The Meaning of Proportion

Welcome to Aquarius, Volume 8 (August 16, 2007), Squaring the Circle Exactly.

What is Proportion, really? (27 pages)

Subsection of The Reality of Proportion and How to Square the Circle Exactly

What is Proportion, really?:

1) Most important, the quadratic construction, not the quadratic equation. (21 pages)

2) Prove or rule out the "ball-bearing" shape of the "planetary" model of the atom.

Revise quantum theory? What is the shape of an atom?

ONE PAGE LINK: [What is the shape of an atom?]

3) The observation of "binary systems" in Nature supports "halving and doubling" in Nature as a process of "sequential halving" and "sequential shaping" which are described here as Nature's technology of "Proportion." See The Pyramid Mark, for description of "sequential halving."

4) The economy of Nature supports "Sufficiency of Proportion Theory."

It is my opinion that the original Pythagorean Theorem and quadratic equation were neither devised nor used to solve algebraic "problems" in mathematics. They originated centuries before the development of the mathematical practice that we call "algebra," and their original use was for geometric constructions. Therefore, I claim that the quadratic equation was originally, and still is of course, the Quadratic Construction.

The Quadratic Construction means that we do not view the equation as an algebraic formula to solve problems. When the quadratic equation is used to solve mathematical problems, we often get a "minus" solution. There is often a minus and a plus solution, and sometimes two plus solutions, but always two solutions. We often get a minus solution that does not constitute a sensible solution. So, in practice, we drop the negative solution and joyfully accept the positive solution as correct. That's fine. However, when we use the quadratic equation as the quadratic construction, all number values are either ratios or line lengths, and in fact we are using the quadratic construction to construct line lengths that have certain significant characteristics. For example, the quadratic construction enables us to construct line lengths of great decimal precision that are exact inverses of one another. The most important reality to recognize about the quadratic construction, in order to understand its original and very powerful meaning, is that because all number values are constructible line lengths, there are really no minus values. This is true for the simple reason that there is no such thing as a minus line length. In reality, in Nature, when we "subtract" a length of 5 from a length of 2, what we are really doing is comparing the two line lengths or "laying" the length of 5 over the length of 2. What we get could be called, mathematically, minus 3, but in reality and in Nature it has to be just 3, always a "plus" value for the line length. If this is still difficult, just think as the 2 minus 5 as having a real concrete result. It is 3, or plus 3, because anything that is physically real has to be "plus" for "present," and it cannot be "minus" for "not present." This is why I make my little joke, a most cosmically serious joke, that...

"There are no minus cows in the real universe."

-b +/- (sqrt[ b^2 - (4*a*c)] ) / 2a = -(b / 2a) +/- ( sqrt[ b^2 - (4*a*c)] ) / 2a )

In school you learned that this formula gives you the two solutions to the typical form of the "second degree" equation: A*X^2 + B*X + C = 0. Note that this equation can be readily changed to represent a somewhat "simpler" reality: A*X^2 + B*X - C = 0. If we change the C value by giving it the minus value with the minus sign, we then get: A*X^2 + B*X = C. In this equation [A*X^2 + B*X = C], if the A and B are both assigned a value of 1, then we have the first equation that can "open one's eyes" to the meaning of proportion: X^2 + X = C. We just need to change one more sign, the plus sign of the X to a minus sign: X^2 - X = C. Now we have X^2 = X + C. So, the reader may ask, what does this prove, or show us? What if we have a target value for C, a number value that we have in mind, so that it is true that X^2 = X + C? If we can do this with the Quadratic Construction, then we might accept that as evidence that Nature can do the same. And this is the beginning of seeing the idea that I propose, that Proportion is the sufficient "number system" that Nature employs to construct all of the shapes of the natural physical world. If this is true, then it is reasonable to conclude that this is the system that produces the shapes of molecules, and the shapes of living molecules, which then makes proportion the sufficient cause of the self-organization of matter, life.

The values assigned to a, b, and c, and the results of the constructions, are values that will result in two solutions that can still be called the Negative solution and Positive solution. This is my way of labeling or "naming" the two solutions.

1) The Negative Solution is the solution [ -b - (sqrt[ b^2 - (4*a*c) ] ) ] / 2a

where the value we take the square root of has a minus sign: - sqrt[ b^2 - (4*a*c) ]

2) The Positive Solution is the solution [ -b + (sqrt[ b^2 - (4*a*c) ] ) ] / 2a

where the value we take the square root of has a plus sign: + sqrt[ b^2 - (4*a*c) ]

The quadratic construction can be "spelled out" to give two solutions, as:

1) -(b / 2a) + ( sqrt[ b^2 - (4*a*c)] ) / 2a), and...

2) -(b / 2a) - ( sqrt[ b^2 - (4*a*c)] ) / 2a)

The second construction is the Negative solution. These "calculations" are certainly real as calculations, but they are also real as constructions that can be performed with a compass and a straightedge. In other words, we can perform each quadratic calculation as a construction.

2) -(b / 2a) - ( sqrt[ b^2 - (4*a*c)] ) / 2a) is the Negative solution, because we have a minus sign assigned to the value of both elements:

sqrt[ b^2 - (4*a*c)] ) / 2a, as well as to (b / 2a)

I call that part of the equation that we "get the square root of, "sqrt[ b^2 - (4*a*c)], the "determinant." We do not want to struggle with getting the square root of a negative value (because there are none among line lengths). Therefore, we give "c" the value of (-1) in the program so that the determinant is always positive: sqrt[ b^2 + (4* a *  )]

We also assign the value of 1 to "a." Therefore:

-(b / 2a) = -(b/2) and ----> sqrt[ b^2 + (4* a *  )] /2a = sqrt[ b^2 + (4* a *  )] / 2

You can see now, quite readily, that when we vary the value of b only,

we get our two solutions in the following form:

1) -(b/2) + (sqrt[ b^2 + (4 *1 *1)] ) /2), and---> -(b/2) - (sqrt[ b^2 + (4 *1 *1)] ) /2)

The first solution gives us, by construction, the line length of the "determinant,"

(sqrt[ b^2 + (4 *1 *1)] ) /2), minus the line length of (b/2): the Positive Solution.

