Construction of pi to infinite precision

Welcome to The Capricorn Construction, Volume 17 of Welcome to Aquarius (also Response #9 at pythagorascode dot org) October 16, 2008.

Claim of Discovery in a Nutshell (The Capricorn Equation): [See also (Capricorn Construction Two) one page.]

Here is the claim of discovery in a nutshell: I, John Manimas Medeiros have discovered how to construct a straight line length of Pi -- meaning infinite Pi -- and how to square the circle exactly using only a compass and straightedge. The ancient statement was: "It is possible to construct squares and circles exactly equal in area using only a compass and straightedge." When the declining human civilization found this difficult, they changed the statement to a question, or a riddle: "Can we construct a circle with exactly the same area as a given square, using only the compass and straightedge?"

The answer is "Yes" as follows:

There is an irrational, endless non-repeating decimal number MQ. MQ is equal to Phi^2, which means (1.618033988...)^2 or 2.618033988...; divided by Pi; times (6/5) or times 1.2.

Also, the square root of (9/5), or square root of 1.8, = 1.341640787..., and this is equal to the secant of a right triangle with a tangent equal to square root of (4/5) or square root of 0.8. Then the sum (9/5) plus the square root of (9/5) = 3.141640787..., and this is Pi times MQ, being the same MQ that is a factor of Phi^2 as shown above. Note that this sum is the same as (9/5) plus (3/sqrt(5)), = [(9/5) + (3 / sqrt(5)].

The essence of my discovery is that the following equations hold true in the perfect calculations and constructions of Nature in the real physical world. Electronic calculators are less than 100% precise, but Nature is 100% perfectly precise.

MQ*5/ sqrt(6) = 22,454/11,000, --> MQ*5 = (22,454 * sqrt(6)) / 11,000

and MQ = (22,454 * sqrt(6)) / 55,000

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

The simplest statement of my claim is The Capricorn Equation, being:

[22,454/11,000] * [sqrt(6)/5] = MQ, and [MQ*Pi*(5/6) = Phi^2 in the perfection of Nature.]

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

(1/ Pi) = [5 * 22,454] / [Phi^2 * sqrt(6) * 55,000]

and Pi = [Phi^2 * sqrt(6) * 55,000] / [5 * 22,454]

and because Phi^2 = 4 * (cosine 36)^2, (1/ Pi) = [5 * 22,454] / [Phi^2 * sqrt(6) * 55,000] =

[5 * 22,454] / [(cosine 36)^2 * sqrt(96) * 55,000], note that [4 * sqrt(6) = sqrt(96)]

and Pi = [Phi^2 * 6 * 55,000] / [5 * 22,454 * sqrt(6)]

This claim of discovery that we can construct MQ, M and infinite pi, published here October 16, 2008, supercedes my previous claim that we can construct pi to the 9th decimal place, which I call the "SF Solution." However, the SF Solution is still fully described at this website because it is a significant contribution to the evidence of the existence of "Proportion" as an independent physical phenomenon pertinent to the shapes and bonds of molecules and the proposition that physical matter is self-organizing.

Any indication on your electronic calculator that the expressions in these equations are precisely equal only to the 9th, or 12th, or 21st decimal place, and therefore not exactly equal, is due to the limitations of the electronic calculator, which is less than 100% accurate. Note that a deficiency of one-ten-millionth of a percent makes a calculator accurate to the 9th decimal place. That is it. If you are still curious, spend more time on this, and consider my argument that there is a phenomenon of "Proportion" that is independent of any number system, and this finite set of proportions is the sufficient cause for matter to be self-organizing.

Please Note: If the Capricorn Equation is true, as I claim it is, then we do produce line lengths of MQ and Pi to infinite precision using only the compass and straightedge in the constructions I describe here under the label "Capricorn Construction."

Link to: (Welcome) or (Geometry Alpha Index) .

Webpage Directions:

Under jmanimas dot com, this file will be the last volume (Volume 17) of Welcome to Aquarius.

Hereafter, I will refer to my postings and publications on my web pages as theJManimas Democracy Magazine. The subject matter will include: education, research in the fields of education, mathematics and science, the definition of mathematics as an individual skill -- and not the language of Nature, democracy, democracy education, the reconciliation of science and religion, the separation of church and state, evolution and intelligent design, culture wars, religious wars, the possibility of creating a democracy in the United States of America.

The Capricorn Construction of Infinite Pi Exactly, with infinite precision.

Presented here, with an introductory narrative of what has passed to the present solution:

A) Introduction: From what has passed to the present.

B) The Mathematical and Geometrical Essence of the Capricorn Construction is embodied in the value of MQ, and the simple, elegant construction of MQ using a compass and straightedge.

C) The Capricorn Construction: The procedure with compass and straightedge that enables construction of MQ, then Pi, followed by the procedure using M (square root of MQ) to construct circles and squares of exactly the same area, and then the procedure that makes use of the law of cosines, already established, to construct circles and squares of the exact same area.

D) What does the Capricorn Construction mean? Squaring the circle is the best pointer to the reality of Proportion, the reality that the shapes of atoms and molecules are the sufficient cause for matter to be self-organizing, meaning the "Secret of Life in the Universe" which John Taylor believed was the message hidden in the proportions of the Great Pyramid. A surprising addition to the facts of Proportion.

Study of the work presented here will always be far more enjoyable if you have and know how to use a scientific calculator. You may have a hand-held scientific calculator. If you do not, you have a scientific calculator on your desktop computer. Usually through "Accessories" you will find a "Calculator." Click on "Calculator" and if that calculator looks like a child's toy and just does addition, subtraction and multiplication, click on "View" and then click on "Scientific." There it is, your scientific calculator that you had all along and didn't know it.

A) (Introduction) From what has passed to the present:

The following web pages are posted/published at jmanimas dot com:

The Precision of the Ancients, Volume 1 (January, 2006)

The Myth of Pi, Volume 2 (February, 2006) How we can construct the area of a circular plane that is exactly the same as the area of a square.

Medicine and the Atom, Volume 3 (March, 2006)

The Amazing LIving Crystal Electromagnetic Atom LICEA (polyhedral shaped atoms)

The following web pages are posted/published at jmanimas.com as Volume 8 of Welcome to Aquarius, and in user friendlier form at pythagorascode.org:

-- Squaring the Circle Exactly (to at least the 9th decimal place precision) (December 2007)

-- Values and Constructions Required (9th decimal precision) (December 2007)

-- What is Proportion, Really? The Ancient Clues Point to "Proportion is Everything" (12/07)

-- Welcome to Aquarius, Volume 11 (10/16/2007): The �Magic� Number MQ

> The �Magic� Number MQ: The value of MQ is 1.000015321... .

The Precision of the Ancients and the other material entered as Volumes of Welcome to Aquarius (jmanimas.com), from December 2000 through March of 2008, as well as the material posted at (pythagorascode.org) through March 2008, should be considered to be notes and essays of John Manimas that illustrate the journey to his discovery of how to square the circle. This material probably represents less than ten percent of all of his notes, but it at least shows that he was doing the work on a regular basis, and searching diligently for a solution that he was certain existed, because he trusted the "Beacon of Light" of the Great Pyramid and the other clues of ancient history that suggested there was an advanced civilization on Earth in ancient times, so long ago as to be deemed by us to be in "pre-history."

Link to: (Welcome) or (Geometry Alpha Index) .

B) The Mathematical and Geometrical Essence of the Capricorn Construction is embodied in the value of MQ, and the simple, elegant construction of MQ using a compass and straightedge.

From JManimas' notes: (Let 22,454 = MN when a variable label is needed in an equation.)

MQ*5/ sqrt(6) = 22,454/11,000, --> MQ*5 = (22,454 * sqrt(6)) / 11,000

and MQ = (22,454 * sqrt(6)) / 55,000, --> therefore:

Computer test #1: (Get Phi^2 by (cosine 36), times 2, squared.)

Phi^2, / by Pi, times (6/5) = MQ = 1.00001532118112944385403915398333

MQ * 5 = 5.00007660590564721927019576991667, / by sqrt(6) means

/ by 2.44948974278317809819728407470589

= 2.04127272654933499508784751800739 due to calculator limitations.

whereas: (22,454/11,000) = 2.0412727272727272727272727272727

and these two are equal but due to calculator limitations they differ by a factor of

0.999999999645617036873889850275288

or the inverse: 1.00000000035438296325169743432326... (note nine zeros)

The next two equations both result in the output for the computed/constructed value

MQ = 1.00001532153551783667130575660811

which differs from our MQ derived from Phi^2 by this same factor of

1.00000000035438296325169743432359... (nine zeros, 9th place precision)

But we also know that Phi^2 = Pi * MQ * (5/6),

therefore, (Phi^2 * 6)/ 5 = Pi * MQ and (Phi^2 * 6)/ (5 * MQ) = Pi (confirmed)

and because MQ = (22,454 * sqrt(6)) / 55,000, then we see that

1/ Pi = [5 * 22,454 * sqrt(6)] / [Phi^2 * 6 * 55,000]

and Pi = [Phi^2 * 6 * 55,000] / [5 * 22,454 * sqrt(6)]

To test these equations, we use:

1/ Pi = [5 * 22,454 * sqrt(6)] / [Phi^2 * 6 * 55,000]

and Pi = [Phi^2 * 6 * 55,000] / [5 * 22,454 * sqrt(6)]

Computer test #2: 1/Pi = 0.318309886183790671537767526745029 (static entry in computer)

[5 * 22,454 * sqrt(6)] / [Phi^2 * 6 * 55,000] = [275004.21342226740508460908306723]

/ by [863951.216287465299907513655340661]

equals 0.318309886296594272235889809168531 *product of construction/computation

differs from computer static (1/Pi)

by a factor of: 1.00000000035438296325169743432326 due to calculator limitations.

Therefore: 275,004.2134 / 863,951.2163 = 0.318309886, and inverse = Pi

Note that equations can be shortened as follows:

1/ Pi = [5 * 22,454] / [Phi^2 * sqrt(6) * 55,000]

Pi = [Phi^2 * sqrt(6) * 55,000] / [5 * 22,454]

and because Phi^2 = 4 * (cosine 36)^2, 1/ Pi = [5 * 22,454] / [Phi^2 * sqrt(6) * 55,000] =

[5 * 22,454] / [(cosine 36)^2 * sqrt(96) * 55,000], note that [4 * sqrt(6) = sqrt(96)]

NOTE ALSO THAT: [(cosine 36)^2 * sqrt(96)] = Pi * (22,454/11,000)

therefore, Pi = [(cosine 36)^2 * sqrt(96) * 11,000] / 22,454

Computer test #2: Pi = 3.1415926535897932384626433832795 (static entry in computer)

and [(cosine 36)^2 * sqrt(96) * 11,000] / 22,454 = 0.65450849718747371205114670859141

* sqrt(96) or * 9.79795897113271239278913629882357, * 11,000

= 70541.3214187065748603885252866913

/ by 22,454 = 3.14159265247646632494827314895748

and this computation/construction differs from the static computer entry Pi by a factor of:

1.00000000035438296325169743432326 due to calculator limitations.

Of course, my proposition here in the Capricorn Construction is that in the perfect Proportions of Nature, the equation is 100% correct and the computer outputs are less than 100% correct. The computer calculations are in error due to the limitations of a binary electronic or semi-conductor based calculator. Such a calculator is, as are all hand-held and desktop computer calculators, less than 100% precise, by a factor of at least one ten-millionth of a percent. That is why the calculations are deemed accurate only to the 9th decimal place.

Link to: (Welcome) or (Geometry Alpha Index) .

The Probable Response of mathematicians:

I believe the most likely response to my work in squaring the circle will be a roaring silence as well as some nasty ridicule and expressions of extreme outrage, and proclamations that my work is the work of a lunatic and should not be taken seriously and is not worth a second of anyone's time.

However, Mathematics, like most institutions, is an authoritarian institution, and authoritarians always break their own rules. My work presents a thorn. It will be hard to ignore completely, and for some, a temptation they cannot resist. For a mathematician to look at my work here and consider it seriously is like a minister or a psychotherapist taking a look at pornography and enjoying it. They must do so secretly. Any authoritarian institution has doctrines that must be defended, and they usually have forbidden doctrines that must be discredited on a regular basis, and whenever necessary. "Squaring the circle" is a forbidden doctrine and for a mathematician to say that mathematicians should look at this work and examine it is likely to destroy or certainly threaten the career of anyone in the mathematical community.

I am not a member of the mathematical community. I do not belong to any mathematical association or organization. I do not plan to apply. Therefore, I am an amateur, an outsider, a "layperson." This means that to mathematicians I am a fool who should not be suffered. They do not want to say that someone who is not one of them did something worthwhile in the field of mathematics. Still, some will stray. Some will look at my work. Someone who is either in a very secure position, or in a state of spiritual freedom, will cry out: THIS COULD BE RIGHT! Then the fun begins.

The Probable Response of journalists, science writers, and the general public:

I believe the most likely response to my work in squaring the circle will be both silence and confusion, as well as an extreme tendency to just laugh or grimace and skip over the subject. Most journalists and science writers think in terms of their job status. They don't want to do anything that could cause them to lose credibility or lose membership in their club. The general public will assume that the work is too difficult for them to understand, even though the level of knowledge required is high school algebra and geometry. The real obstacle to my work being given any serious consideration is that there is a doctrine that "one cannot square the circle." This doctrine is extremely entrenched. It is not only a mathematical or geometrical doctrine. It is also a social and religious doctrine. It is apparently a doctrine in the field of logic. I do believe that someone who is free in mind and spirit will examine my work. They will then compare the outputs of high-precision computers with the Capricorn Equation:

-----> MQ = [22,454/11,000] * [sqrt(6)/5], and Pi = [Phi^2 * sqrt(6) * 55,000] / [5 * 22,454],

and they will see either that the equation is upheld or the factor of deviation is extremely small, beyond the 9th decimal place. They will then explore the question of how and why an electronic calculator does not possess the 100% perfect precision of Nature. Then the fun begins.

The three contributions to Geometry, Number Theory and Mathematics that I claim are:

1) The Super-binary Number Line System and sequential halving and doubling, described in The Pyramid Mark (1 and 2) where the base for the number system is the square root of two.

2) The SF Solution for construction of MQ and then Pi.

3) The Capricorn Construction and Equation: for construction of MQ and or Pi, with Pi being constructed as a product of a trigonometric function (cosine 36) or Phi^2, and an irrational number [sqrt(6)], and a rational fraction, the rational fraction being (22,454/11,000). Also, the discovery of the Capricorn Equation, being the discovery of the Proportional Reality that for the value MQ, the following equation is true: [ (MQ * 5)/ sqrt(6) ] = (22,454/11,000).

Post script: No one has my permission to use my name "JManimas" or "Manimas" or "John Manimas" as the name of any scientific or mathematical theory or natural phenomenon. Any discussion of my work should cite pythagorascode.org and or jmanimas.com as the sources.

Link to: (Welcome) or (Geometry Alpha Index) .

C) The Capricorn Construction: The procedure with compass and straightedge that enables construction of MQ, then Pi, followed by the procedure using M (square root of MQ) to construct circles and squares of exactly the same area, and then the procedure that makes use of the law of cosines, already established, to construct circles and squares of the exact same area.

Summary of Capricorn Construction:

It is assumed the reader knows how to construct lines, perpendiculars, right triangles, ratios, halves, doubles, and sums of line lengths, and how to multiply a line length times a ratio employing the construction of similar right triangles. These types of constructions are described in detail in Precision of the Ancients and Values and Constructions Required, as well as in numerous textbooks on geometry and mathematics. All construction steps must be performed with reference to a fixed line assigned a value of 1 or line UN.

Construction of 1.1 and 2.2454:

Use any satisfactory method to construct a ratio and then line length of 0.8 (4/5). One-half of 0.8 = 0.4. 0.8 plus 0.4 = 1.2. 1.2 = (6/5) and the inverse ratio/line = (5/6). One-half of 1.2 (6/5) = 0.6. Line 0.6 plus 0.5 = Line 1.1. We will use the line length of 1.1 to construct the ratio and then the line length of 22,454/11,000 [2.041272727272727272727...].

There are many ways to construct a line length of 22,454, or 22.454, or 2.2454. To construct a line length of 2.2454, first construct a line length of 0.8. Add a length of 6 to get 6.8. Divide that line length by ten. A reliable way to divide a line by ten is to use the "Tan Two Twice" method described in Values and Constructions Required. We use the tangent = 2 right triangle twice to multiply a line by (1/5), which means the same as divided by 5. Then we divide that line by 2 for the output of our original line divided by 10.

6.8 divided by 10 = 0.68. Add length of 1 to get 1.68. Divide by 2 to get 0.84. Add length of 1 to get new length of 1.84. From a line length of 10, subtract the line length of 1.84. The result is a new line length of 8.16. Divide the line 8.16 by 10 to get 0.816. Add a line length of 9 to get a new length of 9.816. Divide line 9.816 by 10 to get 0.9816. Add a length of 8 to get 8.9816. Divide 8.9816 in half to get 4.4908. Divide 4.4908 in half to get 2.2454, our target line.

Proceeding:

Construct a right triangle with Altitude A = 2.2454 and Base B = 1.1. We now have a right triangle with a tangent (ratio) equal to 2.041272727272727272727... . Using the similar right triangle method, we re-construct this right triangle with a Base B of length 1. Because the tangent equals 2.041272727272727272727..., this value must be the length of side A.

We need to construct a line length and ratio of square root of 6. Construct a square with side equal to 1 and the diagonal is square root of 2. Construct a right triangle with A = 2 and B = sqrt(2). The length of Hypotenuse H = sqrt(6) [2.449489742783178098197...].

For this Sqrt(6) Right Triangle, the facts are:

A = 2, B = sqrt(2), H = sqrt(6)

G (angle of interest) = 54.7356103172453456846229996699812... degrees

Opp (opposite angle) = 35.2643896827546543153770003300188... degrees

Sine = 0.816496580927726... = sqrt(2/3) = sqrt(1.666666666666666...)

---> Cosecant = 1.224744871391589... = sqrt(3/2) = sqrt(1.5)

Cosine = 0.577350269189625... = sqrt(1/3) = sqrt(0.333333333333333...)

---> Secant = 1.732050807568877... = sqrt(3)

Tangent = 1.414213562373095... = sqrt(2)

---> Cotangent = 0.707106781186547... = sqrt(0.5)

Step by step procedure in the Capricorn Construction:

There are four main sections of the Capricorn Construction, each rather simple constructions. The Capricorn solution to the ancient riddle is based on knowledge of 100% accuracy in the Proportions of Nature, and not a torturously complex or "tricky" construction.

1) We construct the ratio and line that equals 22,454/11,000, or 2.2454/1.1

2) We construct a right triangle with A = 2 and B = sqrt(2) and H = sqrt(6)

3) We construct a right triangle with A = sqrt(6) and B = 5 and H = sqrt(31)

After steps 1 through 3, there are steps 4A and 4B:

4A) Construction from the line length MQ, by squaring the rectangle, of a line length of M, and then following a generic procedure to construct squares and circles exactly equal in area.

4B) Construction of squares and circles of the exactly the same area using the procedure already described in the SF Solution, that is using the line lengths of 1 and Pi and the law of cosines.

Step 1) We construct the ratio and line that equals 22,454/11,000, or 2.2454/1.1

Construction of lines 2.2454 and 1.1 is described above. Then construct a right triangle with Altitude A = 2.2454 and Base B = 1.1. We now have a right triangle with a tangent (ratio) equal to 2.041272727272727272727... . Using the similar right triangle method, we re-construct this right triangle with a Base B of length 1. Because the tangent equals 2.041272727272..., this value must be the length of side A.

In Step 1 we construct two right triangles, the 2.2454 right triangle, and the similar right triangle with Base B =1. The drawings presented here are not precisely to scale either when viewed on a monitor screen or when printed out on paper.

The 2.2454 right triangle:

Construction steps with compass and straightedge: Construct perpendiculars intersecting at left side of page. On the vertical, mark off a line length of 2.2454 for Altitude A. On the horizontal, to the right of the Altitude A, mark off a line length of 1.1 for Base B. Construct a straight line from the upper end point of A to lower right end point of B for Hypotenuse H. The purpose of this first step is to construct the tangent = 2.041272727... as a construction tool.

For the 2.2454 right triangle, the facts are:

A = 2.2454, B = 1.1, H = 2.500364205470874... = sqrt(6.25182116)

G = 63.9002080372027965951835617110132... degrees

Opp = 26.0997919627972034048164382889868... degrees

Sine = 0.898029173144854... ---> Cosecant = 1.113549570442181...

Cosine = 0.439935909174017... ---> Secant = 2.273058368609886...

Tangent = 2.041272727272727... ---> Cotangent = 0.489890442682818...

