Updated 2015-04-20 17:37:46 by pooryorick

GS (20150418) Here is a method to visualize fraction thanks to its remainders in any base.

 # fracdraw.tcl
 # Author:      Gerard Sookahet
 # Date:        18 Apr 2015
 # Description: Fraction visualization
 # Reference: A Postmodern View of Fractions and the Reciprocals of Fermat Primes
 #            Mathematics Magazine, Vol. 73(2000), pp. 83-97

 package require Tk
  bind all <Escape> {exit}
  option add *Button.relief flat
  option add *Button.foreground white
  option add *Button.background blue
  option add *Button.width 6
  option add *Label.foreground white
  option add *Label.background black
  option add *Entry.background lightblue
  option add *Entry.relief flat
 
 proc FracDraw {{n 1} {d 13} {b 10} H} {
  .c delete all
  set r1 $n
  set d_1 [expr {$d-1}]
  for {set i 1} {$i <= $d_1} {incr i} {
     set r2 [expr {(1.0*$r1*$b/$d - $r1*$b/$d)*$d}]
     set r2r [expr {$r2 - int($r2)}]
     if {$r2r >= .5} {set r2 [expr {int($r2)+1}]}
     if {$r2r < .5}  {set r2 [expr {int($r2)}]}
     set s [expr {$H/($d-1)}]
     set x1  [expr {$r1*$s}]
     set x2  [expr {$r2*$s}]
     set y1 [expr {$H-$x1}]
     set y2 [expr {$H-$x2}]
     .c create line $x1 $y1 $x1 $y2 -width 2 -fill green
     .c create line $x1 $y2 $x2 $y2 -width 2 -fill green
     if {$r2 == $n} break
     set r1 $r2
  }
 }
 
 wm geometry . +100+1
 set H 400
 set num 1
 set den 37
 set base 35
 pack [canvas .c -width $H -height $H -bg black]
 set f1 [frame .f1 -relief flat -bg black]
 pack $f1 -fill x
 label $f1.l1 -text numerator
 entry $f1.e1 -width 4 -textvariable num
 label $f1.l2 -text denominator
 entry $f1.e2 -width 4 -textvariable den
 label $f1.l3 -text base
 entry $f1.e3 -width 4 -textvariable base
 button $f1.br -text Run -command {FracDraw $num $den $base $H}
 button $f1.bq -text Quit -command exit
 pack {*}[winfo children $f1] -side left -padx 2

AMG: Here's a version that uses spinboxes instead of entrys, automatically updates the screen, and has simpler rounding. There are a few other minor tweaks. I also added an arc visualization mode, and arcs and lines can be separately enabled or disabled. The arc mode seems to show interesting patterns with the perimeter of the "envelope" of the arcs, and these patterns are much harder to see with the lines alone.
# fracdraw.tcl
# Author:      Gerard Sookahet
# Date:        18 Apr 2015
# Description: Fraction visualization
# Reference: A Postmodern View of Fractions and the Reciprocals of Fermat Primes
#            Mathematics Magazine, Vol. 73(2000), pp. 83-97

package require Tk
bind all <Escape> {exit}
option add *Button.relief flat
option add *Button.foreground white
option add *Button.background blue
option add *Button.width 6
option add *Label.foreground white
option add *Label.background black
option add *Spinbox.background lightblue
option add *Spinbox.relief flat
option add *Checkbutton.foreground white
option add *Checkbutton.selectColor blue
option add *Checkbutton.background black
option add *Checkbutton.indicatorOn 0

proc FracDraw {n d b H arc line} {
   .c delete all
   set r1 $n
   set d_1 [expr {$d-1}]
   for {set i 1} {$i <= $d_1} {incr i} {
      set r2 [expr {int(($r1*$b/double($d) - $r1*$b/$d)*$d + 0.5)}]
      set s [expr {$H/($d-1)}]
      set x1 [expr {$r1*$s}]
      set x2 [expr {$r2*$s}]
      set y1 [expr {$H-$x1}]
      set y2 [expr {$H-$x2}]
      if {$line} {
         .c create line $x1 $y1 $x1 $y2 -width 2 -fill green
         .c create line $x1 $y2 $x2 $y2 -width 2 -fill green
      }
      if {$arc} {
         set xc [expr {($x1 + $x2) / 2}]
         set yc [expr {($y1 + $y2) / 2}]
         .c create arc\
               [expr {($x1 - $xc) * sqrt(2) + $xc}]\
               [expr {($y1 - $yc) * sqrt(2) + $yc}]\
               [expr {($x2 - $xc) * sqrt(2) + $xc}]\
               [expr {($y2 - $yc) * sqrt(2) + $yc}]\
               -start [expr {$r1 < $r2 ? 45 : -135}]\
               -outline blue -extent 180 -style arc -tag arc
      }
      if {$r2 == $n} break
      set r1 $r2
   }
   .c raise arc
}

wm geometry . +100+1
wm resizable . 0 0
wm title . "Fraction Visualization"
set H 400
pack [canvas .c -width $H -height $H -bg black]
set f1 [frame .f1 -relief flat -bg black]
pack $f1 -fill x
foreach {var val} {numerator 1 denominator 37 base 35} {
   set $var $val
   label $f1.$var-l -text $var
   spinbox $f1.$var-s -textvariable $var -from 1 -to 99 -width 4\
         -command {FracDraw $numerator $denominator $base $H $arc $line}
}
foreach {var val} {arc 1 line 1} {
   set $var $val
   checkbutton $f1.$var-c -text $var -variable $var\
         -command {FracDraw $numerator $denominator $base $H $arc $line}
}
FracDraw $numerator $denominator $base $H $arc $line
button $f1.quit-b -text Quit -command exit
pack {*}[winfo children $f1] -side left -padx 2

Screenshot of the above: