SOLITU  Part G:  The Unification Construction

Copyright 2015, John Manimas Medeiros


The New Unification Construction That Squares the Circle


Narrative Introduction to the Importance of the Riddle:

The Unification Construction is the successful discovery of the proof of my childhood faith in the Pythagoreans.  The Pythagoreans asked, according to the ancient Greeks:  "Can we construct a circle with exactly the same area as a given square using only the compass and straightedge?"  The ancient Greeks, including Hippocrates, attempted to find a special procedure for a unique geometric construction that resulted in a circle exactly equal in area to a square.  For centuries, geometers and mathematicians searched for a positive solution.  They were right to do so, but they did not understand the true meaning of the riddle, and they did not see the "hidden misleader" that is the mark of all good riddles.  The riddle was never meant to be solved by finding or making up a special construction.  The solution to the riddle is solved by using ordinary standard constructions, the construction tools shown in SOLITU Part D, which are construction tools known to geometers all over the world, and which have been known for centuries.  I understood, since age 14, that the riddle was not intended to have us engage in a game of geometric exercise in order to find a unique construction, but rather was intended to lead us to a much more important understanding of reality, to have us walk out of Plato's cave and see the reality as it really is:  proportion is everything.  


The standard constructions shown in SOLITU Part D show that we can perform all of the necessary mathematical operations using the compass and straightedge:  addition, subtraction, multiplication, division, take the square root, produce the square of a number, square the rectangle, re-construct an angle and a similar right triangle, divide an angle in half, divide a line in half, double a line, double an angle.  We can actually produce the value of pi in a line length, but it is a curved line, the circumference of a circle.  The profoundly important question implied by the riddle is "can we construct a straight line that is equal to pi exactly?"  The misleader in the riddle is the word "exactly."  When we contemplate the meaning of that word, we see that it has astounding implications regarding the meaning of the riddle.  With the word "exactly" it is implied that the geometer who searches for the solution, and the Pythagoreans who composed the question, both know that there is such a thing as a value and a number that is pi exactly.  Thus, once one realizes that the riddle is not about geometric construction, but is about number, and what numbers are, and what mathematics is, then it becomes clear that we are not searching for a unique construction that is different from all other constructions, but we are looking for a set of construction steps, using standard ordinary construction tools, to find the result that yields a straight line equal to pi exactly.  The Institution of Mathematics gave up on this riddle, and lost its treasure, when they listened to a German mathematician in 1884 who said that searching for pi as a straight line was a waste of a mathematician's time.  Since then the mathematicians have behaved like a religious order obligated to defend the religious doctrine that we cannot square the circle. 


They are wrong.  We can square the circle, because proportion is everything.  We can construct a straight line equal to exactly pi, and if we can do so, then Nature can do the same.  And if Nature contains pi exactly in the framework of Evolutionary Proportion, then we are motivated to ask "Why".  Nature does not invent anything that is not useful or necessary, and Nature does invent whatever is useful or necessary.  What then, is the usefulness or necessity of a straight line equal to pi?  It is part of Nature's ruler.  It is a line length in proportion to 1, that enables Nature, or molecules, to measure themselves, which Nature must accomplish in order to assemble molecules into all that there is, all that is in the real, physical universe.  The Unification Construction is therefore the Creator of the Universe, the mechanism that enables matter to be self-assembling and self-organizing.  It is a trinity of constructions, a trinity that is also a unity, a trinity and a unity that is the Creator of Life in the real, physical universe.  That is why it can be called the Secret of Life in the Universe.  It was a secret, but now it is not a secret.  It was lost long ago, at least 2,500 years ago, and probably longer than that.  But now it is found.  I have chosen to work this way, to work alone "secretly" and then reveal the secret all at once, to 71 friends and all the world on the www.  I have done the work this way because I believe that the facts supported by the Unification Construction are very important.  They were once lost and human civilization needs this information.  It is found now, like lost keys found, and my purpose and hope is that the significance of this information will be understood by many and never lost again.  The specific written text here is copyrighted, but you are free to share the information.  Mathematical facts cannot be private intellectual property, cannot be subject to copyright or trademark.  Please share the information as quickly and as widely as you can.  The courtesy of acknowledging that John Manimas www is the source of the information is appreciated. 