The second solution gives us, by construction, the line length of the "determinant,"

(sqrt[ b^2 + (4 *1 *1)] ) /2), plus the line length of (b/2): the Negative Solution.

Notice also that sqrt[ b^2 + (4 *1 *1)] = sqrt[ b^2 + 2^2 ], [b squared + 2 squared].

I have written several computer programs to show how employing the "Quadratic Equation" in this manner as the "Quadratic Construction," suggests how this procedure enables us to produce by both calculation and construction, target proportions.

Probably the first such program I wrote is labeled "AnyNaX01." But even before I wrote this program, I wrote the program it is based on, "uniphi01," which label refers to "The Universality of Phi." That original program, uniphi01, shows that we can calculate and then construct any two target values, "Left Phi" or LP, and "Right Phi," or RP, such that LP*RP = RP+1. Note that this appears to be similar to the "eye opening" equation described above: X^2 = X + C. In our equation LP*RP = RP+1, we have X1 * X2 = X2 + 1. We will soon see that the Quadratic Construction can give us the result X^2 = X + C, but first it is educational to look at our "uniphi01" construction of LP*RP = RP+1, because it employs common trigonometry. We simply make the value of sqrt(LP*RP) the secant, construct the right triangle, and sqrt(RP) is the tangent. The secant squared = (RP+1) for any target value we choose. First we choose an integer (W) for Whole number, which can be zero, and then we choose a Fractional part (F), which could also be zero but only if W is not zero. The value of W plus F is our Phi Number (PN), and the program takes over from there. The calculations adjust to two situations regarding the value of PN, or (W+F), that is: 1) if PN is less than 1, and 2) if PN is greater than 1. My original uniphi01 program did not allow for PN = (W+F) =1. This special case means simply that LP is 2 and RP is 1, and 2*1 = 1+1. The right triangle that incorporates these values is the diagonal half of the square. The secant = (square root of 2)/1, and the tangent = 1/1. The values where PN is not equal to 1 are far more interesting.

The algorithm for this program, uniphi01, is that:

A) When our selected PN (W+F), is less than 1, (PN<1), then our Left Phi (LP) is equal to

(PN+1)/PN and our Right Phi (RP) is equal to PN.

B) When our selected PN (W+F), is greater than 1, (PN>1), then our Left Phi (LP) is equal to

(PN/(PN-1) and our Right Phi (RP) is equal to (PN-1).

THAT'S IT! These simple calculations result in what I call "The Universality of Phi." Here are three examples, where I tested a "series" for each selected PN (W+F) and I simply added the value of 1 for each step to get the next PN: PN2 = PN1+1, PN3=PN2+1, and so on.

Example 1 of uniphi01 (1 to 4 only):

uniphi01.cpp: Universality of Phi 01 (9/6/02)

Get LeftPhi by selecting integer and mantissa.

For initial LeftPhi and RightPhi values:

PN=W+F= 0.000000000 + 0.414213562 = 0.414213562373

Apply rule if PN<1, LP=PN+1, RP=PN:

(PN+1)/PN = 1.414213562373095 / 0.414213562373095

LP= 3.414213562373095, RP= 0.414213562373095

Secant squared (LP*RP)= 1.414213562373095

We construct the secant (sqrt) = 1.189207115003

Then in our constructed right triangle:

Angle= 32.765099739649 degrees Opp= 57.234900260351 degrees

SI= 0.541196100146 CO= 0.840896415254 TA= 0.643594252906

CS= 1.847759065023 SC= 1.189207115003 CT= 1.553773974030

where the tangent squared= 0.414213562373

and the secant squared= 1.414213562373

Generate series, increase LP +1 each step:

1) PN/PN-1 = 1.414213562373095 / 0.414213562373095

LP= 3.414213562373096, RP= 0.414213562373095

Secant squared (LP*RP)= 1.414213562373095

We construct the secant (sqrt) = 1.189207115003

Then in our constructed right triangle:

Angle= 32.765099739649 degrees Opp= 57.234900260351 degrees

SI= 0.541196100146 CO= 0.840896415254 TA= 0.643594252906

CS= 1.847759065023 SC= 1.189207115003 CT= 1.553773974030

where the tangent squared= 0.414213562373

and the secant squared= 1.414213562373

2) PN/PN-1 = 2.414213562373095 / 1.414213562373095

LP= 1.707106781186548, RP= 1.414213562373095

Secant squared (LP*RP)= 2.414213562373095

We construct the secant (sqrt) = 1.553773974030

Then in our constructed right triangle:

Angle= 49.939640973639 degrees Opp= 40.060359026361 degrees

SI= 0.765366864730 CO= 0.643594252906 TA= 1.189207115003

CS= 1.306562964876 SC= 1.553773974030 CT= 0.840896415254

where the tangent squared= 1.414213562373

and the secant squared= 2.414213562373

3) PN/PN-1 = 3.414213562373095 / 2.414213562373095 --->(when printed, p. 5)

LP= 1.414213562373095, RP= 2.414213562373095

Secant squared (LP*RP)= 3.414213562373095

We construct the secant (sqrt) = 1.847759065023

Then in our constructed right triangle:

Angle= 57.234900260351 degrees Opp= 32.765099739649 degrees

SI= 0.840896415254 CO= 0.541196100146 TA= 1.553773974030

CS= 1.189207115003 SC= 1.847759065023 CT= 0.643594252906

where the tangent squared= 2.414213562373

and the secant squared= 3.414213562373

4) PN/PN-1 = 4.414213562373095 / 3.414213562373095

LP= 1.292893218813453, RP= 3.414213562373095

Secant squared (LP*RP)= 4.414213562373095

We construct the secant (sqrt) = 2.101002989615

Then in our constructed right triangle:

Angle= 61.577920865994 degrees Opp= 28.422079134006 degrees

SI= 0.879465224065 CO= 0.475963149478 TA= 1.847759065023

CS= 1.137054624375 SC= 2.101002989615 CT= 0.541196100146

where the tangent squared= 3.414213562373

and the secant squared= 4.414213562373

Example 2 of uniphi01 (1 to 4 only):

uniphi01.cpp: Universality of Phi 01 (9/6/02)

Get LeftPhi by selecting integer and mantissa.

