Dimensions after 1 iteration
length | 4 (times initial side length)
area | 4/3 ≈ 1.33333 (times initial area)

Dimensions after 2 iterations
length | 16/3 ≈ 5.33333 (times initial side length)
area | 40/27 ≈ 1.48148 (times initial area)

Dimensions after 3 iterations
length | 64/9 ≈ 7.11111 (times initial side length)
area | 376/243 ≈ 1.54733 (times initial area)
  
Dimensions after 4 iterations
length | 256/27 ≈ 9.48148 (times initial side length)
area | 3448/2187 ≈ 1.57659 (times initial area)

Dimensions after 5 iterations
length | 1024/81 ≈ 12.642 (times initial side length)
area | 31288/19683 ≈ 1.5896 (times initial area)-

        #  pretty print from autoindent and ased editor occurrence
        #  Koch Snowflake Modeling Growth calculator
        #  written on Windows 10
        #  working under TCL version 8.6 
        #  gold on TCL WIKI, 19aug2021
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        frame .frame -relief flat -bg aquamarine4
        pack .frame -side top -fill y -anchor center
        set names {{} {initial length :} }
        lappend names {iteration N: }
        lappend names {optional switch mode, usually 1 , modes = 2, 3, 4: }
        lappend names {optional constant in  mode, nominal 1 : }
        lappend names {answers,   Koch snowflake perimeter from iteration N    : }
        lappend names { area initial triangle : }
        lappend names { area first iteration  : }
        lappend names { Koch snowflake area from iteration N  : }
        lappend names { limit to Koch snowflake area from infinite series  : }        
        foreach i {1 2 3 4 5 6 7 8 9} {
            label .frame.label$i -text [lindex $names $i] -anchor e
            entry .frame.entry$i -width 35 -textvariable side$i
            grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 }
        proc about {} {
            set msg "Calculator for Koch Snowflake Modeling Growth V2
            from TCL WIKI,
            written on TCL 8.6 "
            tk_messageBox -title "About" -message $msg } 
        proc self_help {} {
            set msg "Calculator for Koch Snowflake Modeling Growth V2
            from TCL ,
            #  self help listing
            #  2 given  follow.
            1) initial length  N1
            2) iteration
            3) optional switch mode, usually 1 for no decay
            #  optional decay modes = 2, 3, 4, experimental use
            4) optional variable decay constant, not used now
            #  This calculator uses Koch snowflake proportions 
            #  based on initial length and iteration number
            #  optional switch mode, usually 1 for no decay, 
            #  optional decay modes = 2, 3, 4
            #  experimental constant_k1 reduces later 
            #  optional rise modes = -2, -3, -4
            #  setting _k1 rise factor and extra length set on time waves, 
            #  but deacceleration of events to person as observer on timeline.
            #  not used & open yet, optional constant in decay mode, nominal 1 
            #  For comparison, TCL code may include redundant paths & formulas.
            #  The TCL calculator normally uses modern
            #  units  for convenience to modern users and textbooks.
            #  Any convenient and consistent in/output units might be used
            #  like inches, feet, nindas, cubits, or dollars to donuts.
            #  Recommended procedure is push testcase and fill frame,
            #  change first three entries etc, push solve,
            #  and then push report. Report allows copy and paste
            #  from console to conventional texteditor. For testcases
            #  testcase number is internal to the calculator and
            #  will not be printed until the report button is pushed
            #  for the current result numbers.
            #  Use one line errorx proc to estimate percent errors. 
            #  errorx proc is used in the report window (console).
            #  Additional significant figures are used to check
            #  the TCL program, not to infer the accuracy
            #  of inputs and product reports.
            #  Proc precisionx <AM> is loaded and may be used to
            #  trim lengthy mantissas.
            #  precisionx used on select numbers only, not used on every number.
            #  Conventional text editor formulas or  grabbed from internet
            #  Screens can be pasted into green report console.
            #  Try copy and paste following into green screen console
            #  set answer \[* 1. 2. 3. 4. 5. \]
            #  returns  120
            #  gold on  TCL Club, 30apr2021 "
            tk_messageBox -title "self_help" -message $msg }
        proc precisionx {precision float}  {
            #   tcl:wiki:Floating-point formatting, <AM>
            #  select numbers only, not used on every number.
            set x [ expr {round( 10 ** $precision * $float) / (10.0 ** $precision)} ]
            #   rounded or clipped to nearest 7nd significant figure
            set x [ format "%#.7g" $x ]
            return $x
        }
        #  Use one line errorx proc to estimate percent errors 
        #  errorx proc is used in the report window (console)
        proc errorx  {aa bb} {expr { $aa > $bb ?   (($aa*1.)/$bb -1.)*100. : (($bb*1.)/$aa -1.)*100.}}
              #  adapted from tcl-wiki Stats 2011-05-22, arithmetic mean  [RLE]
        # 
        #  ::math::snowflake_area --
        # 
        #  Return the division of quantity by  two or more given ratios
        # 
        #  Arguments:
        #     length  first value is length
        #     iteration second value is length
        # 
        #  Results: snowflake_area
        # 