(Check: Secant^2 = Tangent^2 + 1)

The 2.014272727... right triangle:

Construction steps with compass and straightedge: Construct perpendiculars intersecting at left side of page. On the horizontal, to the right of the vertical, mark off a line length of 1 (UN) for Base B. At the lower right end point of B, re-construct the angle G, using the common method for construction of a given angle. The upper side of angle G is then extended to intersect with the vertical perpendicular, which is now Altitude A. Because we know that the tangent of this similar right triangle equals 2.041272727..., and the value of Base B is 1, we know that the length of line A (Altitude) must be equal to the tangent 2.041272727... . The purpose of this step is to construct a line length = 2.041272727... as a construction tool in a following step.

For the 2.014272727... right triangle, the facts are:

A = 2.041272727..., B = 1, H = 2.273058368609... = sqrt(5.166794347107...) = Secant

G = 63.9002080372027965951835617110132... degrees

Opp = 26.0997919627972034048164382889868... degrees

Sine = 0.898029173144854... ---> Cosecant = 1.113549570442181...

Cosine = 0.439935909174017... ---> Secant = 2.273058368609886...

Tangent = 2.041272727272727... ---> Cotangent = 0.489890442682818...

(Check: Secant^2 = Tangent^2 + 1)

Step 2) We construct a right triangle with A = 2 and B = sqrt(2) and H = sqrt(6)

Note that we obtain the line length of sqrt(2) simply by constructing a square with side = 1 (UN) and then constructing the diagonal, which will be exactly sqrt(2) in proportion to UN.

The sqrt(6) right triangle:

Construction steps with compass and straightedge: Construct perpendiculars intersecting at left side of page. On the vertical, mark off a line length of 2 for Altitude A. On the horizontal, to the right of the Altitude A, mark off a line length of sqrt(2) for Base B. Construct a straight line from the upper end point of A to lower right end point of B for Hypotenuse H. Side H now equals the value sqrt(6). The purpose of this step is to construct the line length sqrt(6) as a construction tool for a following step.

For the sqrt(6) right triangle, the facts are:

A = 2, B = sqrt(2), H = sqrt(6)

G (angle of interest) = 54.7356103172453456846229996699812... degrees

Opp (opposite angle) = 35.2643896827546543153770003300188... degrees

Sine = 0.816496580927726... = sqrt(2/3) = sqrt(1.666666666666666...)

---> Cosecant = 1.224744871391589... = sqrt(3/2) = sqrt(1.5)

Cosine = 0.577350269189625... = sqrt(1/3) = sqrt(0.333333333333333...)

---> Secant = 1.732050807568877... = sqrt(3)

Tangent = 1.414213562373095... = sqrt(2)

---> Cotangent = 0.707106781186547... = sqrt(0.5)

Step 3) We construct a right triangle with A = sqrt(6) and B = 5 and H = sqrt(31)

The sqrt(6)/5 right triangle:

Construction steps with compass and straightedge: Construct perpendiculars intersecting at left side of page. On the vertical, mark off a line length of sqrt(6) for Altitude A. On the horizontal, to the right of the Altitude A, mark off a line length of 5 for Base B. Construct a straight line from the upper end point of A to lower right end point of B for Hypotenuse H. Side H now equals the value sqrt(31). The purpose of this step is to construct the tangent equal to sqrt(6) divided by 5 as a construction tool for a following step.

For the sqrt(6)/5 right triangle, the facts are:

A = sqrt(6) = 2.449489742783..., B = 5, H = sqrt(31) = 5.567764362830...

G (angle of interest) = 26.1001387822863844212338418848393... degrees

Opp (opposite angle) = 63.8998612177136155787661581151607... degrees

Sine = 0.439941345064059867508341856513858...

---> Cosecant = 2.2730302828309759821264329253216...

Cosine = 0.898026510133874503567656661115895...

---> Secant = 1.11355287256600438442389425978371...

Tangent = 0.489897948556635619639456814941178... sqrt(6)/5 [squared = 6/25]

---> Cotangent = 2.04124145231931508183107006225491... 5/sqrt(6)

(Check: Secant^2 [= 1.24] = Tangent^2 + 1)

The MQ similar right triangle:

From Step 1 we have a line length of 2.041272727272... .

Construction steps with compass and straightedge: Construct perpendiculars intersecting at left side of page. On the horizontal, to the right of the vertical, mark off the line length taken from Step 1 which is the value 2.041272727272... for the line length of Base B. At the lower right end point of B, re-construct the angle G (26.100138782286...), using the common method for construction of a given angle (described in Values and Constructions Required). The upper side of angle G is then extended to intersect with the vertical perpendicular, which is now Altitude A. Because we know that the tangent of this similar right triangle equals 0.489897948556..., and the value of Base B is 2.041272727272..., we know that the length of line A (Altitude) must be equal to the tangent value (0.489897948...) times the length of B (2.041272727...) . This value is our constructed line length of MQ.

For the MQ similar right triangle, the facts are:

A = MQ, B = 2.041272727272..., H = 2.27306510914518749525946560992572...

[MQ = 1.00001532153551783667130575660811... subject to calculator precision]

[H = sqrt[ MQ^2 + (2.041272727272727..)^2]

G (angle of interest) = 26.1001387822863844212338418848393... degrees

Opp (opposite angle) = 63.8998612177136155787661581151607... degrees

Sine = 0.439941345064059867508341856513858...

---> Cosecant = 2.2730302828309759821264329253216...

Cosine = 0.898026510133874503567656661115895...

---> Secant = 1.11355287256600438442389425978371...