For initial LeftPhi and RightPhi values:

PN=W+F= 0.000000000 + 0.381966011 = 0.381966011250

Apply rule if PN<1, LP=PN+1, RP=PN:

(PN+1)/PN = 1.381966011250105 / 0.381966011250105

LP= 3.618033988749895, RP= 0.381966011250105

Secant squared (LP*RP)= 1.381966011250105

We construct the secant (sqrt) = 1.175570504585

Then in our constructed right triangle:

Angle= 31.717474411461 degrees Opp= 58.282525588539 degrees

SI= 0.525731112119 CO= 0.850650808352 TA= 0.618033988750

CS= 1.902113032590 SC= 1.175570504585 CT= 1.618033988750

where the tangent squared= 0.381966011250

and the secant squared= 1.381966011250

Generate series, increase LP +1 each step:

1) PN/PN-1 = 1.381966011250105 / 0.381966011250105

LP= 3.618033988749895, RP= 0.381966011250105

Secant squared (LP*RP)= 1.381966011250105

We construct the secant (sqrt) = 1.175570504585

Then in our constructed right triangle:

Angle= 31.717474411461 degrees Opp= 58.282525588539 degrees

SI= 0.525731112119 CO= 0.850650808352 TA= 0.618033988750

CS= 1.902113032590 SC= 1.175570504585 CT= 1.618033988750

where the tangent squared= 0.381966011250

and the secant squared= 1.381966011250

2) PN/PN-1 = 2.381966011250105 / 1.381966011250105

LP= 1.723606797749979, RP= 1.381966011250105

Secant squared (LP*RP)= 2.381966011250105

We construct the secant (sqrt) = 1.543361918427

Then in our constructed right triangle:

Angle= 49.613822440803 degrees Opp= 40.386177559197 degrees

SI= 0.761694642423 CO= 0.647936163294 TA= 1.175570504585

CS= 1.312862063490 SC= 1.543361918427 CT= 0.850650808352

where the tangent squared= 1.381966011250

and the secant squared= 2.381966011250

3) PN/PN-1 = 3.381966011250105 / 2.381966011250105

LP= 1.419821271704536, RP= 2.381966011250105

Secant squared (LP*RP)= 3.381966011250105

We construct the secant (sqrt) = 1.839012237928

Then in our constructed right triangle:

Angle= 57.059338539233 degrees Opp= 32.940661460767 degrees

SI= 0.839234175062 CO= 0.543770171495 TA= 1.543361918427

CS= 1.191562533695 SC= 1.839012237928 CT= 0.647936163294

where the tangent squared= 2.381966011250

and the secant squared= 3.381966011250

4) PN/PN-1 = 4.381966011250105 / 3.381966011250105

LP= 1.295685999407889, RP= 3.381966011250105

Secant squared (LP*RP)= 4.381966011250105

We construct the secant (sqrt) = 2.093314599206

Then in our constructed right triangle:

Angle= 61.463971401761 degrees Opp= 28.536028598239 degrees

SI= 0.878516893077 CO= 0.477711281610 TA= 1.839012237928

CS= 1.138282038604 SC= 2.093314599206 CT= 0.543770171495

where the tangent squared= 3.381966011250

and the secant squared= 4.381966011250

Note that in this Example 2, our #1) right triangle, where LP = [1+(phi)^2], is the left right triangle in our initial steps for constructing the pentagon:

Angle= 31.717474411461 degrees Opp= 58.282525588539 degrees

SI= 0.525731112119 CO= 0.850650808352 TA= 0.618033988750

CS= 1.902113032590 SC= 1.175570504585 CT= 1.618033988750

If 58.282525588539 degrees is the "angle of interest," then our facts for the right triangle are consistent with a tangent = 1.618033989..., and A=1, B=0.618033989..., and H=1.175570505..., which is the square root of 1.381966011... . This is our "left right triangle" constructed by "laying down the hypotenuse" of our tangent=2 "right right triangle." In this uniphi01 printout, the next right triangle, #2), where LP = [2+(phi)^2], the next constructed right triangle with tangent = sqrt(1.381966011250...), the tangent = 1.175570505...), the side of the pentagon.

Example 3 of uniphi01 (1 to 4 only): (Where I selected 1 for W and 0.78775 for F.)

uniphi01.cpp: Universality of Phi 01 (9/6/02)

Get LeftPhi by selecting integer and mantissa.

For initial LeftPhi and RightPhi values:

PN=W+F= 1.000000000 + 0.787750000 = 1.787750000000

Apply rule if PN>1, LP=PN, RP=PN-1:

PN/(PN-1) = 1.787750000000000 / 0.787750000000000

LP= 2.269438273563948, RP= 0.787750000000000

Secant squared (LP*RP)= 1.787750000000000

We construct the secant (sqrt) = 1.337067687142

Then in our constructed right triangle:

Angle= 41.590747772847 degrees Opp= 48.409252227153 degrees

SI= 0.663805448195 CO= 0.747905292766 TA= 0.887552815330

CS= 1.506465490333 SC= 1.337067687142 CT= 1.126693513589

where the tangent squared= 0.787750000000

and the secant squared= 1.787750000000

Generate series, increase LP +1 each step:

1) PN/PN-1 = 2.787750000000000 / 1.787750000000000

LP= 1.559362326947280, RP= 1.787750000000000

Secant squared (LP*RP)= 2.787750000000000

We construct the secant (sqrt) = 1.669655653121

Then in our constructed right triangle:

Angle= 53.206991062177 degrees Opp= 36.793008937823 degrees

SI= 0.800804456082 CO= 0.598925891174 TA= 1.337067687142

CS= 1.248744300066 SC= 1.669655653121 CT= 0.747905292766

where the tangent squared= 1.787750000000

and the secant squared= 2.787750000000

2) PN/PN-1 = 3.787750000000000 / 2.787750000000000

LP= 1.358712223119003, RP= 2.787750000000000

Secant squared (LP*RP)= 3.787750000000000

We construct the secant (sqrt) = 1.946214273917

Then in our constructed right triangle:

Angle= 59.081516310879 degrees Opp= 30.918483689121 degrees

SI= 0.857899192035 CO= 0.513818038128 TA= 1.669655653121

CS= 1.165638118422 SC= 1.946214273917 CT= 0.598925891174

where the tangent squared= 2.787750000000

and the secant squared= 3.787750000000

3) PN/PN-1 = 4.787750000000000 / 3.787750000000000 --->(when printed, p. 10)

