Koch Snowflake Modeling Growth and TCL demo example calculator, numerical analysis

Koch Snowflake Modeling Growth and TCL demo example calculator, numerical analysis V2

This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, gold 20Aug2021


Preface

gold 15aug2021 Here are some calculations for Koch Snowflake Modeling Growth. High school geometry class was to the rescue on constructing the Universe From scratch, Bruce Camber,2016,. Note. When I started this page, I did not know about L-system 2D on this TCL wiki, but still think I could learn some. Strange that some search engines are not pulling this remarkable page L-system 2D on this wiki.


Introduction



Fractal Limits of the simulated reality from Koch snowflake modeling growth


Fractal Code of the simulated reality is similar to repetitive patters in our lives, per lecture of Gregg Braden. Repetitive patterns in our lives hows fractal growth is limited. Koch snowflake. Area or growing curve of Koch snowflake has 1.6X limit. Koch snowflake has 1.6X limit to area growth of fractal pattern. First area increase of Koch snowflake is 1/3, 2nd iteration is 4/27, 3ird iteration is 16/243, 4th iteration is 64/2187. The area limit is 1.6X. Perimeter or edge of fractal simulation can change to infinite complexity. How long fractal pattern or simulation lasts? How complex edge or perimeter can become? The limiting scale of universe may called the Planck length, which means there may be a limit to how large the universe may become, quoting Steven Hawking.



Considerations for Koch Snowflake into TCL code


Most of the online references are calculating the area ratio of the Koch_snowflake to the initial equilateral triangle of side 1. The TCL calculator was calculating the area of the Koch_Snowflake per se. Its a little difficult to find reference testcases or spreadsheets of the Koch_Snowflake_Area(N) to check my area proc and printout in TCL.



Pushbutton Operation

For the push buttons, the 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. For testcases in a computer session, the eTCL calculator increments a new testcase number internally, eg. TC(1), TC(2) , TC(3) , TC(N). The testcase number is internal to the calculator and will not be printed until the report button is pushed for the current result numbers. The current result numbers will be cleared either on the next clear button or on the next solve button. All results will be lost on program exit, unless the report text is saved from the report screen on console.



Conclusions


The perimeter calculations were accurate. Proc :math::snowflake_area needs some work and check on upper bounds of operation.


References:






Screenshots Section


figure 1a.Screenshot, Koch_Snowflake_Modeling_Growth_screenshot


Koch_Snowflake_Modeling_Growth_koch_snowflake_screenshot


figure 1b. Initial Vaporware screenshot, Koch_Snowflake_Modeling_Growth_screenshot


Koch_Snowflake_Modeling_Growth_screenshot



figure 2. report_card,Koch_Snowflake_Modeling_Growth_report


Koch_Snowflake_Modeling_Growth_report


figure 3. Credit. wikipedia/commons/,Koch_Snowflake_Modeling_Growth_patternx


Koch_Snowflake_Modeling_Growth_patternx



figure 4. Credit. wikipedia/commons/, Koch_Snowflake_Modeling_Growth_patterns


Koch_Snowflake_Modeling_Growth_patterns



figure 5. Credit Romero Schmidtke on es.wikipedia/commons, Koch_Snowflake_Modeling_Growth_flake


Koch_Snowflake_Modeling_Growth_flake


figure 6. Credit. wikipedia/commons/f/f0/Flocke.PNG, Koch_Snowflake_Modeling_Growth_pattern_more


Koch_Snowflake_Modeling_Growth_pattern_more



figure 7. Credit. L-system 2D on TCL wiki GS, Koch_Snowflake_Modeling_Growth_pattern_more


Koch_Snowflake_Modeling_Growth_flake_L-system 2D


figure 8. Credit. onlinefractaltools.com/draw-koch-fractal, Koch_Snowflake_Modeling_Growth_flake_brown


Koch_Snowflake_Modeling_Growth_flake_brown


figure 9. Credit. onlinefractaltools.com/draw-koch-fractal, Koch_Snowflake_Modeling_Growth_flake_brown_2


Koch_Snowflake_Modeling_Growth_flake_brown_2



figure 10. Credit. opentextbc.ca/calculusv2openstax, Koch_Snowflake_Modeling_Growth_koch_snowflake_


Koch_Snowflake_Modeling_Growth_koch_snowflake_


Testcases Section


In planning any software, it is advisable to gather a number of testcases to check the results of the program.


Testcase 1, Equilateral triangle of side length 1


Testcase 1. Equilateral triangle has side of length 1. The area of equilateral triangle is expr {1.*1.*1.732/4.}, 0.433. The snowflake formula is expr {(1.*1.*1.732*2) /5.}, 0.692799. Check: the ratio of the two areas approximates expr {0.692/.433}, ~1.598. The exact ratio is 1.6. TCL calculator returns 0.69282032302755092. The exact ratio expr {0.69282032302755092/0.4330127018922193}, returns 1.600000000000000.


table 1 printed in tcl wiki format
quantity value value comment, if any
1:testcase_number
1.0 :initial length
1.0 :iteration :
1.0 :option switch mode, usually 1 , modes = 2, 3, 4:
1.0 :optional constant, nominal 1 ::
1. : experimental decay constant_K1 :
Note : experimental constant_k1 not used :
4.0 : Koch snowflake perimeter from iteration N :
0.4330127018922193 : area initial triangle :
0.57735026918962573 : area first iteration :
0.57735026918962573 : Koch snowflake area from iteration N :
0.69282032302755092 : limit to Koch snowflake area from infinite series :

