**Time Fractals in Golden Ratio Proportions and TCL demo example calculator, numerical analysis**

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] 30Apr2021
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<<TOC>>
*** Introduction***
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[gold] Here are some calculations on time fractal windows. This calculator uses golden ratio proportions to predict successive time windows or successive time fractals of similar probable occurrences based a seed time or initial age in decimal years. There is plenty of uncertainty about probable occurrences events after the seed time in decimal years, but the the probable occurrences are largely based on growth, accumulation, and succession following the golden ratio proportions. Not all events in time have  golden ratio proportions.
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*** Body ***
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The golden ratio constant is 1.6180339887… As used in the TCL program, the golden ratio conjugate is 0.6180339887… In some circles, a peak is considered 1.6X and a dip is considered 0.6X. The most commonly used Fibonacci ratios as dips include the  23.6%, 38.2%, 50%, 61.8%, and 78.6% shorts. A version of 61.8% is loaded in the TCL program as 0.618... Not sure these Fibonacci ratios apply on all occasions, but there is considerable interest in predicting peaks and dips in Bitcoin cryptocurrency.
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#; derivation of analyst Fibonacci constants
set $g_constant1 1.61803398874989484820
set result [ expr {  1./ ($g_constant1 * $g_constant1 ) }]
#; result out = 2.6180339887498945 
set  g_constant2  1.61803398874989484820
expr {  1./ ($g_constant * $g_constant )  }
#; result out = 0.38196601125010515
======
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The golden time point in any duration is defined 61.8% in its entire length of time. Golden time segments may defined as multiple and alternating time segments in proportion to Fibonacci series constants. An example is 3:2:5:2 (here, 4 time segments approximating golden ratio equivalence 1.618:1:1.618:1). The reported testcase  is roughly 3:2:2:5. 
 
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*** Conclusions ***
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The TCL calculator seems to be working as it stands and matches the scanty textbook examples. The calculator carries the numbers out to the TCL 8.6 maximum (17 places), but suggest there is about a 5 per cent accuracy inherent in most inputs and the probable event outputs. One relative error calculation in TCL notation was vis expr {(1 -(28.797 / 27.506))* 100. } >>   4.69 accuracy no units.
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*** Pseudocode, Equations, and Wiki Page Checklist ***
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     #pseudocode can be developed from rules of thumb.
     #pseudocode: some problems can be solved by proportions (rule of three), to some order of magnitude
     #pseudocode: enter quantity1,  quantity2, quantity3 and expected output (quantity4) for testcases.
     #pseudocode: enter time in years, number of remaining items
     #pseudocode: output fraction of (remaining items) over (items at time zero)
     #pseudocode: ouput remaining items as fraction or percent
     #pseudocode: output fraction of (quantity4 ) over ( quantity1 at time zero)
     #pseudocode: output fraction of (quantity2) * (quantity3 ) over (quantity1 at time zero)
     #pseudocode: outputs should be in compatible units.
     #pseudocode: rules of thumb can be 3 to 15 percent off, partly since g..in g..out.
     #pseudocode: need test cases > small,medium, giant
     #pseudocode: need testcases within range of expected operation.
     #pseudocode: are there any cases too small or large to be solved?
     #
     # F(N+1) == F(N) + F(N-1)
     # F(1) == 1
     # F(2) == 1
     set Grade_School 6
     set Middle_School 2
     set High_School 4
     set College 4
     set Graduate_School 2
     set values [list 6 8 12 16 18]
     set fractals [list 9.7 12.94 19.4 25.888 29.14]
     partials, f(1)= .5 
     f(2) = (1/4) * .5
     f(3) = (1/6) * .5

