[Keith Vetter] 2003-01-26

Start with an equilateral triangle then draw lines connecting the midpoints of each side. Now how many triangles are there? Answer: 5 -- three small "up" triangles, one "down" triangle plus the outer, larger triangles. Repeat the subdivision on all the smaller triangles and now how many triangles are there?

Counting the number of triangles in this type of figure is an old puzzle. Finding a general formula is even trickier.

This little program is a visualization of this problem. I found it while cleaning up my hard drive and I thought I'd share it here.

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 ##+##########################################################################
 # 
 # triags -- Draws nested triangles and displays the total number of
 # triangles there are and also shows how many triangles of each size.
 # 
 # Keith Vetter Jun 24, 1998 - initial revision
 #
 
 ##+##########################################################################
 # 
 # DoDisplay -- puts up our interface
 # 
 proc DoDisplay {} {
    global w h
    
    canvas .c -width $w -height $h -bd 2 -relief ridge
    frame .bottom
    scale .s -orient h -from 1 -to 50 -command DrawTriangles -showvalue 0 \
        -variable S(n) -highlightthickness 0
    set ::S(n) 3
    focus .s
    label .lcntU -text "Up triangles: "
    label .lcntD -text "Down triangles: "
    label .lcntT -text "Total triangles: "
    label .cntU -textvariable S(up)
    label .cntD -textvariable S(down)
    label .cntT -textvariable S(total)
    catch {image create photo ::img::blank -width 1 -height 1}
    button .about -image ::img::blank -highlightthickness 0 -command {
        tk_messageBox -title "About" -message "by Keith Vetter\nJanuary, 2003"}
    pack .c -side top -fill both -expand 1
    place .s -in .c -x 10 -y 5 -anchor nw
    pack .bottom -side bottom -fill x
 
    grid .lcntU .cntU -in .bottom
    grid .lcntD .cntD -in .bottom
    grid .lcntT .cntT -in .bottom
    grid configure .lcntU .lcntD .lcntT -sticky e
    grid configure .cntU  .cntD  .cntT  -sticky w
    grid columnconfigure .bottom 1 -weight 1
    place .about -in .c -relx 1 -rely 1 -anchor se
    update
    bind . <Configure> {triag $S(n)}
 }
 ##+##########################################################################
 # 
 # DrawTriangles -- called when the number of sides changes.
 # 
 proc DrawTriangles {n} {
    global S CNT
    
    .s config -label "Sides: $n"
    triag $n
    set S(up) [CountUp $n]
    set S(down) [CountDown $n]
    set S(total) [expr {$CNT(up) + $CNT(down)}]
 }
 ##+##########################################################################
 # 
 # triag -- draws an N sided triangle
 # 
 proc triag {n} {
    global w h alt len
 
    set w [winfo width .c]
    set h [winfo height .c]
    
    set alt [expr {($h - 20.0) / $n}]           ;# Scale to fit window
    set len [expr {($w - 20.0) / $n}]
    
    .c delete all
    set b [GetBottom $n]                       ;# Bottom coords
    set l [GetLeft $n]                         ;# Left side coords
    set r [GetRight $n]                        ;# Right side coords
    
    foreach {x1 y1} $l {x2 y2} $r {             ;# Parallel to bottom
        .c create line $x1 $y1 $x2 $y2 -width 2
    }
    foreach {x1 y1} $b {x2 y2} $r {             ;# Parallel to left side
        .c create line $x1 $y1 $x2 $y2 -width 2
    }
    foreach {x1 y1} [reverse $b] {x2 y2} $l {   ;# Parallel to right side
        .c create line $x1 $y1 $x2 $y2 -width 2
    }
 }    
 ##+##########################################################################
 # 
 # GetXY -- returns x,y coordinates for this triangle
 # 
 proc GetXY {row col} {
    global w len alt
 
    set y [expr {round(10 + $row * $alt)}]
    set x [expr {round($w/2.0 + $len * ($col - $row / 2.0))}]
    return [list $x $y]
 }
 ##+##########################################################################
 # 
 # GetBottom -- returns coordinates of all bottom side vertices
 # 
 proc GetBottom {n} {
    for {set c 0} {$c <= $n} {incr c} {
        eval lappend result [GetXY $n $c]
    }
    return $result
 }
 ##+##########################################################################
 # 
 # GetLeft -- returns coordinates of all left side vertices
 # 
 proc GetLeft {n} {
    for {set r 0} {$r <= $n} {incr r} {
        eval lappend result [GetXY $r 0]
    }
    return $result
 }
 ##+##########################################################################
 # 
 # GetRight -- returns coordinates of all right side vertices
 #
 proc GetRight {n} {
    for {set r 0} {$r <= $n} {incr r} {
        eval lappend result [GetXY $r $r]
    }
    return $result
 }
 ##+##########################################################################
 # 
 # reverse -- reverses a list (by pairs)
 # 
 proc reverse {l} {
    set r ""
    foreach {x y} $l {
        set r "$x $y $r"
    }
    return $r
 }
 ##+##########################################################################
 # 
 # CountUp -- counts how many upright triangles there are.
 # 
 proc CountUp {n} {
    global CNT
 
    set CNT(up) 0                               ;# Reset all values to zero
    for {set i 1} {$i <= $n} {incr i} {
        set CNT(up,$i) 0
    }
    
    for {set row 0} {$row < $n} {incr row} {
        set size 1
        set num [expr {$n - $row}]              ;# How many size 1 triangles
        while {$num > 0} {
            incr CNT(up,$size) $num
            incr CNT(up) $num
            incr num -1                         ;# One less triangle at size+1
            incr size
        }
    }
    set v "$CNT(up): "
    for {set i 1} {$i <= $n && $CNT(up,$i) > 0} {incr i} {
        append v "$CNT(up,$i) "
    }
    return $v
 }
 ##+##########################################################################
 # 
 # CountDown -- same as count but for upside down triangles
 # 
 proc CountDown {n} {
    global CNT
 
    set CNT(down) 0                             ;# Reset all values to zero
    for {set i 1} {$i <= $n} {incr i} {
        set CNT(down,$i) 0
    }
 
    for {set row 0} {$row < $n-1} {incr row} {
        set size 1
        set num [expr {$n - $row - 1}]          ;# How many size 1 triangles
        while {$num > 0} {
            incr CNT(down,$size) $num
            incr CNT(down) $num
            incr num -2                         ;# Two less triangles at size+1
            incr size
        }
    }
    set v "$CNT(down): "
    for {set i 1} {$i <= $n && $CNT(down,$i) > 0} {incr i} {
        append v "$CNT(down,$i) "
    }
    return $v
 
 }
 
 set w 276
 set h 276
 DoDisplay
 update
 wm geom . [wm geom .]                           ;# Keeps window from growing
 .c config -width 0 -height 0                    ;# Better resizing behavior
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[Category Graphics] | [Category Application]