box.  People employed in a wide variety of fields use binomial and normal distributions to make precise calculations about the likelihood of events or groups of events occurring.
 

HOW TO USE IT                    
-------------
Click the SETUP to set up the rows of the triangle, the number of balls, and other parameters.  Click to GO button to begin the simulation.

The SLOW-DOWN slider controls how fast the model is running.  The larger the value, the slower the model runs.  

The SHADE-PATH? button gives the user the option to have the balls chart their path as they fall down the triangle.  The more balls that cover a particular path, the darker the path will become.  This option provides a graphical tool that helps the user define the most used path.

The PILE-UP? button controls if the balls create piles or simply disappear when they reach the bottom of the triangle.  If PILE-UP? is on and the pile of balls reaches the bottom of the triangle, the model will stop.  Note: If you are running a simulation with a  large number of balls you might want to turn PILE-UP? off.


THINGS TO NOTICE
----------------
With a small number of balls it is hard to notice any patterns as the balls pile up.

As you increase the number of balls, clear patterns and distributions start to form.  By adjusting the CHANCE-OF-BOUNCING-RIGHT you can see how different factors can change the distribution of balls.  What types of distributions form when the CHANCE-OF-BOUNCING-RIGHT is set at 20, 50, or 100?

This model is a good example of an independent trials process.  Each ball has a probability of turning one way, and its decision is unrelated to that of any of the other balls.  The large number of rows the balls must travel affects the amount of variation present in a run of the model.


THINGS TO TRY
-------------
Change the NUMBER-OF-BALLS and NUMBER-OF-ROWS sliders.  How does varying numbers alter how balls stack up?

Change the CHANCE-OF-BOUNCING-RIGHT slider as balls have begun to fall.  What kinds of ball distributions can you produce?

Change the NUMBER-OF-BALLS slider.  What is the best way to produce a standard binomial distribution (or approximate a bell curve)?

Set a CHANCE-OF-BOUNCING-RIGHT then try to predict the resulting stacks of balls.  How would you calculate the mean and variances of a given stack for a given setting?


EXTENDING THE MODEL
-------------------
Modify the program to allow independent adjustment of each peg, so that they can adjust their own orientation, rather than having all the pegs synchronized.

Change the shape of the board.  Maybe flip the triangle upside down.  How does this effect how the balls get distributed?

In addition to changing the shape of the board, change the direction balls can go.  Maybe allow balls to go in all directions.

Make it so you can select a specific peg.  If a ball bounces off that peg, stop the ball.  Keep track of how many balls are stopped.  What specific insight does this provide about the independent trials process and ball distributions?.


RELATED MODELS
--------------
Binomial Rabbits, Random Walk Left Right


NETLOGO FEATURES
----------------
SHADE-PATH? enables balls to increment the shade of the path they travel.  You can easily modify this function to allow different shades.  You might want to try setting a tolerance, so when a specific path is traveled a certain number of times it turns a different color.


CREDITS AND REFERENCES
----------------------
To refer to this model in academic publications, please use: Wilensky, U. (2001).  NetLogo Galton Box model. http://ccl.northwestern.edu/netlogo/models/GaltonBox. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

In other publications, please use: Copyright 2001 by Uri Wilensky.  All rights reserved.  See http://ccl.northwestern.edu/netlogo/models/GaltonBox for terms of use.
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NetLogo 2.0beta5
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