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***Testcases Section***  

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

**** Testcase 1 ****
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%|table 1|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 1:|testcase_number | |&
&| 11.0 :|initial age decimal years  |   |&
&| 17.798373876248842 :|answers: probable  2nd next occurrence, decimal years  : | |& 
&| 24.596747752497684 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 31.395121628746526 :|probable 4  next  occurrence, decimal years  :   | |&
&| 38.193495504995369 :|probable 5  next  occurrence, decimal years  :   | |&
&| 44.991869381244214 :|probable 6  next  occurrence, decimal years  :   | |&
&| 51.790243257493053 :|probable 7  next  occurrence, decimal years  :   | |&
&| 58.588617133741892 :|probable 8  next  occurrence, decimal years  :   | |&
---- 
======
;# printout
start study of initial conditions 
observer value 17.792999999999999 returns prior reverse
returns prior reverse 10.996678761826878 
prior reverse rnd 10.997 
 seed event rounded 17.793 
end study of initial conditions 
%|table 1|printed in| tcl wiki format|% 
======
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**** Testcase 2 ****
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%|table 2|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 2:|testcase_number | |&
&| 16.0 :|initial age decimal years  |   |&
&| 25.888543819998318 :|answers: probable  2nd next occurrence, decimal years  : | |& 
&| 35.777087639996637 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 45.665631459994955 :|probable 4  next  occurrence, decimal years  :   | |&
&| 55.554175279993274 :|probable 5  next  occurrence, decimal years  :   | |&
&| 65.442719099991592 :|probable 6  next  occurrence, decimal years  :   | |&
&| 75.331262919989911 :|probable 7  next  occurrence, decimal years  :   | |&
&| 85.219806739988229 :|probable 8  next  occurrence, decimal years  :   | |&

