[Arjen Markus] (25 august 2003) I got to know them during the hype in the late 1980s: pictures of the Mandelbrot set -- beautiful, enigmatic and so easy to create! 

Later, I learned their counterpart too: Julia sets. And now, more than a decade later, I have finally taken the time to do them in Tcl too.

Here is a quickly written script - I will use it in one of the chapters for the young programmers' tutorial/booklet ...

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 # Mandelbrot and Julia pictures
 #
 proc det_iteration { zx zy cx cy maxiter } {
    set znx $zx
    set zny $zy
    set noiter -1

    for { set i 0 } { $i < $maxiter } { incr i } {
       if { hypot($znx,$zny) > 2.0 } {
          set noiter $i
          break
       }
       set znnx [expr {$znx*$znx-$zny*$zny+$cx}]
       set znny [expr {2.0*$znx*$zny+$cy}]
       set znx  $znnx
       set zny  $znny
    }

    return $noiter
 }

 proc setpixel { image xpix ypix noiter } {
    if { $noiter >= 0 } {
       set noiter [expr {$noiter%11}]
       set colour [lindex {white lightblue blue green yellow orange red purple magenta black} $noiter]
    } else {
       set colour black
    }
    $image put $colour -to $xpix $ypix
 }

 #
 # Create the canvas and the image and fill it
 #
 set width  200
 set height 200

 canvas .c -width $width -height $height
 pack   .c -fill both

 set image [image create photo -width $width -height $height]
 $image blank

 .c create image 0 0 -image $image -anchor nw

 set cdx [expr {4.0/$width}]
 set cdy [expr {4.0/$height}]
 proc mandelbrot_row {cdx cdy row width height} {
    set cy [expr {-2.0+$cdy*$row}]
    for {set col 0} {$col < $width} {incr col} {
       set cx [expr {-2.0+$cdx*$col}]
       set zx 0.0
       set zy 0.0
       set noiter [det_iteration $zx $zy $cx $cy 200]
       setpixel $::image $col $row $noiter
    }
    if { $row < $height } {
       after 1 [list mandelbrot_row $cdx $cdy [incr row] $width $height]
    }
 }
 proc julia_row {cx cy zdx zdy row width height} {
    set zy [expr {-2.0+$zdy*$row}]
    for {set col 0} {$col < $width} {incr col} {
       set zx [expr {-2.0+$zdx*$col}]
       set noiter [det_iteration $zx $zy $cx $cy 200]
       setpixel $::image $col $row $noiter
    }
    if { $row < $height } {
       after 1 [list julia_row $cx $cy $zdx $zdy [incr row] $width $height]
    }
 }

 set row 0
 after 1 [list mandelbrot_row $cdx $cdy $row $width $height]

 #set cx  0.0
 #set cy  0.75
 #after 1 [list julia_row $cx $cy $cdx $cdy $row $width $height]

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[[
[Category Mathematics]
| [Category Graphics]
]]