SOLITU  Part F:  Construction of the Pi-lines

As stated previously, a pi-line is a constructible line that is slightly less than or slightly more than the value of pi exactly.  Here is a table of 26 pi-lines that I constructed years ago in the process of searching for the solution to the Pythagorean riddle:

PL = 3.125;             // pi * 0.99471839432434584855552352107821

PL = 3.136773748694594219681304368847;   // G[ sr(5) + 2.2] / sr(2)]

PL = 3.1399111679090046934014154186896;  // Q[sr(Phi*8) * 48] / 55 ]

PL = 3.1406228624894574694952341031297;  // A[2 + sqrt(5) ] - sqrt(6/5)

PL = 3.141320000468716979810539729062;   // B[sqrt(37) - 1] * (1/Phi)

PL = 3.1414803893897418379555528189649;  // C[sqrt(pi*B* pi *D]

PL = 3.1415964392353511387409825201656;  //              [pi*sr(J*G)]

PL = 3.1416407864998738178455042012388;  // D[(9/5) + sqrt(9/5)] (pi*MQ)

PL = 3.1418181818181818181818181818181;  // E[(1/0.55)*1.728]  Pi*MQ*TN

PL = 3.1426968052735445528926416093549;  // V(40/9)*sr(0.5)

PL = 3.1446055110296931442782343433718;  // F[sqrt(1/Phi) * 4 ] (pi*Ho)

PL = 3.1464265445104546409743605410398;  // J[sr(9.9)]

PL = 3.1465153760315117097549686764014;  // H[pi*G*MN*MN], (pi*H)

PL = 3.1466042100605056464977059018508;  //  [pi*sr(G*MX)], GX

PL = 3.14928;       //  pi * MT * 1.002375

PL = 3.1512640759198245358269325763168;  //  (pi*J) * sr(J/G)]

PL = 3.1514420146250458834693530842498;  //  [pi * sr(J * MX)]

PL = 3.1561090448809076493016604787293;  // K[(9.9) / (pi*G) ]

PL = 3.1562872571612667838672356476287;  // L[(pi*H)^2/ (pi*G)]

PL = 3.1564654795045281880219131572653;  // MX[ L^2 / K]

PL = 3.2;     // pi*MT*(55/54)

PL = 3.2360679774997896964091736687313;  // [sr(5)+1]

PL = 3.24;      // pi*MT*1.03125 or (33/32)

PL = 3.4909090909090909090909090909; // 192/55

PL = 3.6;    // pi * (55/48)

PL = 3.84;     // pi*MQ*SE, or pi*MQ*(11/9)*TN

[The value TN = PL / PL and = (secant 18)^4 * (9/11), or

TN = 1.0000564658184571247007181063881]

I used different labels for some of these at different stages of research.  For example: PL was also labelled (pi*MT), and PL was labelled (pi*MQ) at first, then (pi*D).  The other capital letters to the right were used as labels for the pi-line "PL" to the left.  The figures and formulas in brackets are the source of each pi-line, or the procedure with compass and straightedge that produces the pi-line value.  For example:  we get line length PL by constructing the line length of [ sqrt(5) + 2.2] and then dividing that line by the square root of 2.  Try it on your scientific calculator.  You will get 3.136773748694594219681304368847 or some different decimal digital values after the 9th decimal place to the right of the decimal point IF your calculator is precise beyond the 9th decimal place.

I constructed these pi-lines and experimented with many processes, hundreds if not thousands of procedures over the years.  But one small set of pi-lines enables construction of the solution, which I have named the Unification Construction because it unifies squares and circles.  The set of five pi-lines I used are:

The five pi-lines I studied that lead to the solution:

PL = 3.1399111679090046934014154186896;  // Q[sr(Phi*8) * 48] / 55 ]

PL = 3.1416407864998738178455042012388;  // D[(9/5) + sqrt(9/5)] (pi*MQ)

PL = 3.1418181818181818181818181818181;  // E[(1/0.55)*1.728]  Pi*MQ*TN, also called pi*MT

PL = 3.1426968052735445528926416093549;  // V(40/9)*sr(0.5)

PL = 3.1446055110296931442782343433718;  // F[sqrt(1/Phi) * 4 ] = (pi*F) also used to call this pi-line "(pi*Ho)"

Note that there are some different ways to construct each pi-line, which is certainly logical because each "pi-line" is neither more nor less than a constructible ratio value, which is constructed as a line by way or our Similar Right Triangle procedure:  construct the ratio times a line length of 1 and we have converted the ratio value to a line length.  Otherwise, we sometimes construct the line length first, before the ratio.  All of the pi-lines are related in interesting ways.  For example:  sqrt(2) * 3 = sqrt(18) = 4.2426406871192851464050661726291, and divide that value by [sqrt(5) + 2] or by 4.2360679774997896964091736687313.  The result is the value that I labelled as "MN" which is: 1.00155160626656770318654828804.  PL is equal to PL times MN, and PL is equal to PL times MN.  Look again at PL which I had also labelled (pi * H):   3.1465153760315117097549686764014.  That pi-line squared equals the interesting value 9.9005590116027255345371092532424, which is equal to 9.9 times the value I named "TN" described above:  TN = 1.0000564658184571247007181063881.  And, the square root of 9.9 is a pi-line, equals 3.1464265445104546409743605410398 which is PL.  I am not reciting these facts just to torture the reader or to make small talk.  These relationships are all part of the phenonmenon of Evolutionary Proportion.  By identifying and manipulating pi-lines, I was in fact moving toward the solution all along, because I was exploring proportion in depth and in detail.  One might say that I was a scientist collecting proportions.  It has been said that the first step in any science is to collect, to collect examples of the phenomenon that is the subject of study.  So, since I was aware that I was studying proportion, I collected proportions.  The significance of my research was that I was aware that I was not studying pi in order to search for pi.  I understood from the beginning and continuously that I was studying proportion in order to find pi, exactly pi.

Construction of the five pi-lines:

Here I will describe each of the five pi-lines and how they are constructed.  If you have experience making constructions with the compass and straightedge, you will soon see that these are easy constructions.  There are no tricks or special procedures involved, just our toolbox of construction tools.  Here are the five in the order of research history, meaning the order in which I constructed them and treated them as significant and useful.  Remember always, a pi-line is a constructible proportion, a proportional value, also a numerical value.  We are working with proportions because that is the whole point of the riddle, that pi is found as a proportional value that can be constructed the same as other proportional values.  This means a change in number theory, because this means pi is not a "transcendental" number.

1)  PL = 3.1416407864998738178455042012388;  equal to [(9/5) + sqrt(9/5)]

also named (pi*D) or (pi *MQ).  Hereafter I refer to this pi-line as (pi*D)

2)  PL = 3.1446055110296931442782343433718;  equal to [sqrt(1/Phi) * 4 ]

also named (pi*F) or (pi*Ho).  Hereafter I refer to this pi-line as (pi*F)

3)  PL = 3.1418181818181818181818181818181;  equal to [(1/0.55)*1.728]

also named (Pi*MQ*TN) or (pi*MT) or (pi*E) Note that this very interesting pi-line is equal to (20/11) times 1.2, times 1.2, times 1.2.  Hereafter I refer to this pi-line as (pi*E)

4)  PL = 3.1426968052735445528926416093549;   equal to  [(40/9)*sr(0.5)] . . . here is an EXAMPLE: = (4.444444444... * 0.707106781...)

I named this pi-line (pi*V) because I believed I was close to victory.  I was.

5)  PL = 3.1399111679090046934014154186896;  equal to [sqrt(Phi*8) * 48] / 55 ]

this fascinating pi-line I named (pi*Q) for some obscure reason, maybe because I was running out of alphabet.  It is in fact equal to the ratio described above, but it is also equal to (pi*V) times the ratio of (pi*E)/(pi*F).  Another neuron in the brain of proportion.

Narrative Summary of the Unification Construction

1)  (pi*D)  The first pi line = 3.141640786.  Our D value, the pi multiplier, is 1.000015321.  All values, unless indicated otherwise, are endless, non-repeating decimals.  In Secrets of the Great Pyramid, Peter Tompkins provides a detailed history of studies of the Great Pyramid at Giza.  The ancient builders used a value for pi, but there has been continuous disagreement among historians, and mathematicians, as to whether the ancient builders had only a working approximation.  For example, did they use (22/7) or 3.16?  Some historians have suspected they knew a much more precise approximation for pi.  Tompkins wrote that there is evidence they understood the value of Phi (1.618033989) and they attached some importance to the value of (6/5) or 1.2.  Our (pi*D) is equal to Phi^2 times (6/5).  Also, a value that plays a special role in the phenomenon of proportion is (11/9) or 1.222222222 …  forever.  Our (pi*D) is also equal to the square root of (9/5) plus (9/5), or 1.341640786 plus 1.8.

2) (pi*F)  This pi-line is easy, the square root of 0.618033989 is 0.786151377.  That value multiplied by 4 equals 3.144605511.  Our multiplier is 1.000959022.  For years I struggled with a search for a means to construct the inverse of a multiplier.  This did not work for my list of 26 pi-lines.

3)  (pi*E)  This pi-line is fascinating and turns up in the phenomenon of proportion in many complex ways.  It could be called "a number in the (6/5) series."  (20/11) = (100/55) = 1.818181818, and that times 1.2 = 2.181818181, that times 1.2 = 2.618181818, and that times 1.2 = 3.141818181.  Is it rash to suspect that the ancients were aware of this series?  Our E multiplier = 1.000071788.  Throughout my research, my brain swam through a sea of pi-line multipliers.

4)  (pi*V)  A later discovery, but named appropriately (V for Victory) because it is the key pi-line value in the algorithm that is the solution.  Two other pi-lines play a role in the trinity that is the solution.  However, so long as one uses this "pi-line" system to search for the solution, seeing (pi*V) is required.  Our (pi*V) is the ratio (40/9) divided by the square root of 2.  Or, multiplied by the square root of (1/2).  Notice also that 40 times the square root of (1/2) equals the sqrt(800).  Therefore, (pi*V) = [ sqrt(800) / 9 ] = 3.142696805.  The multiplier is 1.000351462.  The (pi*V) pi-line possesses attractive and seductive wonders, like Venus on the Half Shell.  (pi*V) squared equals 9.87654321.  Does that look like the decimal digit number system?    And, the inverse of (pi*V)^2 equals 0.10125, which equals (81/800).

This is a good place to note that my style of research looks like "numerology," a neurotic obsession with numbers suggesting a belief in magic.  Also suggesting an imagination descending into delusion.  I have known all along that one has to careful searching for pi.  I had to search quietly.  That is one reason, but not the only reason, that I was committed to working alone.  There was a short period when I tried to discuss my work with mathematicians.  The primary response was prompt and total and angry rejection.  This is the way the institution of mathematics has treated "circle squarers" for centuries, supported by a conforming and gullible public.  It is as though mathematics has been the true religion that could not be questioned.  The institution of mathematics stepped into the shoes of the Medieval Church.  They have defended a false doctrine and severely punished dissenters.  They have locked a door and obstructed the way to the truth.  This is evidence that when Jesus articulated his critique of authoritarian behavior, as in Matthew Chapter 23, he was not addressing a single culture but rather he was addressing all human beings in all places and at all times.  Like now.  Right now is usually a time when humans are being authoritarian and conforming to doctrines.

5) (pi*Q)  This one is a wonder to behold, and entails a character that cannot be justly explored in a few words.  It is related to the other pi-lines and proportions in complex and elegant ways.  This pi-line is equal to a complex ratio.  The numerator is the sqrt(Phi*8) * 48.  The denominator is 55.  Note that Phi times 8 is the same as (cosine 36) * 16.  In my exploration of proportion, certain number values came up repeatedly.  The values 16 and 55 are two such numbers.  Numerology again.  But for me, numerology is a swear word that really means the same thing as Evolutionary Proportion.  When one does not see it clearly, it is "numerology."  When one sees it more precisely, and its consistency with other evidence in electronics, genetics and physics, it becomes the most fundamental of Nature's forces:  Evolutionary Proportion.  (Pi*Q) = 3.139911168 and the multiplier is 0.999464766.  It is not difficult to construct because [ (pi*E) / (pi*F) ] * (pi*V) = (pi*Q).  This is of course the same as:

(pi*E)  * (pi*V)  =  (pi * Q)  = 9.873781963

(pi*F)                              3.144605511

The Unification Construction Procedures

Here you can follow using a hand-held scientific calculator or your computer calculator.  These are usually precisely accurate (for fractional values) to the 9th decimal place.  My favorite hand held calculator has been the Casio fx-115MS because the operational keystrokes seem easiest to use and it allows for saving 6 values in memory (A B C D E F).  If you only have access to a calculator that does not save values in memory, you can still check this process by writing down the values needed to complete the mathematical expression that embodies the construction.

Referring to our set of construction tools (SOLITU Part D), one should see that anyone can verify the outcome of the constructions using a scientific calculator.  It is not necessary to take the time to produce each construction drawing on paper with a compass and straightedge.  Of course, anyone can take the time to complete many constructions on paper, as I did, if they wish.  The point of my work is to show that "mathematics" is "speed geometry."  We can perform all of the mathematical operations with the compass and straightedge:  addition, subtraction, division of a line in half, division of an angle in half, reconstruction of an angle, reconstruction of an angle and of a right triangle, construction of similar right triangles -- which means with the same angles but different proportional lengths of the three sides.  Using the similar right triangle system, and the SRT Table, we can multiply any two numbers (lines) or divide one line by another.  We can take the square root of a number (a line), and construct the square of a number (a line) by multiplication.  We can convert a rectangular area to a square area.  And now, you will see that using the Unification Construction, we can construct squares and circles exactly equal in area.

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