For me, the construction of pi exactly is proof that an advanced civilization existed on Earth in pre-history, named "Atlantis" by some.  Whatever it is named, it is real, and they sent us the Pythagorean riddle so that we would know that we are not alone, and that we have parents and they are smart and they wish us the best.  They want us to succeed, like Jesus who also taught us what we need to know in order to succeed.


The Pi-line Constructions and then Quick Verification:

The five pi-lines that lead to the solution:

1)  PL[8] = 3.1416407864998738178455042012388;  (pi*D)

There are two easy constructions for (pi*D):


A)  Construct 1.8, equal to (9/5).  Use the square-the-rectangle procedure to get the square root of 1.8 (The rectangle has L = 1.8, W = 1).  Add the side of the square, sqrt(1.8) to 1.8.

B)  Use the Phi value from our construction of the pentagon.  Phi is the tangent of the left right triangle.  Use the Similar Right Triangle (SRT) method to multiply the line length of Phi times the ratio of Phi to get Phi^2 (2.618033989).  Then use the Similar Right Triangle method to multiply Phi^2 times 1.2 (6/5).  The ratio (6/5) is in our example of a construction of a similar right triangle.


2)  PL[11] = 3.1446055110296931442782343433718;  (pi*F)

Use the phi value (0.618033989) from our left right triangle in the construction of the pentagon.  That value is the cotangent (inverse of tangent).  It is also 2 times the sine of 18 degrees.  Use the square the rectangle procedure (L = phi, W = 1) to get the square root of 0.618033989, which is

0.786151377.  Multiply that line by 4 (double, double again).


3)  PL[9] = 3.1418181818181818181818181818181;  (pi*E)

Construct the ratio (20/11) which is the same as (10/5.5) and (5/2.75) and (4/2.2).  We can see that since 2.2 = (1.2 + 1) the construction of (4/2.2) is probably the most convenient method.  Then, to get 1.2 squared, multiply the ratio 1.2 times the line 1.2 which equals 1.44.  Then multiply line 1.44 by the ratio 1.2 to get (1.2)^3 which equals 1.728.  Multiply the line 1.728 times the ratio of (4/2.2).  Bazinga.  This line equals (864/275).


4)  PL[10] = 3.1426968052735445528926416093549;  (pi*V)

The ratio (40/9) equals 4.444444444, the same as (4/0.9).  Construct the ratio of (9/10) and the ratio (10/9).  Multiply the ratio (10/9) times 4, or just construct the line length (10/9) and then double, double again.  Construct the square root of 2 (diagonal of 1 squared), and then construct the inverse ratio, square root of (0.5).  Then multiply the ratio of sqrt(0.5) times the line (40/9), which is the same as times 4.444444444, using the reliable SRT construction tool.


5)  PL[3] = 3.1399111679090046934014154186896;  (pi*Q)

As stated previously, this construction could be completed by construction of the elaborate ratio

[sqrt((Phi*8) * 48] / 55.  But there is an easier method based on a generic algorithm.



We can create a new ratio with any pair of pi-lines.  Think of each pi multiplier being named "M" or "N."  Then pi-line (pi*M) divided by pi-line (pi*N) equals the ratio (M/N).  Then ratio (M/N) times any pi-line creates a new pi-line equal to:

(pi*Z) * M  which also = pi * (Z* M) 

              N                                    N


Note that if we use the same pi-line in place of our (pi*Z) and our (pi*M) such as (pi*V), then


we get (pi * V^2) , a new pi-line.



In this particular case where we want to get (pi*Q), we multiply the pi-line (pi*V) times the ratio of (pi*E)/(pi*F), which is the same as:

(pi*V) * (pi*E)



I used this algorithm to create hundreds of pi-line ratios and new pi-lines.  Fun, but not the solution.  It was useful for the long-haul learning.


So now we have our five useful pi-lines.

NEXT:  We construct two new special pi lines, pi-line A, the Father (larger), and pi-line B, the Son, (slightly smaller), and we will subtract pi-line B from pi-line A.  The difference, which is in a sense the same thing as the Father and the Son, we could say is, like a "Ghost" because it is small and hidden, but is very important, because it is what enables molecules to measure themselves, which they must do in order to be able to assemble their molecular parts.