Tangent = 0.489897948556635619639456814941178... sqrt(6)/5 [squared = 6/25]

---> Cotangent = 2.04124145231931508183107006225491... 5/sqrt(6)

(Check: Secant^2 [= 1.24] = Tangent^2 + 1)

Step 4A) Construction from the line length MQ, by squaring the rectangle, of a line length of M, and then following a generic procedure to construct squares and circles exactly equal in area.

First, construct a "rectangle" with width (vertical) of 1 and length (horizontal) of MQ. The area of this rectangle is MQ (1.00001532153551783667130575660811...). Square this rectangle. The length of the side of the constructed square is the square root of MQ, which is the line length equal to M (1.00000766073841546180058678869611...).

[Imperfection of electronic calculations] Note that you can obtain a slightly different electronic calculation of M by the following procedure: Get the cosine of 36 degrees. Multiply by 2, to get Phi (1.618033988... or 1.618033989...). Divide Phi by the square root of Pi, which on my calculator is (1.77245385090551602729816748334115...). Then multiply by the square root of 1.2, which is 1.0954451150103322269139395656016... . The result will most likely deviate from exactly equal to our previous M, such as: 1.00000766056122262280424348031297..., which differs from our previous electronic calculation of M by a factor of

0.999999999822808518421246514565663..., or the inverse value, being

1.00000000017719148161015030658397... ( nine zeros). Your calculator may indicate a slightly different deviation. My position is that the geometric equation is correct, the deviation in the electronic calculations is wrong.

Capcon15, The Capricorn Construction:

This "capcon15" program employs the M value, which is the square root of the MQ value that is derived from Phi^2, which is of course the same as [2 * (cosine 36)]^2 . To construct a radius, we take any line length value (RX) that we can construct and multiply that line length by the square root of (10/3) times M. We then use that value as the radius (Rd) of a circle, which we can readily construct with a compass. We then take that same line (RX) and employ similar right triangle method to multiply RX by Phi and then times 2, or 4 times (cosine 36). This product is our Side S which we use to construct a square, and the square resulting has exactly the same area as the previously constructed circle.

Using the line length of M, we can construct squares and circles exactly equal in area by using a simple formula or algorithm. This method is easier than that described in 4B, where we construct a line length of Pi, then Pi/2, and use the law of cosines. The use of a line with value M works as follows in computer programs I have labeled "capcon15" and "capcon16."

First look at the "circle areas" and squares that are exactly equal in the short list of four examples from capcon15. This brief printout of the program is then explained in detail.

Program capcon15.cpp: Constructs RX*sqrt(10/3)*M, RX*Phi*2

RX*sqrt(10/3)*M for Rd of circle area, RX*Phi*2 for S of equal square.

Here, RQ = Rd^2, and AC (Area of Circle) = Pi * RQ.

Radius Rd (1) = RX (0.894427190999916) * sqrt(10/3)*M:

Rd= 1.633005671500 RQ= 2.666707523150 AC= (8.377708764000)

S(1) = RX (0.894427190999916) *Phi*2, S = 2.894427190999916

Square equal in area = S * S = [8.377708764000]

Radius Rd (2) = RX (1.118033988749895) * sqrt(10/3)*M:

Rd= 2.041257089374 RQ= 4.166730504921 AC= (13.090169943749)

S(2) = RX (1.118033988749895) *Phi*2, S = 3.618033988749895

Square equal in area = S * S = [13.090169943749]

Radius Rd (3) = RX (0.447213595499958) * sqrt(10/3)*M:

Rd= 0.816502835750 RQ= 0.666676880787 AC= (2.094427191000)

S(3) = RX (0.447213595499958) *Phi*2, S = 1.447213595499958

Square equal in area = S * S = [2.094427191000]

Radius Rd (4) = RX (2.236067977499790) * sqrt(10/3)*M:

Rd= 4.082514178749 RQ= 16.666922019685 AC= (52.360679774998)

S(4) = RX (2.236067977499790) *Phi*2, S = 7.236067977499791

Square equal in area = S * S = [52.360679774998]

Example of step-by-step constructions for Step 4A), equal squares and circles using M:

First example in program printout:

Radius Rd (1) = RX (0.894427190999916) * sqrt(10/3)*M:

Rd= 1.633005671500 RQ= 2.666707523150 AC= (8.377708764000)

S(1) = RX (0.894427190999916) *Phi*2, S = 2.894427190999916

Square equal in area = S * S = [8.377708764000]

Step-by-step construction of the "M" method for equal area circles and squares:

To get a line length of M, we construct a rectangle with width of 1 and length of MQ. The differences between these two lines will not be visible to the human eye, but is visible to the Eye of Ra. Square the rectangle. The length of the Side S of the square is M. Since the constructions required here do not involve any new inventions, the commonly known procedures are described in narration only, without drawings.

1) Construct a right triangle with tangent ratio of (10/3), which would be A=10 and B=3, or A=5 and B=1.5. Re-construct this right triangle, using the similar right triangle method where the angle of interest [tangent=(10/3)] is reconstructed with the Base B equal to 1. Then the Altitude A in the similar right triangle must have a line length of (10/3).

2) Construct a rectangle with length (10/3) and width of 1. Square the rectangle. The line length of the side of the square is square root of (10/3) or 1.82574185835055371152323260933601... .

3) Construct a right triangle with A=sqrt(10/3) and B=1. The tangent ratio is sqrt(10/3).