LP= 1.264008976305195, RP= 3.787750000000000

Secant squared (LP*RP)= 4.787750000000000

We construct the secant (sqrt) = 2.188092776826

Then in our constructed right triangle:

Angle= 62.805083526063 degrees Opp= 27.194916473937 degrees

SI= 0.889456925469 CO= 0.457019012444 TA= 1.946214273917

CS= 1.124281537830 SC= 2.188092776826 CT= 0.513818038128

where the tangent squared= 3.787750000000

and the secant squared= 4.787750000000

4) PN/PN-1 = 5.787750000000000 / 4.787750000000000

LP= 1.208866377734844, RP= 4.787750000000000

Secant squared (LP*RP)= 5.787750000000000

We construct the secant (sqrt) = 2.405774303629

Then in our constructed right triangle:

Angle= 65.438698183657 degrees Opp= 24.561301816343 degrees

SI= 0.909517062148 CO= 0.415666589543 TA= 2.188092776826

CS= 1.099484596406 SC= 2.405774303629 CT= 0.457019012444

where the tangent squared= 4.787750000000

and the secant squared= 5.787750000000

The Pyramid points to Phi and the Universality of Phi:

The point to be made with the output of the uniphi01 algorithm and program is that the Great Pyramid points to both Phi and the Universality of Phi, if it is as I believe it is, a Phi Pyramid. Because, if it is a Phi Pyramid, then the height is equal to the square root of (Phi/2) or (0.636009824...), and the side of the base is 1. The arguments around this issue appear in Volume 1, The Precision of the Ancients. This makes the right triangle that is one-half of the section of the face of the Great Pyramid possess the dimensions A = 0.636009824..., B = 0.5, and H = 0.809016994... (or H = [Phi/2] = [secant 36]). This is similar to A = sqrt(Phi), B = 1, H = Phi. The facts for this right triangle are:

Facts for right triangle, one-half of section of face of the Great Pyramid:

G = 51.82729238..., OPP = 38.17270762...

Sine = 0.786151377... sqrt(phi), Cosecant = 1.27201965... [sqrt(Phi)]

Cosine = 0.618033989... (phi), Secant = 1.618033989... (Phi)

Tangent = 1.27201965... [sqrt(Phi)], Cotangent = 1.27201965...[sqrt(Phi)]

The point to be made with the output of uniphi01 is that Phi is a special case for a universal phenomenon. It is the single instance in the universe of number where LP = RP or Left Phi is Equal to Right Phi (1.618033989...) and this right triangle is the only right triangle in the universe of right triangles where the secant is equal to the tangent squared. Because in all cases the secant squared is equal to (the tangent squared plus 1), we know that it is not possible for there to be any other case where this is true and it is also true that the tangent squared, which always equals (the secant squared minus 1) could possibly equal the secant.

While bending your mind around that, we need to move on to AnyNaX01 (Any N and X), the algorithm and program that is our next step in our journey from the Universality of Phi to the Quadratic Construction.

AnyNaX01.cpp: For N(value added) and X(RP), get LP*RP = RP+N (12/18/05)

To get LP such that LP*RP = RP+N for pre-selected N, and X, where RP = X.

Spelled out: We select a value N to be added to the Right Phi (RP) number, and we select the value X, that is the RP number, and then we calculate, or construct, the Left Phi (LP) number, such that LP*RP = RP+N. We see, through the program, that this works, but this stimulates one's curiosity as to whether we can construct number values that satisfy our version of the quadratic equation: X^2 = X + C (AX^2+BX +C, where A=1, C= -1 and B= -1). This we shall see shortly, but take a look at examples of AnyNaX01.

The algorithm for AnyNaX01 requires that we adjust to four possible conditions:

1) N is not equal to 1, and N is greater than X (X<N). Here, LP= (X+N)/X, and RP= X.

2) N is equal to 1, ......and N is greater than X (X<N). Here, LP= (X+N)/X, and RP= X.

3) N is not equal to 1, and N is smaller than X (X>N). Here, LP= (X/(X-N) and RP= (X-N).

4) N is equal to 1,.......and N is smaller than X (X>N). Here, LP= (X/(X-N) and RP= (X-N).

Here are four examples:

1) The N in LP*RP = RP+N is 1.000000000000000

The X(RP) in LP*RP = RP+N is 0.789000000000000

Apply rule for N=1 and X<N in this calculation.

LeftPhi = (X+N)/X and RtPhi = X:

(LP)2.267427122940*(RP)0.789000000000 = 1.789000000000

The product LP*RP = RP+N = 1.789000000000

And our constructed right triangle is:

Angle= 41.613299391002 degrees Opp= 48.386700608998 degrees

SI= 0.664099772283 CO= 0.747643961023 TA= 0.888256719648

CS= 1.505797836013 SC= 1.337535046270 CT= 1.125800658616

where the tangent squared= 0.789000000000

and the secant squared= 1.789000000000

2) The N in LP*RP = RP+N is 1.000000000000000

The X(RP) in LP*RP = RP+N is 1.435921100000000

Apply rule for N=1 and X>N in this calculation.

LeftPhi = X/(X-N) and RtPhi = X-N:

(LP)3.293993110221*(RP)0.435921100000 = 1.435921100000

The product LP*RP = RP+N = 1.435921100000

And our constructed right triangle is:

Angle= 33.434516885281 degrees Opp= 56.565483114719 degrees

SI= 0.550983580312 CO= 0.834516083863 TA= 0.660243212763

CS= 1.814936117394 SC= 1.198299253108 CT= 1.514593381149

where the tangent squared= 0.435921100000

and the secant squared= 1.435921100000

3) The N in LP*RP = RP+N is 1.750000000000000

The X(RP) in LP*RP = RP+N is 2.000000000000000

Apply rule for N/=1 and X>N in this calculation.