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======
;# printout
start study of initial conditions 
observer value 16.0 returns prior reverse
returns prior reverse 9.8885438199983167 
prior reverse rnd 9.8885 
 seed event rounded 16.000 
end study of initial conditions 
#####
start study of initial conditions 
observer value 25.888000000000002 returns prior reverse
returns prior reverse 15.999663900757277 
prior reverse rnd 16.000 
 seed event rounded 25.888 
end study of initial conditions 
======
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**** Testcase 3 ****
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%|table 3|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 3:|testcase_number | |&
&| 10.0 :|initial age decimal years  |   |&
&| 16.180339887498949 :|answers: probable  2nd next occurrence, decimal years  : | |& 
&| 22.360679774997898 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 28.541019662496847 :|probable 4  next  occurrence, decimal years  :   | |&
&| 34.721359549995796 :|probable 5  next  occurrence, decimal years  :   | |&
&| 40.901699437494742 :|probable 6  next  occurrence, decimal years  :   | |&
&| 47.082039324993694 :|probable 7  next  occurrence, decimal years  :   | |&
&| 53.262379212492647 :|probable 8  next  occurrence, decimal years  :   | |&
---- 
======
;# printout
start study of initial conditions 
observer value 10.0 returns prior reverse
returns prior reverse 6.1803398874989481 
prior reverse rnd 6.1803 
 seed event rounded 10.000 
end study of initial conditions 
start study of initial conditions 
observer value 16.18 returns prior reverse
returns prior reverse 9.9997899379732971 
prior reverse rnd 9.9998 
 seed event rounded 16.180 
end study of initial conditions 
======
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**** Testcase 4 ****
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%|table 8|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 8:|testcase_number | |&
&| 15.0 :|initial age decimal years  |   |&
&| 24.270509831248422 :|answers: probable  2nd next occurrence, decimal years  : | |& 
&| 33.541019662496844 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 42.811529493745269 :|probable 4  next  occurrence, decimal years  :   | |&
&| 52.082039324993687 :|probable 5  next  occurrence, decimal years  :   | |&
&| 61.352549156242105 :|probable 6  next  occurrence, decimal years  :   | |&
&| 70.623058987490538 :|probable 7  next  occurrence, decimal years  :   | |&
&| 79.893568818738956 :|probable 8  next  occurrence, decimal years  :   | |&
---- 
======
;# printout
start study of initial conditions 
observer value 15.0 returns prior reverse
returns prior reverse 9.2705098312484218 
prior reverse rnd 9.2705 
 seed event rounded 15.000 
end study of initial conditions 
start study of initial conditions 
observer value 24.27 returns prior reverse
returns prior reverse 14.999684906959946 
prior reverse rnd 15.000 
 seed event rounded 24.270 
end study of initial conditions 
======
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**** Testcase 5, successive Fibonacci numbers ****
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%|table 10|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 10:|testcase_number | |&
&| 144.0 :|initial age decimal years  |   |&
&| 232.99689437998484 :|answers: probable  2nd next occurrence, decimal years  : | |& 
&| 321.99378875996968 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 410.99068313995451 :|probable 4  next  occurrence, decimal years  :   | |&
&| 499.98757751993935 :|probable 5  next  occurrence, decimal years  :   | |&
&| 588.98447189992419 :|probable 6  next  occurrence, decimal years  :   | |&
&| 677.98136627990903 :|probable 7  next  occurrence, decimal years  :   | |&
&| 766.97826065989386 :|probable 8  next  occurrence, decimal years  :   | |&
---- 
======
;# printout
start study of initial conditions 
observer value 144.0 returns prior reverse
returns prior reverse 88.996894379984852 
prior reverse rnd 88.997 
seed event rounded 144.00 
end study of initial conditions
 start study of initial conditions 
observer value 232.99600000000001 returns prior reverse
returns prior reverse 143.99944724277049 
prior reverse rnd 144.00 
seed event rounded 233.00 
end study of initial conditions 
======
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If the entry is a Fibonacci number greater than zero, the TCL calculator should approximate the next   successive Fibonacci numbers as reals, but need to round to nearest integer. The On-Line Encyclopedia of Integer Sequences A000045 gives the Fibonacci numbers as follows 
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269.
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%|table 5|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 5:|notes on Elliott Wave Principle, possible peak curve rises and falls from Fibonacci numbers | |&
&| wave number | entry (144)  is  Fibonacci number  |   |&
&| wave 1  :|161.8%   |new bull (+) or bear (-) market and is usually accompanied by sentiment extremes, possible Fibonacci 161.8% plus|& 
&| wave 2 :|61.8%   | possible Fibonacci 61.8% or 78.6% retrenchment|& 
&| wave 3 :|161.8%   | possible Fibonacci  161.8% advance|& 
&| wave 4:|38.2%   | possible Fibonacci 38.2% retrenchment, sideways market|&
&| wave 5:|32.6%   | possible Fibonacci final leg in the direction of the dominant trend|&
&| wave 5:|50.0%   | possible Fibonacci final leg,  50.0% used as analyst midpoint and lost shirts|&
&| Elliot wave analyst Fibonacci constants :|161.8%.,61.8%, 38.2%, 23.6%, and 50.0 %  | apparently from OEIS A000045 Fibonacci series, <3,5,8>|&
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**** Testcase 6, life cycle of cicada insect  and separate cicada species****
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The cicada insect in the USA has a life cycle of 17 years, believed to be based loosely based on the solar cycle of eleven years. Enter 10 decimal years in the TCL calculator, result rounds down to 17 years. The cicada insects emerge in May of the seventeenth year,  so the hand TCL expression was  expr { 17 + 5./12 } as  17.4166 or rounded 17.4 decimal years. Comment that time coincidence is not proof of causation.
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Another cicada insect species in the USA has a life cycle of 13 years, possibly an unknown factor may be involved. From the Elliot wave theory, there are other Fibonacci constants for breakpoints. Skipping a number in the Fibonacci sequence (55/144 and 144/55) produces 2 more Fibonacci  breakpoint constants,   38.2 percent  and 261.8%. In this case the Fibonacci break constant  38.2% is of interest, so the hand TCL expression was  expr { 10. * ( 1. + 100.* 38.2% ) } as expr {  10. *1.382 } to 13.819 or rounded 13.8 decimal years. In this problem, we used reverse logic or the reverse golden ratio proc. What number of years would return a life cycle of 13 years? So the hand TCL expression was  expr {  $age_years * (1./1.618)   } as    or rounded 8 decimal years.
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======
,# printout
start study of initial conditions 
observer value 13.0 returns prior reverse
returns prior reverse 8.034441853748632 
prior reverse rnd 8.0344 
 seed event rounded 13.000 
======
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%|table 6a |printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 6a :|testcase_number | |&
&| 11.0 :|initial age decimal years  | solar cycle of eleven years   |&
&| 17.798373876248842 :|answers: probable  2nd next occurrence, decimal years.  life cycle of cicada insect : | |& 
&| 24.596747752497684 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 31.395121628746526 :|probable 4  next  occurrence, decimal years  :   | |&
&| 38.193495504995369 :|probable 5  next  occurrence, decimal years  :   | |&
&| 44.991869381244214 :|probable 6  next  occurrence, decimal years  :   | |&
&| 51.790243257493053 :|probable 7  next  occurrence, decimal years  :   | |&
&| 58.588617133741892 :|probable 8  next  occurrence, decimal years  :   | |&
----
---- 
======
;# printout
start study of initial conditions 
observer value 11.0 returns prior reverse
returns prior reverse 6.798373876248843 
prior reverse rnd 6.7984 
 seed event rounded 11.000 
end study of initial conditions 
start study of initial conditions 
observer value 17.0 returns prior reverse
returns prior reverse 10.506577808748212 
prior reverse rnd 10.507 
 seed event rounded 17.000 
end study of initial conditions 
%|table 14|printed in| tcl wiki format|% 
======
 