4) Re-construct the right triangle with tangent = sqrt(10/3) with a Base B equal to M. Then the line length of A is M*sqrt(10/3).

5) Construct a right triangle with Altitude A equal to line length M*sqrt(10/3). This is line length 1.82575584455783631107165421274361... . Construct the Base B equal to 1. Construct the hypotenuse. The tangent of this right triangle is M*sqrt(10/3).

6) Now, using the right triangle we have constructed in step 5, we can re-construct this right triangle with a Base B that is any line value RX, and then the line length of Altitude A must be equal to that line length RX, times M*sqrt(10/3). This is our radius: RX * M * sqrt(10/3).

-----> If the length of our RX is sqrt(0.8), as in our first example, then our radius Rd is equal to line length 1.63300567149954458354558880759468... . This Rd squared is equal to the value 2.66670752314967851694410441062222..., times Pi is the area of the circle and that area has a value of 8.37770876399966351425467786997004... .

7) To take that same RX value of sqrt(0.8) and multiply it by 2 times Phi, we simply need to look at the right triangle that gives us a ratio of Phi, and this is the 36-54 right triangle that we obtain when we construct a pentagon. The pentagon is comprised of five equilateral triangles that have center angles of 72 degrees (360/5 = 72). We bisect that angle, a basic and well-known procedure, and obtain a pair of right triangles in each of which one angle is 36 degrees and the other is 54 degrees. This 36-54 right triangle was deemed of special or "sacred" or "magical" importance to the ancient Egyptians.

8) Reconstruct the 36/54 right triangle with a hypotenuse having a line length of 1. One way to do this is to re-construct the 36-degree angle on a horizontal line and mark off a line length of 1 on the upper "hypotenuse" side of that 36-degree angle. At the upper end point of that Hypotenuse equal to 1, drop a perpendicular down to the horizontal "Base." When the Hypotenuse is equal to 1, the longer side of the right triangle, the Base B, must be equal to the cosine, which means B has a line length of 0.809016994374947424102293417182819... .

This (cosine 36) value is one-fourth of two times Phi.

9) Reconstruct the 36/54 right triangle with the Hypotenuse equal to RX sqrt(0.8). Since the Hypotenuse is now sqrt(0.8)*1, the Base B is equal to sqrt(0.8)*(cosine 36), and that line length of B times 4 = sqrt(0.8) * Phi * 2. Use that line length as S the Side of a square, and the result is a square with an area equal to 8.37770876399966351425467786997004..., exactly equal to the area of our previously constructed circle.

This is a "generic" algorithm. For any RX, radius Rd = RX*sqrt(10/3)*M produces a circle exactly equal in area to a square with Side S = RX * Phi * 2. Now we know why the ancients, and allegedly the Freemasons and the Rosicrucians, told us to construct a pentagon.

Capcon15 and capcon16:

This "capcon15" program employs the M value, which is the square root of the MQ value that is derived from Phi^2, which is of course the same as [2 * (cosine 36)]^2 . To construct a radius, we take any line length value (RX) that we can construct and multiply that line length by the square root of (10/3) times M. We then use that value as the radius (Rd) of a circle, which we can readily construct with a compass. We then take that same line (RX) and employ similar right triangle method to multiply RX by Phi and then times 2, or 4 times (cosine 36). This product is our Side S which we use to construct a square, and the square resulting has exactly the same area as the previously constructed circle.

I ran this same program using the M calculator value obtained from the Capricorn Equation, and I re-named this program "capcon16." The deviation, on my calculator, between constructions of capcon15 and capcon16 is 1.00000000035438806523786039888671... on my calculator, which is consistent with the previous deviation of 1.00000000035438296325169743432326... between the MQ value derived from the calculator value of Phi^2, [2*(cosine 36)]^2, and the value of MQ that results from the Capricorn Equation. This method reveals the reality which I suspected since 1990, that the original purpose of the ancient "squaring the circle" riddle was to point to the relationship between Phi and Pi that is embodied in the "magic number" MQ, or M. I expect that any mathematician that wants to dismiss my work and the Capricorn Equation, for whatever reason, will state authoritatively that the electronic calculator is correct and the Capricorn Equation is not infinitely precise. I will stick to my proposition that the Capricorn Equation is precisely correct, infinitely correct, until there is a proof, understandable to the general public, as to why and how a particular electronic calculator is infinitely precise and does not deviate from the 100% perfect precision of Nature.

Link to: (Welcome) or (Geometry Alpha Index) .

Identifying the limitations of binary electronic calculators:

I predict that any electronic engineer who takes this challenge seriously will find, and agree with me, that in order for a binary electronic calculator to comply with the perfection of Nature, the value of all rational fractions would have to be matched or "keyed" perfectly to the cosine of 36 degrees and all trigonometric ratios, and all endless, non-repeating decimal (irrational) ratios, INCLUDING the value of [Pi*(5/6)*MQ] as the proper endless, non-repeating decimal value for Phi squared. I believe that the only "number system" required by Nature is Proportion and Addition. And, to the extent that there is a "binary" phenomenon in Nature, where numerical values are involved, the "binary" system of Nature uses the square-root-of-two as the number base in what I call the "super-binary number system," and not two. If semi-conductors or any other form of electronic device could be keyed to a fixed value for the square-root-of-two, which is a trigonometric function, so that all trigonometric functions, and all rational and irrational fractional values were "keyed" to the same square-root-of-two, then there would be found only one exact value for MQ in both Phi^2 and in the Capricorn Equation. The "super-binary number line system" is described in Pyramid Mark (2) at jmanimas dot com.