LeftPhi = X/(X-N) and RtPhi = X-N:

(LP)8.000000000000 *(RP)0.250000000000 = 2.000000000000

The product LP*RP = RP+N = 2.000000000000

And our constructed right triangle is:

Angle= 26.565051177078 degrees Opp= 63.434948822922 degrees

SI= 0.447213595500 CO= 0.894427191000 TA= 0.500000000000

CS= 2.236067977500 SC= 1.118033988750 CT= 2.000000000000

where the tangent squared= 0.250000000000

and the secant squared= 1.250000000000

4) The N in LP*RP = RP+N is 2.130000000000000

The X(RP) in LP*RP = RP+N is 1.700000000000000

Apply rule for N/=1 and X<N in this calculation.

LeftPhi = (X+N)/X and RtPhi = X:

(LP)2.252941176471 *(RP)1.700000000000 = 3.830000000000

The product LP*RP = RP+N = 3.830000000000

And our constructed right triangle is:

Angle= 52.513056879560 degrees Opp= 37.486943120440 degrees

SI= 0.793492047616 CO= 0.608580619450 TA= 1.303840481041

CS= 1.260252075625 SC= 1.643167672515 CT= 0.766964988847

where the tangent squared= 1.700000000000

and the secant squared= 2.700000000000

In each of the examples, we see it is true that LP*RP = RP+N, where we selected, in advance, the values for RP and N. We can perform the calculations of AnyNaX01 by construction. The right triangles in the print out are not needed for the constructions, but are present just for information. They show in examples 3 and 4 that although we have chosen the N (value added) to be not equal to 1, we still have the fixed relationship of the secant^2 and tangent^2 that gives us always our "Phi" value of RP + 1.

Now we can look at the AnyNrt01 program, meaning we want to find the X such that X^2 = X+N for an N value that we select in advance. Here we begin to use the quadratic construction to get the results of X^2 = X+N for a pre-selected N. X = the "root" or rt of X^2. We do this by using the quadratic equation and having the "C" value be equal to our pre-selected N value, and A=1 and B=1.

AnyNrt01.cpp For N, construct X^2 = X+N (12/18/2005) [Three examples.]

To get X such that X^2 = X + N for a pre-selected N...

0.5 is added to the N value to get the N results.

and is discussed in detail below.

1) The N in this X^2 = X + N is 2.350000000000000

For this calculation A=1.000000000000 B=1.000000000000

+Neg Solution = 2.112451549660 sqd= 4.462451549660 *

Pos Solution = 1.112451549660 sqd= 1.237548450340

(NS^2) 4.462452 - (NS) 2.112452 = 2.350000000000 *

(PS^2) 1.237548 - (PS) 1.112452 = 0.125096900681

We construct the secant (sqrt) = 1.453427517856

Then in our constructed right triangle:

Angle= 46.525726360159 degrees Opp= 43.474273639841 degrees

SI= 0.725683375755 CO= 0.688028806194 TA= 1.054728187572

CS= 1.378011448808 SC= 1.453427517856 CT= 0.948111572046

where the tangent squared= 1.112451549660

and the secant squared= 2.112451549660

2) The N in this X^2 = X + N is 1.236067977499790

For this calculation A=1.000000000000 B=1.000000000000

+Neg Solution = 1.719043878414 sqd= 2.955111855914 *

Pos Solution = 0.719043878414 sqd= 0.517024099085

(NS^2) 2.955112 - (NS) 1.719044 = 1.236067977500 * [precision limit]

(PS^2) 0.517024 - (PS) 0.719044 = -0.202019779329

We construct the secant (sqrt) = 1.311123136252

Then in our constructed right triangle:

Angle= 40.296763054906 degrees Opp= 49.703236945094 degrees

SI= 0.646746691264 CO= 0.762704869093 TA= 0.847964550211

CS= 1.546200411238 SC= 1.311123136252 CT= 1.179294582245

where the tangent squared= 0.719043878414

and the secant squared= 1.719043878414

3) The N in this X^2 = X + N is 1.789000000000000 --->(when printed, p. 15)

For this calculation A=1.000000000000 B=1.000000000000

+Neg Solution = 1.927935572776 sqd= 3.716935572776 *

Pos Solution = 0.927935572776 sqd= 0.861064427224

(NS^2) 3.716936 - (NS) 1.927936 = 1.789000000000 *

(PS^2) 0.861064 - (PS) 0.927936 = -0.066871145553

We construct the secant (sqrt) = 1.388501196534

Then in our constructed right triangle:

Angle= 43.928919177584 degrees Opp= 46.071080822416 degrees

SI= 0.693765427204 CO= 0.720201035834 TA= 0.963294125787

CS= 1.441409388227 SC= 1.388501196534 CT= 1.038104534462

where the tangent squared= 0.927935572776

and the secant squared= 1.927935572776

In this AnyNrt01 program, we have stepped away from Left Phi and Right Phi and into the world of the quadratic construction. The most likely explanation for why this algorithm works by adding 0.5 to our pre-selected N value, appears immediately below. It is not easy to follow, and I question whether it fully explains what is happening and why. The algorithm works.

In this AnyNrt01 program, we select our N value and this same value is assigned to C and the amount of 0.5 is added to C. B =1 and B^2 is of course 1. A=1 and the denominator (2*A) therefore equals 2. B/(2*A) is always 0.5, and the determinant is always sqrt[b^2 - 4*A*C] /2 except that we use the absolute value of the determinant. Therefore, the value of the determinant, the value that we take the square root of, is: plus 1, minus [4*1*N], as a positive value equal to [4*1*N] minus 1. But we add 0.5 to our N, which has the effect of making the determinant = [4*(N+0.5)] minus 1. But 4 times 0.5 = 2, and therefore, the net effect is to make the determinant always equal to [(4*N)+2] minus 1, equal to [4*N] plus 1. But we take the square root of that, and divide by 2, sqrt[1 + 4*(C+0.5)] / 2, of course, and then to get our two solutions: 1) we add (B/2) or we add 0.5, and 2) we subtract (B/2) or we subtract 0.5. The end result is that the difference between our two solutions is always 1. We are seeing once again our secant squared and our tangent squared. Look at the examples again and you will see that in every case, the solution to our target value X such that X^2 = X + (pre-selected N), turns out to be a secant squared, and the equation X^2 = X + N is (secant)^4 = (secant)^2 + N. I have not yet explored this program further to find if any other relationships apply with regard to the pre-selected N value.