We will have line A minus line B, which difference we can call line C.  We will find that our line C equals pi exactly times a very small but constructible number, which I have named "HG" -- HG = 0.1118 exactly.  To accomplish our original goal we will not need to construct a very small line value or ratio value because we can multiply by 10 or 5 or any value we need to for the convenience and practical visibility of the construction. 



The validity of the Unification Construction can be completed in a few minutes using a scientific calculator, either a hand-held calculator or a desktop computer calculator in scientific view.  I am using my Casio fx-115MS that allows saving six values in memory (A, B, C, D, E, F)

1)  PL[8] = 3.1416407864998738178455042012388;  (pi*D)

2)  PL[11] = 3.1446055110296931442782343433718;  (pi*F)

3)  PL[9] = 3.1418181818181818181818181818181;  (pi*E)

4)  PL[10] = 3.1426968052735445528926416093549;  (pi*V)

5)  PL[3] = 3.1399111679090046934014154186896;  (pi*Q)

6)  The inverse of HG: (1 / 0.1118) =  8.9445438282647584973166368515206

            or (1 / 1.118)  =  0.89445438282647584973166368515206

            same as sqrt(0.8) * 1.0000304013863102398967681881625


This Quick Verification by calculator takes me less than two minutes:


A)  By hand-held calculator:

I use my hand held Casio fx-115MS calculator.  The price of this available calculator is about $20.  Similar to a home computer calculator, the numbers are precisely accurate to the 9th decimal place.  If you want to use your computer calculator, it is one of the applications that comes with Accessories.


B)  By computer calculator:

Open Accessories and send "Calculator" to your desktop.  Open Calculator and select View,

then select "scientific."  You find your answer to a trigonometric operation, such as construct

the cosine of 36 degrees by pressing 36 then press "cos" and you will get the answer with about 30 decimal digits.  Those 30 decimal digits are not all correct because a computer math processor is usually precisely accurate only to the 9th decimal place.  Since you can save only one value in memory, you can create a document page named "calc01" and copy the numbers you need to save from the calculator to the document.  Then to perform the final calculations you can copy the numbers from the document and enter (paste) them in the calculator to perform the necessary operations.   


A)  Again using the hand-held calculator, press the following keys, including the parentheses.

Press keys indicated to the vertical bar "|" then press the save key or keys.  I saved to the letters shown below but any order is fine.


1)  1.8, square root, + sqrt(1.8) = | pi-line (pi*D) press save key, save to "C"


2)  sine 18, times 2, sqrt, * 4 = | pi-line (pi*F) press save key, save to "A"


3)  0.55, press (1/X) key [inverse], times 1.44, times 1.2 = | pi-line (pi*E) save to "D"


4)  40 / 9 = , that times sqrt(0.5) = | pi-line (pi*V) press save key, save to "F"


5)  (pi*E) [D key] times (pi*V) [F key] / (pi*F) [A key] = | (pi*Q) save to "B"


6)  (1/0.1118)  There are several ways to construct this number, but the procedure

            shown below seems most convenient:  [HG = 0.1118]


            0.9 * 0.8 = 0.72,          + 4 = 4.72  

            / 5 = 0.944,      + 8 = 8.944

            / 8 = 1.118,      inverse = (1/1.118) = 0.894454382826475

            press save key, save to "E" and THEN PROCEED:


Line A:  (pi*V)^2 / (pi*E) / sqrt(cos 36)


            = 3.49498323774898999309129431119694  (Line A)  ***


            [ / by sqrt(cos 36) is the same as times sqrt(sec 36)]


minus Line B: - [ (pi*V)^2 / (pi*D) ] is the same as

            minus 3.14375317901321112588817420275095  (minus Line B) ***


That is:  3.494983238 minus 3.143753179


= 0.351230058735778867203120108445054  = (Line C)  or [ = pi * HG exactly]  ***


times 5 = 1.75615029367889433601560054222527


times 1/1.118 =  1.57079632708308974598890925064872


same as times 0.89445438282647584973166368515206  ***


times 2 = pi exactly, 3.141592654 16617949197781850129744  ***  to 9th place here

            see important note below on calculator precision!


OR, 0.351230058735778867203120108445054 times 10, / pi

=  1.118000000 20511883699932630279168  good to 9th place.