        proc ::math::snowflake_area {length iteration} {
            set area_eq_triangle [ expr { ($length*$length*sqrt(3.))/4.} ] 
            set area_snowflake_nth [ expr { ($area_eq_triangle / 5.) * (8.-3.*(4./9.)**$iteration)}]
            return $area_snowflake_nth
        }
        proc calculate {     } {
            global side1 side2 side3 side4 side5
            global side6 side7 side8 side9 
            global side10 side11 side12
            global g_constant gr age_years reference_year
            global testcase_number golden_time_cycle  
            global option_mode optional_constant seed_date
            incr testcase_number 
            set side1 [* $side1 1. ]
            set side2 [* $side2 1. ]
            set side3 [* $side3 1. ]
            set side4 [* $side4 1. ]
            set side9 1.
            set side10 1.
            set side11 1.
            set option_mode [ expr { $side3*1.0 } ]
            set optional_constant [ expr { $side4*1.0 } ]
            set seed_date  [ expr { $side1 + $side2 } ]
            #  golden conjugate is  0.61803398874989484820
            #  alternate test value for 
            #  golden conjugate is 0.618
            #  but do not see much difference on small numbers
            #  under 20
            #  initialize golden ratio constants
            set gr 1.61803398874989484820
            set g_constant 0.61803398874989484820
            #  experimental constant_k1 reduces later 
            #  time waves in length after 1st occurence
            set constant_k1 1.
            #  setting 
            if { $side3 == 1. } {set constant_k1 1. }
            set side12 $constant_k1
            set length $side1 
            set iteration $side2 
            set area_initial_triangle [ expr { ($side1*$side1*sqrt(3.))/4.} ]
            set area_first_iteration [ expr { ($side1*$side1*sqrt(3.))/3.} ]
            set side6 $area_initial_triangle
            set side7 $area_first_iteration
            set koch_snowflake_perimeter [ expr { 3.*$side1*((4./3.)**$iteration)} ]
            #  koch_snowflake_area is sum of infinite series of Nth areas
            set koch_snowflake_perimeter [ expr { 3.*$side1*((4./3.)**$iteration)} ]
            set side5 $koch_snowflake_perimeter 
            set side8 [::math::snowflake_area $length $iteration]
            set koch_snowflake_area [ expr {(2.*$side1*$side1*sqrt(3.))/5.} ]
            set side9 $koch_snowflake_area 
            #  we have computed the set 
             }
        proc fillup {aa bb cc dd ee ff gg hh ii} {
            .frame.entry1 insert 0 "$aa"
            .frame.entry2 insert 0 "$bb"
            .frame.entry3 insert 0 "$cc"
            .frame.entry4 insert 0 "$dd"
            .frame.entry5 insert 0 "$ee"
            .frame.entry6 insert 0 "$ff" 
            .frame.entry7 insert 0 "$gg"
            .frame.entry8 insert 0 "$hh" 
            .frame.entry9 insert 0 "$ii" 
             }
        proc clearx {} {
            foreach i {1 2 3 4 5 6 7 8 9 } {
                .frame.entry$i delete 0 end } }
        proc reportx {} {
            global side1 side2 side3 side4 side5
            global side6 side7 side8 side9 gr
            global side10 side11 side12  
            global testcase_number golden_time_cycle
            global g_constant gr age_years reference_year
            global option_mode optional_constant
            console show;
            puts "%|table $testcase_number| || printed in tcl wiki format|% "
            puts "&| quantity| value  |value| comment, if any|& "
            puts "&| $testcase_number:|testcase_number || |&"
            puts "&| $side1 :|initial length |   |   |&"
            puts "&| $side2 :|iteration : | | |& "  
            puts "&| $side3 :|option switch mode, usually 1 , modes = 2, 3, 4:  | | |& "
            puts "&| $side4 :|optional constant, nominal 1 :: | | |& "
            puts "&| $side12 :| experimental decay constant_K1 : | | |& "
            puts "&| Note :| experimental constant_k1 not used : | | |& "
            puts "&| $side5 :|   Koch snowflake perimeter from iteration N  : |  | |& "
            puts "&| $side6 :|  area initial triangle  : |  | |& "
            puts "&| $side7 :|  area first iteration  :  |  | |&"
            puts "&| $side8 :|  Koch snowflake area from iteration N : |  | |&"
            puts "&| $side9 :| limit to Koch snowflake area from infinite series  : |   | |&"
            } 
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 1. 2. 1. 1.  5.33 0.433 0.577 5.33 0.69    }
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 1. 3. 1. 1. 7.11 0.433 0.577 7.11 0.69  }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 1. 4. 1. 1. 9.48 0.433 0.577 9.48 0.69}
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command {about}
        ::ttk::button .self_help -text self_help -command {self_help }
        ::ttk::button .cons -text report -command { reportx }
        ::ttk::button .exit -text exit -command {exit}
        pack .calculator  -in .buttons -side top -padx 10 -pady 5
        pack  .clearallx .cons .self_help .about .exit .test4 .test3 .test2   -side bottom -in .buttons
        grid .frame .buttons -sticky ns -pady {0 10}
               . configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
        wm title . "Koch Snowflake Modeling Growth Calculator V2 " 
        # end of working deck
        # add cosmetics below to bottom of file 
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . " Report for Koch Snowflake Modeling Growth Calculator V2  "}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
        puts "  Console wrapper for solution proc"
        puts "  ***************************"
        puts "  ***************************"        