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%|table 6b|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 6b:|testcase_number | |&
&| 8.0 :|initial age decimal years  ||unknown factor for separate species   |&
&| 12.944271909999159 :|answers: life cycle of cicada insect species, decimal years  : | |& 
&| 17.888543819998318 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 22.832815729997478 :|probable 4  next  occurrence, decimal years  :   | |&
&| 27.777087639996637 :|probable 5  next  occurrence, decimal years  :   | |&
&| 32.721359549995796 :|probable 6  next  occurrence, decimal years  :   | |&
&| 37.665631459994955 :|probable 7  next  occurrence, decimal years  :   | |&
&| 42.609903369994115 :|probable 8  next  occurrence, decimal years  :   | |&
&| 12.944 :|probable  1st next occurrence, rounded or clipped : | life cycle of cicada insect, separate species |& 
&| Elliot wave analyst Fibonacci constants :|161.8%.,61.8%, 38.2%, 23.6%, and 50.0 %  | apparently from OEIS A000045 Fibonacci series, <3,5,8>|&
&| breakpoint from alternate Elliot wave theory :|expr { 10. * ( 1. + 100.* 38.2% ) }  13.8 decimal years  | Elliot wave theory , not proof of causation. |&
----
======
;# printout
start study of initial conditions 
observer value 8.0 returns prior reverse
returns prior reverse 4.9442719099991583 
prior reverse rnd 4.9443 
seed event rounded 8.0000 
end study of initial conditions 
start study of initial conditions 
observer value 13.0 returns prior reverse
returns prior reverse 8.034441853748632 
prior reverse rnd 8.0344 
 seed event rounded 13.000 
end study of initial conditions 
%|table 16|printed in| tcl wiki format|% 
======
======
;# wave analyst Fibonacci constants
;# using Fibonacci series
expr 5./11 = 0.4545 ( ~ .5)
expr 5./13 = 0.38461538461538464
expr 5./21 = 0.23809523809523808
expr 5./34 = 0.14705882352941177
;# wave analyst Fibonacci constants
;# using golden ratio
expr { 1/.618  } =  1.618
expr { 1/1.618 } = 0.6180469715698392
expr { .618 *.618 } = 0.381924
expr { .618 *.618 *.618 } = 0.236029032
expr { .618 *.618 *.618*.618 }  =  0.14586594177599999
======

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**** Testcase 7 ****
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Most agree that humans start puberty at 13 years old and become mature at 21 years. Enter 13 years for puberty in  calculator  and receive rounded 21 years for human maturity.
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%|table 17|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 17:|testcase_number | |&
&| 13.0 :|initial age decimal years  |   |&
&| 21.034441853748632 :|answers: probable  2nd next occurrence, decimal years  : | |& 
&| 29.068883707497264 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 37.103325561245896 :|probable 4  next  occurrence, decimal years  :   | |&
&| 45.137767414994528 :|probable 5  next  occurrence, decimal years  :   | |&
&| 53.17220926874316 :|probable 6  next  occurrence, decimal years  :   | |&
&| 61.206651122491792 :|probable 7  next  occurrence, decimal years  :   | |&
&| 69.241092976240424 :|probable 8  next  occurrence, decimal years  :   | |&