Note that to multiply by 5 can be done by adding the line 5 times or use the

SRT method to multiplie by the ratio sqrt(5) and then again by sqrt(5).


Important note on the precision of the constructions and calculations:

I hope you have an opportunity to test for the infinite precision of this Unification Construction of pi exactly.  Your hand-held or computer calculator is probably accurately precise only to the 9th decimal place. As one uses a more precise electronic math processor, the number of zero digits to the right of the decimal point will increase.  Only Nature has a math processor of infinite precision.  What value should we accept as "pi-exactly"?  Due to pi-mania of some people, including some mathematicians, pi has been calculated, using high-precision computers, to 50 decimal places, then 100, then 1,000 and far more than a million decimal places.  Those calculations did not accomplish anything useful.  The precision of calculation is meaningful in the real, physical universe only to the 12th or 13th decimal place, because 600 picometers, or 600 times one-trillionth of a meter, which equals 0.000 000 000 600 of a meter, IS THE MAXIMUM WIDTH OF AN ATOM.  That means that if we had a metal rod of a measured length that was 1.000 000 000 600 meters, then we could not change any of the decimal digits following the 6 unless we added another atom of length, which would then make the measurement 1.000 000 001 200 meters.  It is clear that we cannot change any of the digits following the six  in the ending 600 because that would mean we added less than an atom of length to the rod, which we cannot do.  Any process that results in precision to the 12th decimal place has reached not only our technical capacity to measure, but Nature's limit of the smallest possible particle of matter.  There may be smaller particles, but they cannot be added to or subtracted from a physical object to change that object's physical size. 

It is also important to understand that in order for ANY computer or calculator to be accurately precise beyond the 9th decimal place that device must have a special math co-processor. Such a higher accuracy co-processor costs about $4,000.00 USD. You WILL NOT obtain greater precision by downloading a "high precision" calculator from a seller on the Internet. Those applications simply write out more decimal digits to the right of the decimal point but they DO NOT PROVIDE MORE PRECISELY ACCURATE DECIMAL DIGITS. Although they may print out 30, 50 or 100 decimal digits, the true precision is still only accurate to the 9th decimal place.




440,000                   .               = our Line A

139,968 * sr(cos 36)               


.                        440,000                                                 .

     139,968 * 0.899453719973933636130613791812128


= .                         440,000                             .



=  3.49498323774898999309129431119871, our Line A


[inverse = 0.286124405175708052686203980028091]



.  1,000                   .                = our Line B

486 * (cos 36)^2                    


.                        1,000                                                 .

     486 * 0.65450849718747371205114670859141


= .                             1,000                             .



=  3.14375317901321112588817420275195, our Line B


[inverse = 0.318091129633112224056857300375425]



AND:  common denominator is:  139,968 * (cos 36)^2


sqrt(cos 36) * (cos 36) * 440,000        = our Line A

            139,968 * (cos 36)^2


=  0.727673345112677404061330919854849 * 440,000






=  3.49498323774898999309129431119871             CONFIRMED A


minus  .                    288 * 1,000                          .      our Line B



=  3.14375317901321112588817420275203             CONFIRMED B


= 0.351230058735778867203120108445968            = (A minus B)


=  pi * 0.1118000000 20511883699932630279 476   10th place, THE SOLUTION



Same as: 


320176.271849578057786985604736133 - 288,000


or:  32176.2718495780577869856047361


/ by      91610.2453343363205283749025081224


=  0.351230058735778867203120108446405           CONFIRMED


minor differences in extended digits due to calculator limitations


=  pi * 0.1118000000 20511883699932630279 615


and note proportional value for 139,968 * (cos 36)^2


= 6 * 9 * 1.44 * 1.8 = 139.968  * (cos 36)^2


as   54 * 2.592  or  8.64 * 16.2  = 139.968,  and 139.968


times (cos 36)^2 = 91.6102453343363205283749025081224


CONFIRMED, end of calculation by proportions only (not "pi-lines")


What about "Show your work?"