Testcase 2.length 10


The snowflake starts with equilateral triangle, which has side of length 10. The TCL calculator returns snowflake area of 69.2820


table 2 printed in tcl wiki format
quantity value value comment, if any
2:testcase_number
10.0 :initial length
1.0 :iteration :
1.0 :option switch mode, usually 1 , modes = 2, 3, 4:
1.0 :optional constant, nominal 1 ::
1. : experimental decay constant_K1 :
Note : experimental constant_k1 not used :
40.0 : Koch snowflake perimeter from iteration N :
43.301270189221931 : area initial triangle :
57.735026918962575 : area first iteration :
57.735026918962575 : Koch snowflake area from iteration N :
69.282032302755084 : limit to Koch snowflake area from infinite series :

Testcase 3.length 100


The snowflake starts with equilateral triangle, which has side of length 100. The TCL calculator returns snowflake area of 6928.2032.


table 3 printed in tcl wiki format
quantity value value comment, if any
3:testcase_number
100.0 :initial length
1.0 :iteration :
1.0 :option switch mode, usually 1 , modes = 2, 3, 4:
1.0 :optional constant, nominal 1 ::
1. : experimental decay constant_K1 :
Note : experimental constant_k1 not used :
400.0 : Koch snowflake perimeter from iteration N :
4330.1270189221932 : area initial triangle :
5773.5026918962576 : area first iteration :
5773.5026918962576 : Koch snowflake area from iteration N :
6928.2032302755088 : limit to Koch snowflake area from infinite series :

Testcase 4. Koch_Snowflake_area(N)


The snowflake starts with initial equilateral triangle of sides = 1.


Table Koch_Snowflake_Area(N)with Snowflake_Perimeter(N)
counter Koch_Snowflake_Area(N) Snowflake_Perimeter(N) area ratio to initial triangle area initial triangle comment, if any
0 0.4330127018922193 3.0 1.0 0.4330127018922193 initial equilateral triangle of sides = 1.
1 0.57735026918962584 4.0 1.3333333333333335 0.4330127018922193
2 0.64150029909958417 5.333333333333333 1.4814814814814816 0.4330127018922193
3 0.67001142350401022 7.1111111111111089 1.5473251028806587 0.4330127018922193
4 0.68268303435042177 9.4814814814814792 1.5765889346136261 0.4330127018922193
5 0.68831486139327125 12.641975308641971 1.5895950820505005 0.4330127018922193
6 0.69081789563453777 16.855967078189295 1.5953755920224448 0.4330127018922193
7 0.69193035529732283 22.474622770919058 1.5979447075655311 0.4330127018922193
8 0.69242478181411626 29.966163694558741 1.5990865366957918 0.4330127018922193
9 0.69264452693269107 39.954884926078321 1.5995940163092408 0.4330127018922193
10 0.6927421914298354 53.273179901437764 1.599819562804107 0.4330127018922193
11 0.69278559787301064 71.030906535250338 1.5999198056907142 0.4330127018922193
12 0.69280488962553299 94.707875380333789 1.5999643580847618 0.4330127018922193
13 0.6928134637377652 126.27716717377838 1.5999841591487831 0.4330127018922193
14 0.69281727445431285 168.36955623170448 1.5999929596216815 0.4330127018922193
15 0.69281896810611177 224.49274164227265 1.5999968709429695 0.4330127018922193
16 0.69281972084024457 299.32365552303014 1.5999986093079863 0.4330127018922193
17 0.69282005538874813 399.09820736404026 1.5999993819146607 0.4330127018922193
18 0.69282020407697187 532.13094315205353 1.5999997252954048 0.4330127018922193
19 0.69282027016062697 709.50792420273808 1.5999998779090689 0.4330127018922193
20 0.69282029953114022 946.01056560365078 1.5999999457373639 0.4330127018922193
21 0.69282031258470178 1261.3474208048674 1.599999975883273 0.4330127018922193
22 0.69282031838628466 1681.7965610731567 1.5999999892814547 0.4330127018922193
23 0.69282032096476598 2242.3954147642085 1.5999999952362023 0.4330127018922193
24 0.6928203221107575 2989.8605530189443 1.5999999978827564 0.4330127018922193
25 0.69282032262008719 3986.4807373585927 1.599999999059003 0.4330127018922193
26 0.69282032284645589 5315.3076498114569 1.5999999995817791 0.4330127018922193
27 0.69282032294706419 7087.0768664152747 1.5999999998141239 0.4330127018922193
28 0.69282032299177909 9449.4358218870329 1.5999999999173886 0.4330127018922193
29 0.69282032301165231 12599.247762516045 1.5999999999632839 0.4330127018922193
30 0.69282032302048491 16798.997016688056 1.5999999999836818 0.4330127018922193

Math checks are from www.wolframalpha.com snowflake calculator. The snowflake starts with equilateral triangle of sides = 1. Most of the online references are calculating the area ratio of the Koch_snowflake to the initial equilateral triangle of side 1. The TCL calculator was calculating the area of the Koch_Snowflake per se.


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)-

Appendix TCL programs and scripts

Pretty Print Version


        #  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 "  ***************************"        

Table for Koch_Snowflake_Area(N)


        # 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}


gold 8/20/2021. Fractal geometry updates to Compendium


    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


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Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Thanks, gold 12Aug2021



gold 9/27/2021. Switched some comment signs ;# to #. This a big file. Check earlier editions, if not compatible. Maybe obvious, but this page was written on Windows10 Tcl ports including ActiveTCL. I assume reader can cut and paste on screen, what the reader needs, and tootle on to his own project and own contribution pages to the TCL Wiki.



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