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======
#;
start study of initial conditions 
observer value 13.0 returns prior reverse
returns prior reverse 8.034441853748632 
prior reverse rnd 8.0344 
 seed event rounded 13.000 
end study of initial conditions 
======
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**** Testcase 8 ****
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Most of the golden ratio problems I can find deal with dimensions rather than time. According to Nick Braden in the book Fractal Time , one complete turn of a DNA strand is 34 angstrom units  in length and 21 angstrom units wide. Enter 21 angstrom units  for DNA width  in TCL calculator  and receive rounded 34 angstrom units  for DNA height.
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Perhaps an algorithm for the reverse progress would be useful, Height >> Width. The TCL expression would be expr { 34. * ( 1/ 1.618  ) } or 21.013   angstrom units width.
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%|table 18|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 18:|testcase_number | |&
&| 21.0 :|angstroms width |   |&
&| 33.978713763747791 :|answers: probable  2nd next occurrence, angstroms  : | |& 
&| 46.957427527495582 :|probable 3  next  occurrence, decimal years  :   | |& 
&| 59.936141291243374 :|probable 4  next  occurrence, decimal years  :   | |&
&| 72.914855054991165 :|probable 5  next  occurrence, decimal years  :   | |&
&| 85.893568818738956 :|probable 6  next  occurrence, decimal years  :   | |&
&| 98.872282582486747 :|probable 7  next  occurrence, decimal years  :   | |&
&| 111.85099634623454 :|probable 8  next  occurrence, decimal years  :   | |&
 
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======
start study of initial conditions 
observer value 21.0 returns prior reverse
returns prior reverse 12.978713763747791 
prior reverse rnd 12.979 
 seed event rounded 21.000 
end study of initial conditions 
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======
            proc reverse_time_fractal {age_years} {
            set g_constant 1.61803398874989484820
            set reverse_time_fractal  [ expr {  $age_years * (1./$g_constant)   } ]
            ;#return $reverse_fractal
            }
======
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**** Testcase 10 ****
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The starfish has 5 arms and the  dimensions of a pentagon in length, rather than time units. Enter 1 inch  for  width between two arms in TCL calculator  and receive rounded 1.816 units  for maximum width or length across two alternate arms. The units cancel out in the calculator, so one may enter 8 centimeters and receive proportional  centimeters across 2 alternate arms, using   proportional calculations with the golden mean. 
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%|table 8|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 8:|testcase_number | |&
&| 8.0 :|aa quantity , centimeters  | between 2 arms   |&
&| 2.0 :|bb quantity   | |& 
&| 3.0 :|cc quantity   | |& 
&| 4.0 :|dd quantity   | |&
&| 20.943391999999999 :|probable 2nd next occurrence, decimal years  : | |&
&| 33.886408255999996 :|3rd next occurrence, decimal years : |  |&
&| 12.943999999999999 :|probable  1st next occurrence, decimal years : |  |&
&| 12.944 :|centimeters  : | units cancel out, maximum starfish dimensions |&
----
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**** Testcase 11 ****
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Ratio of male bees to female bees in hive, 1:1.618.  A single hive can have from 10,000 to well over 60,000 bees. Female should be fraction expr { 1.618/ (1.+1.618) } or 0.618.

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***Screenshots Section***
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****figure 1.Golden_Ratio_screenshot****
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[Golden_Ratio_screenshot]
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****figure 2.time_fractals_equation****
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[time_fractals_equation]
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****figure 3.time_fractal_dummy_curve****
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[time_fractal_dummy_curve]
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****figure 4. Time_Fractal_Starfish****
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[Time_Fractal_starfish]
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****figure 5.Golden_Ratio_Proportions****
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[Golden_Ratio_Proportions]
----****figure 6.Golden_Ratio_wavdesfinition****
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[Golden_Ratio_definition][Golden_Ratio_waves]
----****figure 7.Golden_Ratio_waves****
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[Golden_Ratio_more_waves]
 