Next we will look at a series of programs that begin with InvXpN01. The "Inv" means inverse, because in their original and simplest form the Negative Solution and Positive Solution are inverses of one another. These values can be constructed. But there is more to what they mean in terms of "proportion."

InvXpN01.cpp Inverse of X = X + N (2/6/2006)

Based on X^2 + B*X - 1 = 0. Yields solution to

(1/X) = X + B, where B only is varied, 1, 2, 3, 4, ... and then 1/2, 1/3, 1/4, ...

Pos = -(B/2) + sqrt[B^2 + 4 ]/2, Neg = -(B/2) - sqrt[B^2 + 4 ]/2 (as a line)

PS = Positive solution, NS = Negative solution, TR = absolute value of determinant

BD = (B/2), TD = sqrt[B^2 + 4]/2. FOR EACH VALUE B, THE INVERSE OF THE POSITIVE SOLUTION IS THE NEGATIVE SOLUTION, AND THE NEGATIVE SOLUTION IS EQUAL TO THE POSITIVE SOLUTION PLUS B, AS IN (1/X) = X + B.

AND, THE NEGATIVE SOLUTION SQUARED IS EQUAL TO THE NEGATIVE SOLUTION, TIMES B, PLUS 1: NS^2 = (B*NS) + 1. (Run for B = 1 to 6, 1/1 to 1/6)

PS = X^2 + 1.00*X -1 = 0.618033988750

TR=5.000000000000 TD=1.118033988750 -BD=0.500000000000

NS = X^2 - 1.00*X -1 = 1.618033988750

TR=5.000000000000 TD=1.118033988750 +BD=0.500000000000

NS^2= 1.00 * 1.618033988750 +1 = 2.618033988750

PS = X^2 + 2.00*X -1 = 0.414213562373

TR=8.000000000000 TD=1.414213562373 -BD=1.000000000000

NS = X^2 - 2.00*X -1 = 2.414213562373

TR=8.000000000000 TD=1.414213562373 +BD=1.000000000000

NS^2= 2.00 * 2.414213562373 +1 = 5.828427124746

PS = X^2 + 3.00*X -1 = 0.302775637732

TR=13.000000000000 TD=1.802775637732 -BD=1.500000000000

NS = X^2 - 3.00*X -1 = 3.302775637732

TR=13.000000000000 TD=1.802775637732 +BD=1.500000000000

NS^2= 3.00 * 3.302775637732 +1 = 10.908326913196

PS = X^2 + 4.00*X -1 = 0.236067977500

TR=20.000000000000 TD=2.236067977500 -BD=2.000000000000

NS = X^2 - 4.00*X -1 = 4.236067977500

TR=20.000000000000 TD=2.236067977500 +BD=2.000000000000

NS^2= 4.00 * 4.236067977500 +1 = 17.944271909999

PS = X^2 + 5.00*X -1 = 0.192582403567

TR=29.000000000000 TD=2.692582403567 -BD=2.500000000000

NS = X^2 - 5.00*X -1 = 5.192582403567

TR=29.000000000000 TD=2.692582403567 +BD=2.500000000000

NS^2= 5.00 * 5.192582403567 +1 = 26.962912017836

PS = X^2 + 6.00*X -1 = 0.162277660168

TR=40.000000000000 TD=3.162277660168 -BD=3.000000000000

NS = X^2 - 6.00*X -1 = 6.162277660168

TR=40.000000000000 TD=3.162277660168 +BD=3.000000000000

NS^2= 6.00 * 6.162277660168 +1 = 37.973665961010

Change to fractions: (The same algorithm works for B<1.)

PS = X^2 + 1.000000000*X -1 = 0.618033988750

TR=5.000000000000 TD=1.118033988750 -BD=0.500000000000

NS = X^2 - 1.000000000*X -1 = 1.618033988750

TR=5.000000000000 TD=1.118033988750 +BD=0.500000000000

NS^2= 1.000000000 * 1.618033988750 +1 = 2.618033988750

PS = X^2 + 0.500000000*X -1 = 0.780776406404

TR=4.250000000000 TD=1.030776406404 -BD=0.250000000000

NS = X^2 - 0.500000000*X -1 = 1.280776406404

TR=4.250000000000 TD=1.030776406404 +BD=0.250000000000

NS^2= 0.500000000 * 1.280776406404 +1 = 1.640388203202

PS = X^2 + 0.333333333*X -1 = 0.847127088383

TR=4.111111111111 TD=1.013793755050 -BD=0.166666666667

NS = X^2 - 0.333333333*X -1 = 1.180460421716

TR=4.111111111111 TD=1.013793755050 +BD=0.166666666667

NS^2= 0.333333333 * 1.180460421716 +1 = 1.393486807239

PS = X^2 + 0.250000000*X -1 = 0.882782218537

TR=4.062500000000 TD=1.007782218537 -BD=0.125000000000

NS = X^2 - 0.250000000*X -1 = 1.132782218537

TR=4.062500000000 TD=1.007782218537 +BD=0.125000000000

NS^2= 0.250000000 * 1.132782218537 +1 = 1.283195554634

PS = X^2 + 0.200000000*X -1 = 0.904987562112

TR=4.040000000000 TD=1.004987562112 -BD=0.100000000000

NS = X^2 - 0.200000000*X -1 = 1.104987562112

TR=4.040000000000 TD=1.004987562112 +BD=0.100000000000

NS^2= 0.200000000 * 1.104987562112 +1 = 1.220997512422

PS = X^2 + 0.166666667*X -1 = 0.920132881566

TR=4.027777777778 TD=1.003466214899 -BD=0.083333333333

NS = X^2 - 0.166666667*X -1 = 1.086799548233

TR=4.027777777778 TD=1.003466214899 +BD=0.083333333333

NS^2= 0.166666667 * 1.086799548233 +1 = 1.181133258039

THIS IS THE KIND OF OUTPUT THAT HAS PERSUADED ME THE ORIGINAL PYTHAGOREAN THEOREM WAS UNDERSTOOD AND USED AS A PROCEDURE FOR THE CONSTRUCTION OF NUMBER VALUES AS LINES WITH INTERESTING PROPORTIONS, SUCH AS: X, SUCH THAT (1/X) = X + B, AND NS^2 = (B*NS) +1.