        # pretty print from autoindent and ased editor occurrence
        # Koch Snowflake Modeling Growth table
        # written on Windows 10 ActiveTcl
        # working under TCL version 8.6
        # gold on TCL WIKI, 19aug2021
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        console show
        set tcl_precision 17
        puts "  Console wrapper for solution proc"
        puts "  ***************************"
        puts "  ***************************"
        # ::math::snowflake_area --
        #
        # Return the division of quantity by  two or more given ratios
        #
        # Arguments:
        #    length  first value is length
        #    iteration second value is length
        #
        # Results: snowflake_area
        #
        proc koch_snowflake_perimeter  { side1 iteration }  {return [ expr { 3.*$side1*((4./3.)**$iteration)}]  }
        proc ::math::snowflake_area {length iteration} {
            set area_eq_triangle [ expr { ($length*$length*sqrt(3.))/4.} ]
            set area_snowflake_nth [ expr { ($area_eq_triangle / 5.) * (8.-3.*(4./9.)**$iteration)}]
            return $area_snowflake_nth
        }
        proc precisionx {precision float}  {
            #  tcl:wiki:Floating-point formatting, <AM>
            # select numbers only, not used on every number.
            set x [ expr {round( 10 ** $precision * $float) / (10.0 ** $precision)} ]
            #  rounded or clipped to nearest 7nd significant figure
            set x [ format "%#.9g" $x ]
            return $x}
        proc expansion { length } {
            set counter -1
            puts "%| Table Koch_Snowflake_Area(N)|with Snowflake_Perimeter(N) |||||% "
            puts "&| counter | Koch_Snowflake_Area(N)  | Snowflake_Perimeter(N)  | area ratio to initial triangle| area initial triangle    | comment, if any   |& "
            while { $counter < 30. } {
                incr counter
                puts "&| $counter |   [::math::snowflake_area 1 $counter]  |   [  koch_snowflake_perimeter 1  $counter] |     [/  [::math::snowflake_area 1 $counter]  [::math::snowflake_area 1. 0]  ] |        [::math::snowflake_area 1. 0]     |    |& "
            }  }
        # printout follows
        expansion 1
        # end of working deck
        # add cosmetics below to bottom of file
        console eval {.console config -bg palegreen}
        console eval {.console config -font {fixed 20 bold}}
        console eval {wm geometry . 40x20}
        console eval {wm title . " Report for Koch Snowflake Modeling Growth Calculator V2  "}
        console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}

    proc koch_snowflake_perimeter  { side1 iteration }  {return [ expr { 3.*$side1*((4./3.)**$iteration)}]  }
    # Usage using set answer
    set answer3 [ koch_snowflake_perimeter 1. 5. ]
    proc area_eq_triangle { length } { return [ expr { ($length*$length*sqrt(3.))/4.} ] }
    # area_eq_triangle has to be initial triangle of length XX, if not equilateral triangle of 1.
    # Usage using set answer
    set area_eq_triangle [ area_eq_triangle 1. ]
    proc koch_snowflake_nth  { iteration area_eq_triangle } {return  [ expr { ($area_eq_triangle / 5.) * (8.-3.*(4./9.)**$iteration)}]}
    # Usage using set answer
    set area_eq_triangle [ area_eq_triangle 2. ] ;# answer = 1.7320508075688772
    set answer2  [ koch_snowflake_nth 1 $area_eq_triangle ] ;# answer2 = 2.3094010767585034