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***References:***
   * Wikipedia search engine < time  >
   * Wikipedia search engine < golden ratio proportions  >
   * Wikipedia search engine <  Fibonacci  >
   * Google search engine < fractal time calculator Braden Greg  >
   * Book >> Fractal Time: The Secret of 2012 and a New World Age
   * Paperback – Illustrated, February 1, 2010
   * book(s) by Braden Gregg
   * www.greggbraden.com/fractal time calculator
   * Website articles by Tony Spilotro
   * Bitcoin Mathematics: Why 21 Million BTC May Have Been Chosen
   * Extreme interest in trading Bitcoin cryptocurrency [golden ratio tops ] 
   * Fibonacci Day: How To Use Math To Trade Bitcoin And Altcoins
   * Web article Mathematical Mystery: Why Did The Bitcoin Rally Stop At The Golden Ratio?
   * Crypto Calculated: How Ancient Math Predicts Bitcoin’s Next Top At $270K
   * Fibonacci Day: How To Use Math To Trade Bitcoin And Altcoins
   * by Tony Spilotro
   * Understanding Bitcoin’s Market Cycles: 3 Simple indicators for future tops and bottoms
   * Collected Works of R. N. Elliot
   * The Wave Principle. Nature's Law: The Secret of the Universe. R. N. Elliot
   * Series of Articles Published in 1939 by  Ralph Nelson Elliott.
   * Elliott Wave Principle by A.J. Frost and Robert Prechter
   * Elliott Wave Principle: Key To Market Behavior
   * Elliott, Ralph Nelson, Frost, Alfred John, Prechter, Robert Rougelot
   * R.N. Elliott's Masterworks: The Definitive Collection
   * 318 Pages · 1994 English
   * by R. N. Elliott & Robert R. Prechter & Jr.
   * Fractal Time.  coded in python ,  sourceforge.net_projects_fractaltimecalc
   * Golden Ratios in Energy Radiation and Vibrations
   * May 23, 2012 by Gary Meisner 1.6180339887498948420
   * Terence McKenna.   1998, TimeWave Zero Software
   * Essay: Timewave Zero .Pdf by Terence McKenna
   * contributions on Time Code Software by Peter Meyer.
   * Peter J. Meyer , Peter Johann Gustav Meyer, born 1946 
   * Mathematics of Timewave Zero by Peter Meyer.
   * appeared in the Invisible Landscape, 2nd edition, HarperCollins, 1993
   * www.science20.com/hammock_physicist/fibonacci_butterflies
   * by Johannes Koelman, August 6th 2009  
   * en.wikipedia.org/wiki/Anosov_diffeomorphism,  Dmitri Victorovich Anosov
   * Fractal universe and the speed of light: Revision of the universal constants
   * Antonio Alfonso-Faus
   * Weinberg, S. (1972) Gravitation and Cosmology. Wiley,
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**Appendix Code**