So, some dense mathematicians will say I have not shown my special geometric construction, my Unification Construction, in sufficient detail.  I have not shown each step clearly with only the compass and straightedge.  But that is precisely the point and was the point all along, which I suspected since I was fourteen years old and first heard the riddle.  I suspected immediately that the purpose of the riddle was not to lead toward a special construction, but to lead the student toward a new understanding of what number is and how the universe works, how everything in the universe is subject to the structure of Evolutionary Proportion.  Of course the details did not occur to me way back then in my youth, but came gradually, in pieces, hesitantly, but with repeated reinforcement of the concept that "proportion is everything."  Therefore, I have shown my work, because the work of finding the solution IS NOT AND NEVER WAS A SPECIAL GEOMETRIC TRICK OR GEOMETRIC CONSTRUCTION.  The solution is and always was finding pi-exactly using only the compass and straightedge which means using only the standard and well-known geometric constructions.  The solution that I found and the way to show it is what I have done here, to show that the solution is in fact a calculation, a computation of specific ratios that result in the specific product of pi-exactly times a constructible value X exactly, and that X value I call HG and it is 0.1118.  One of the persistent traits of human beings is that they see what they expect to see and they find what they expect to find.  I expected to find the solution to the riddle and I expected to find the truth in the form of Evolutionary Proportion.


I found the solution for many reasons:  because I was an only-child for nearly 11 years and I developed an independent mind and a high tolerance for aloneness, for doing things by myself and for myself and my way.  That does not mean I did it all alone.  No one accomplishes anything alone.  My childhood included what some people call "poverty" but I was never poor.  I was always surrounded by people who were interesting or very smart.  I attended the best schools in the world with the best teachers in Fairfield, Connecticut:  St. Anthony's K to 4th, Mckinley 5th to 8th, and Andrew Warde (now Fairfield High School) 9th to 12th, with a school year in Italy as an AFS exchange student.  Then Brandeis University for four years and later UMass at Amherst.  I became a "bureaucrat" in 1977, a social worker for the State of Vermont and combined with intentional and planned personal study devoted myself to learning how people learn.  That is what this work is about.  This is not about mathematics and mathematicians.  This is about human psychology and anthropology and how people learn, how much of human knowledge is like an imprint on the mind of a duck (Lorenz).  This is about the New Testament because Jesus, God bless him, tried to teach us that we have a problem with knowledge and authority.  We have difficulty, sometimes great difficulty, changing our mind, which means changing our understanding of reality.  Sometimes, we prefer death and cosmic destruction to learning something new.  Thus, I have shown my work.  Evolutionary Proportion is here, visible to those who have eyes to see.  Evolutionary Proportion is the mechanism that enables molecules to measure and assemble themselves.  Isaac Newton is back.  The universe is a mechanism driven by Evolutionary Proportion.  The universe does not unfold by a random process or by "probability waves" or strings or by whatever the human mind imagines.  The universe has a structure, and sub-structures within larger structures, and it is all structured by Evolutionary Proportion.  A door that was closed to us is now open.  Go into that room now and probe and sniff.  Collect pi-lines and study them to see Evolutionary Proportion unfold.        


The clues I followed  (Narrow is the way…):

I followed many clues (see SOLITU Part B), and the most important scientific clue came from Jesus:  "Narrow is the way of life, and few there are who find it."  I understood that this statement from Jesus is not about moral behavior, but is cold, hard scientific fact.  It is a reference to how difficult it is find pi exactly hidden within the wide body of Evolutionary Proportion.  Jesus taught much science, including both physical science and social science.  The critique of authoritarian behavior in Matthew Chapter 23 is the most important sociological treatise in all of human history.  It is addressed to all people in all cultures in all times.  It tells us what our problem is.  If you want to learn more about the science Jesus taught, get my book and read it:  The Primacy of Stewardship.  Not just one but many icons are broken today. 


My book is sold legally online by Amazon and Barnes and Noble, and well known affiliates in the United Kingdom and Australia.  Independent local bookstores can order the book for you through Ingram, at similar prices when shipping is considered.  Other online offers, such as a PDF file, are illegal and may be a dangerous scam to obtain your personal identifying information.   


Do I know you?

For safety and other reasons, I do not plan to communicate with you by phone or email unless I first receive a signed letter informing me who you are and the nature of your interest or purpose.  I do not want to talk about this material with anyone who has not studied SOLITU first.


Link back to: (Journey List) or (Welcome) page links or (Mindstream) of J. Manimas or (JM Magazine 2015) or back to (SOLITU Contents).