To continue the discovery of "interesting proportions," by performing the Quadratic Construction, we next take a look at InvXpN02, where I made C = -0.8 instead of 1, and the "calculation" of the construction is "more refined."

InvXpN02.cpp Inverse of X = X + N (3/17/2006)

Based on X^2 + B*X - C = 0. Yields solution for X such that (1/X) = X + B.

Range= 5, multipliers for this run are:

MV=1.000000000000 CV=0.800000000000 [meaning C is -0.8 instead of -1]

PS= 1.000000*X^2 + 1.000000*X -0.800000= 0.524695076596

NS= 1.000000*X^2 - 1.000000*X -0.800000= 1.524695076596

NS^2=(1/1.000000)*([1.000000*1.524695]+0.800000)=2.324695076596

TR=4.200000000000 TM=2.049390153192 DE=2.000000000000

TD=1.024695076596 +/- BD(0.500000000000) = PS/NS

PS= 1.000000*X^2 + 2.000000*X -0.800000= 0.341640786500 ***

NS= 1.000000*X^2 - 2.000000*X -0.800000= 2.341640786500

NS^2=(1/1.000000)*([2.000000*2.341641]+0.800000)=5.483281573000

TR=7.200000000000 TM=2.683281573000 DE=2.000000000000

TD=1.341640786500 +/- BD(1.000000000000) = PS/NS

*** PLEASE NOTICE this fascinating result: Here, where B = 2 and C = [-] 0.8, first note that the "more refined" calculation of NS^2 is: (1/A)*( [B*NS] + C ). NOTE ALSO, that for this particular calculation and construction, the Negative Solution is equal to (Pi*MQ) -0.8. And, NS^2 is equal to the Negative Solution [NS] plus (Pi*MQ) as well as to (2*NS)+0.8. And, the secant of the right triangle is (NS-1) when the tangent is equal to sqrt(0.8).

THIS IS THE KIND OF OUTPUT THAT HAS PERSUADED ME THAT THERE IS SOMETHING OCCURRING HERE THAT POINTS TO "PROPORTION" AS BEING A DISTINCT PHYSICAL PHENOMENON THAT CAN BE EXPRESSED AND PERCEIVED IN NUMBER VALUES (AND CONSTRUCTIBLE LINES) BUT WHICH IS IN REALITY DIFFERENT AND SEPARATE FROM BOTH NUMBER THEORY AND MATHEMATICS AS KNOWN AND PRACTICED BY MATHEMATICIANS.

*** --->(when printed, p. 20)

PS= 1.000000*X^2 + 3.000000*X -0.800000= 0.246424919657

NS= 1.000000*X^2 - 3.000000*X -0.800000= 3.246424919657

NS^2=(1/1.000000)*([3.000000*3.246425]+0.800000)=10.539274758972

TR=12.200000000000 TM=3.492849839315 DE=2.000000000000

TD=1.746424919657 +/- BD(1.500000000000) = PS/NS

PS= 1.000000*X^2 + 4.000000*X -0.800000= 0.190890230021

NS= 1.000000*X^2 - 4.000000*X -0.800000= 4.190890230021

NS^2=(1/1.000000)*([4.000000*4.190890]+0.800000)=17.563560920083

TR=19.200000000000 TM=4.381780460041 DE=2.000000000000

TD=2.190890230021 +/- BD(2.000000000000) = PS/NS

PS= 1.000000*X^2 + 5.000000*X -0.800000= 0.155183609470

NS= 1.000000*X^2 - 5.000000*X -0.800000= 5.155183609470

NS^2=(1/1.000000)*([5.000000*5.155184]+0.800000)=26.575918047352

TR=28.200000000000 TM=5.310367218941 DE=2.000000000000

TD=2.655183609470 +/- BD(2.500000000000) = PS/NS

Change to fractions: (Algorithm works for B<1.)

PS= 1.000000*X^2 + 1.000000*X -0.800000= 0.524695076596

NS= 1.000000*X^2 - 1.000000*X -0.800000= 1.524695076596

NS^2=(1/1.000000)*([1.000000*1.524695]+0.800000)=2.324695076596

TR=4.200000000000 TM=2.049390153192 DE=2.000000000000

TD=1.024695076596 +/- BD(0.500000000000) = PS/NS

PS= 1.000000*X^2 + 0.500000*X -0.800000= 0.678708781050

NS= 1.000000*X^2 - 0.500000*X -0.800000= 1.178708781050

NS^2=(1/1.000000)*([0.500000*1.178709]+0.800000)=1.389354390525

TR=3.450000000000 TM=1.857417562101 DE=2.000000000000

TD=0.928708781050 +/- BD(0.250000000000) = PS/NS

PS= 1.000000*X^2 + 0.333333*X -0.800000= 0.743156270930

NS= 1.000000*X^2 - 0.333333*X -0.800000= 1.076489604264

NS^2=(1/1.000000)*([0.333333*1.076490]+0.800000)=1.158829868088

TR=3.311111111111 TM=1.819645875194 DE=2.000000000000

TD=0.909822937597 +/- BD(0.166666666667) = PS/NS

PS= 1.000000*X^2 + 0.250000*X -0.800000= 0.778119593409

NS= 1.000000*X^2 - 0.250000*X -0.800000= 1.028119593409

NS^2=(1/1.000000)*([0.250000*1.028120]+0.800000)=1.057029898352

TR=3.262500000000 TM=1.806239186819 DE=2.000000000000

TD=0.903119593409 +/- BD(0.125000000000) = PS/NS

PS= 1.000000*X^2 + 0.200000*X -0.800000= 0.800000000000

NS= 1.000000*X^2 - 0.200000*X -0.800000= 1.000000000000

NS^2=(1/1.000000)*([0.200000*1.000000]+0.800000)=1.000000000000

TR=3.240000000000 TM=1.800000000000 DE=2.000000000000

TD=0.900000000000 +/- BD(0.100000000000) = PS/NS

Next, in InvXpN03, I fixed B = 1, and C = -0.8, and varied A, A = 1 to 5.

The result is a variation of X such that (1/X) = X+B, in that we get X [NS], such that

NS = PS + (1/A). NS^2 still is equal to (1/A) * ( [B*NS] + C ).