***appendix TCL programs and scripts ***

======
        ;# pretty print from autoindent and ased editor occurrence
        ;# Time Fractal Proportions calculator
        ;# written on Windows 
        ;# working under TCL version 8.6 
        ;# gold on TCL WIKI, 30apr2021
        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 age decimal years  :} }
        lappend names {answers:  probable 2nd  next occurrence, decimal years  : }
        lappend names {probable 3  next  occurrence, decimal years  : }
        lappend names {probable 4  next  occurrence, decimal years  : }
        lappend names {probable 5  next  occurrence, decimal years  : }
        lappend names {probable 6  next  occurrence, decimal years  : }
        lappend names {probable 7  next  occurrence, decimal years  : }
        lappend names {probable 8  next  occurrence, decimal years  : }  
        foreach i {1 2 3 4 5 6 7 8} {
            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 Time Fractal Proportions V2
            from TCL WIKI,
            written on TCL 8.6 "
            tk_messageBox -title "About" -message $msg } 
        proc self_help {} {
            set msg "Calculator for Time Fractal Proportions V2
            from TCL ,
            ;# self help listing
            ;# 1 given  follow.
            1) initial age decimal years  N1
            ;# This calculator uses golden ratio proportions 
            ;# to predict successive time windows or time fractals 
            ;# of similar probable occurrences based 
            ;# a seed time or initial age decimal years.
            ;# There is plenty of uncertainty about probable occurrences
            ;# and events after the seed time, but the 
            ;# the probable occurrences are largely based on growth,
            ;# accumulation, and succession
            ;# following the golden ratio proportions.
            ;# Not all events in time have  golden ratio proportions.
            ;# 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.
            ;# This posting, screenshots, and TCL source code is
            ;# copyrighted under the TCL/TK 8.6 license terms.
            ;# Editorial rights retained under the TCL/TK license terms
            ;# and will be defended as necessary in court.
            Conventional text editor formulas or  grabbed from internet
            screens can be pasted into green 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 5ird significant figure
            set x [ format "%#.5g" $x ]
            return $x
        }
        proc time_fractal {age_years} {
            set g_constant .6180339887498948420
            ;# golden ratio is 1.6180339887498948420
            set year_occurrence  [ expr {  $age_years + $g_constant * $age_years  } ]
            return $year_occurrence
            }
        proc reverse_time_fractal {age_years} {
            set g_constant 1.61803398874989484820
            set reverse_time_fractal  [ expr {  $age_years * (1./$g_constant)   } ]
            ;#return $reverse_fractal
            }
       proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8 
            global testcase_number 
            incr testcase_number 
            set side1 [* $side1 1. ]
            set side2 [* $side2 1. ]
            set side3 [* $side3 1. ]
            set side4 [* $side4 1. ]
            set age_years [ expr { $side1*1.0 } ]
            ;# 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
            set g_constant 0.61803398874989484820
            set year_occurrence [    time_fractal $age_years   ] 
            set side2  $year_occurrence
            ;# start study of initial conditions on seed event
            puts "start study of initial conditions "
            ;# reverse operation is of interest in
            ;# observer problem finding seed event value if unknown
            puts "observer value $age_years returns prior reverse"
            puts "returns prior reverse [reverse_time_fractal $age_years] "
            puts "prior reverse rnd [ precisionx 6 [reverse_time_fractal $age_years]] "
            puts " seed event rounded [precisionx 5  $age_years ] " 
            puts "end study of initial conditions "
            set time_cycle [ expr { $year_occurrence - $age_years      }  ]
            set side3 [ expr { $age_years + $time_cycle  + $time_cycle    }  ]
            set side4 [ expr { $age_years + 3 * $time_cycle  }  ]
            set side5 [ expr { $age_years + 4 * $time_cycle  }  ]
            set side6 [ expr { $age_years + 5 * $time_cycle  }  ]
            set side7 [ expr { $age_years + 6 * $time_cycle  }  ]
            set side8 [ expr { $age_years + 7 * $time_cycle  }  ]
             }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$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 "$hh" 
             }
        proc clearx {} {
            foreach i {1 2 3 4 5 6 7 8 } {
                .frame.entry$i delete 0 end } }
        proc reportx {} {
            global side1 side2 side3 side4 side5
            global side6 side7 side8 
            global testcase_number
            console show;
            puts "%|table $testcase_number|printed in| tcl wiki format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| $testcase_number:|testcase_number | |&"
            puts "&| $side1 :|initial age decimal years  |   |&"
            puts "&| $side2 :|answers: probable  2nd next occurrence, decimal years  : | |& "  
            puts "&| $side3 :|probable 3  next  occurrence, decimal years  :   | |& "
            puts "&| $side4 :|probable 4  next  occurrence, decimal years  :   | |&"
            puts "&| $side5 :|probable 5  next  occurrence, decimal years  :   | |&"
            puts "&| $side6 :|probable 6  next  occurrence, decimal years  :   | |&"
            puts "&| $side7 :|probable 7  next  occurrence, decimal years  :   | |&"
            puts "&| $side8 :|probable 8  next  occurrence, decimal years  :   | |&" 
            }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 11. 17.79  24.59  31.39  38.19  44.99  51.79 58.58}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 10.0 16.18 22.36 28.54  34.72  42.360  47.08 53.26 }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 15.0 24.27 33.54  42.81   52.08  61.35  70.62 79.89 }
        ::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 . "Time Fractal Proportions Calculator V2 "            
======
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*** 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  TCL 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.   
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**Comments Section**

<<discussion>>
Please place any comments here, Thanks, [gold] 30Apr2021

                              
<<categories>> Numerical Analysis | Toys | Calculator | Mathematics| Example| Toys and Games | Games | Application | GUI
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<<categories>> Development | Concept| Algorithm