(Interpretive aside:)

It is fair to conclude therefore, that construction of line M is the easiest way to construct squares and circles that are exactly equal in area. This is what I believe to be the meaning of the Eye of Ra on the Great Pyramid, in ancient Egyptian texts and in the symbols of the Freemasons. The Eye of Ra means that an extremely high level of precision, perhaps only 100% precision or infinite precision, enables one to construct squares and circles exactly equal in area, using only the compass and straightedge. This could be taken to mean that only Nature can construct squares and circles exactly equal in area, and we cannot because we cannot achieve the same infinite precision that Nature achieves, not even with our electronic calculators. If this is the case, why should this be important? It may be of the highest importance because if Nature does construct squares (or cubes and other polyhedrons) that are exactly equal in area to circles (or spheres), we are naturally stimulated to ask why. If Nature can construct polyhedrons that are exactly equal in area to circles, it may be that the ability to do so, using the real physical building blocks of the universe, atoms and molecules, as the "compass and straightedge," then I believe this is the key to how matter is self-organizing.

Please Note: If the Capricorn Equation is true, as I claim it is, then we do produce line lengths of MQ and Pi to infinite precision using only the compass and straightedge in the constructions I describe here under the label "Capricorn Construction."

Step 4B) Construction of squares and circles of the exactly the same area using the procedure already described in the SF Solution, that is using the line lengths of 1 and Pi and the law of cosines. This method is described in detail at the end of Values and Constructions Required which was previously associated with The SF Solution.

D) What does the Capricorn Construction mean?

To me, the discovery of the Capricorn Construction means that Proportion is a phenomenon that is not only worthy of investigation, but which must be investigated so that it can be either confirmed or ruled out. This is the only way that we can assure ourselves that we are understanding the physical universe as it really is.

Here is a surprising addition to the concept that there is this phenomenon of Proportion that is:

1) a finite set of ratios;

2) independent of and different from number theory;

3) independent of and different from the infinite number of trigonometric ratios;

4) the sufficient cause for atoms and molecules to possess "electronic geometry" and enabling matter to be self-organizing.

Look again at the 1.2 series or the "Point Five" or "Eighteen Series:

PF=0.55, ES= (1/0.55)=1.81818181818181818181818181818182...

*1.2 = 2.18181818181818181818181818181818...

*1.2 = 2.61818181818181818181818181818181...

*1.2 = 3.14181818181818181818181818181818...

*1.2 = 3.77018181818181818181818181818181...

*1.2 = 4.52421818181818181818181818181818...

*1.2 = 5.42906181818181818181818181818181...

*1.2 = 6.51487418181818181818181818181818...

And, 2.618181818... = 432/165, and 3.618181818... = 864/275

We also can use the value of (11/9) to produce the value TN, as follows:

LN= (11/9) = 1.22222222222222222222222222222222... (eLeven Nines)

NL = 0.818181818181818181818181818181818... (Nine eLevenths, eighteen again)

* 1.2 = 0.981818181818181818181818181818182...

SF = (secant 18)^4 = 1.22229123600033648574532213002996...

inverse of SF = FS = 0.818135621484342140063933385739262...

sqrt of that = (cosine 18)^2 = 0.90450849718747371205114670859141...

and that = (Phi/4) + 0.5 [0.5 + 0.40450849718747371205114670859141...]

TN= SF/LN = 1.00005646581845712470071810638815...

Why does the second step [2.618181818...] = Phi^2 * TN?

Why does the third step = [3.141818181...] = Pi * MQ * TN?

Is this meaningless? If this reality has meaning, what is the meaning?

MQ = 1.00001532118112944385403915398333 "approximately"

TN = 1.00005646581845712470071810638815 "approximately"

(10/3) = 3.33333333333333333333333333333333

So, where does (10/3) come in? It comes in through a complex seemingly related set of values and proportions related to Pi. The most obvious is as follows:

(4/Pi) is the ratio of the area of a square to its "inner circle." That means if we construct a circle and then a square that has the same "center" with the sides equal to the diameter of the circle, and of course the sides are tangent to the circumference of the circle, that square is the "outer square" of the circle. The area of that outer square is (4/Pi) times the area of the "inner circle."

1/Phi^2 = LPQ = 0.381966011250105151795413165634362... and (sine 18) * 2

= phi (0.618033988...) and phi^2 = 0.381966011250105151795413165634362... = LPQ

and MQ * LPQ * (10/3) = (4/Pi).

Let us try (10/3) - (Pi*MQ*TN) = 0.1915151515151515151515151515151...

You can do this on your calculator.

0.1915151515151515151515151515151...

* 2 = 0.383030303030303030303030303030...

* 2 = 0.766060606060606060606060606060...

* 2 = 1.532121212121212121212121212121...

/100,000 = 1.5321212121212121212121212121... * 10^(-5)

plus 1 = 1.00001532121212121212121212121212... (Look familiar?)

Therefore, one conclusion is that this is just coincidence, "numerology" or some form of neurosis that involves obsession with numbers, fractions and ratios. Another conclusion is that it means something, and it would be good for us to know with some degree of certainty exactly what does this mean, this wispy vision of Proportion?

John Manimas