InvXpN03.cpp Inverse of X = X + N (3/21/2006)

Based on A*X^2 + B*X - C = 0, but here in InvXpN03 A is the variable of interest.

A varies A = 1 to 5, (1/1) to (1/5).

Range= 5, multipliers for this run are:

MV=1.000000000 AM=1.000000000 BM=1.000000000 CM=0.800000000

PS= 1.000000*X^2 + 1.000000*X -0.800000= 0.524695076596

NS= 1.000000*X^2 - 1.000000*X -0.800000= 1.524695076596

NS^2=(1/1.000000)*([1.000000*1.524695]+0.800000)=2.324695076596

TR=4.200000000000 TM=2.049390153192 DE=2.000000000000

TD=1.024695076596 +/- BD(0.500000000000) = PS/NS

PS= 2.000000*X^2 + 1.000000*X -0.800000= 0.430073525437

NS= 2.000000*X^2 - 1.000000*X -0.800000= 0.930073525437

NS^2=(1/2.000000)*([1.000000*0.930074]+0.800000)=0.865036762718

TR=7.400000000000 TM=2.720294101747 DE=4.000000000000

TD=0.680073525437 +/- BD(0.250000000000) = PS/NS

PS= 3.000000*X^2 + 1.000000*X -0.800000= 0.375960686537

NS= 3.000000*X^2 - 1.000000*X -0.800000= 0.709294019870

NS^2=(1/3.000000)*([1.000000*0.709294]+0.800000)=0.503098006623

TR=10.600000000000 TM=3.255764119220 DE=6.000000000000

TD=0.542627353203 +/- BD(0.166666666667) = PS/NS

PS= 4.000000*X^2 + 1.000000*X -0.800000= 0.339354390525

NS= 4.000000*X^2 - 1.000000*X -0.800000= 0.589354390525

NS^2=(1/4.000000)*([1.000000*0.589354]+0.800000)=0.347338597631

TR=13.800000000000 TM=3.714835124201 DE=8.000000000000

TD=0.464354390525 +/- BD(0.125000000000) = PS/NS

PS= 5.000000*X^2 + 1.000000*X -0.800000= 0.312310562562

NS= 5.000000*X^2 - 1.000000*X -0.800000= 0.512310562562

NS^2=(1/5.000000)*([1.000000*0.512311]+0.800000)=0.262462112512

TR=17.000000000000 TM=4.123105625618 DE=10.000000000000

TD=0.412310562562 +/- BD(0.100000000000) = PS/NS

Change to fractions: (Algorithm works for A<1.)

PS= 1.000000*X^2 + 1.000000*X -0.800000= 0.524695076596

NS= 1.000000*X^2 - 1.000000*X -0.800000= 1.524695076596

NS^2=(1/1.000000)*([1.000000*1.524695]+0.800000)=2.324695076596

TR=4.200000000000 TM=2.049390153192 DE=2.000000000000

TD=1.024695076596 +/- BD(0.500000000000) = PS/NS

PS= 0.500000*X^2 + 1.000000*X -0.800000= 0.612451549660

NS= 0.500000*X^2 - 1.000000*X -0.800000= 2.612451549660

NS^2=(1/0.500000)*([1.000000*2.612452]+0.800000)=6.824903099319

TR=2.600000000000 TM=1.612451549660 DE=1.000000000000

TD=1.612451549660 +/- BD(1.000000000000) = PS/NS

PS= 0.333333*X^2 + 1.000000*X -0.800000= 0.656385865285

NS= 0.333333*X^2 - 1.000000*X -0.800000= 3.656385865285

NS^2=(1/0.333333)*([1.000000*3.656386]+0.800000)=13.369157595854

TR=2.066666666667 TM=1.437590576857 DE=0.666666666667

TD=2.156385865285 +/- BD(1.500000000000) = PS/NS

PS= 0.250000*X^2 + 1.000000*X -0.800000= 0.683281573000 ***

NS= 0.250000*X^2 - 1.000000*X -0.800000= 4.683281573000

NS^2=(1/0.250000)*([1.000000*4.683282]+0.800000)=21.933126291999

TR=1.800000000000 TM=1.341640786500 DE=0.500000000000

TD=2.683281573000 +/- BD(2.000000000000) = PS/NS

*** PLEASE NOTE THAT there is an interesting relationship between this construction, where the NS = 4.683281573... and the construction that I "tagged" and commented on previously under InvXpN02, where NS = 2.3416407865..., and NS^2 = 5.483281573... . First, 4.683281573... +0.8 = 5.483281573..., and 4.683281573... = 2*2.341640786... or 2 times the previous NS. Also, here, the TD value is 2.683281573..., which is equal to [sqrt(0.2) * 6], which is equal to the most efficient, most conveniently constructed, value that we chose, using the start-of-the-pentagon construction, to get (UN+B2), our value that is 1.2 in proportion to UN, being the same as the proportion of (6/5). And, of course, 2.683281573... = 2*1.341640786..., where 1.341640786... is the secant for the right triangle with a tangent = sqrt(0.8).

***

PS= 0.200000*X^2 + 1.000000*X -0.800000= 0.701562118716

NS= 0.200000*X^2 - 1.000000*X -0.800000= 5.701562118716

NS^2=(1/0.200000)*([1.000000*5.701562]+0.800000)=32.507810593582

TR=1.640000000000 TM=1.280624847487 DE=0.400000000000

TD=3.201562118716 +/- BD(2.500000000000) = PS/NS

The outcomes of these programs, and my observations on the number values, will probably be labeled by some if not all conforming mathematicians and physicists as "coincidence," just the same old reality about the quadratic equation. But I see this as the most dangerous and most unscientific attitude of turning away from a path that has not been explored. It is not rational, and not good science, to conclude "There is nothing worthwhile here," and turn away. There is something here, something subtle, something which involves the numbers of very great precision and their proportional relationships.

I have written and run several other programs that explore the realities of proportion. There are too many "coincidences," to be only coincidences.

Another example: sqrt(5), plus 5, divided by 10, squared, = that same value -0.2, as follows:

2.236067978..., 7.236067978..., 0.7236067978..., squared = 0.5236067978...

Exploration of the quadratic construction, and how the proportions of the solutions and their squares consistently suggests that proportion is the natural origin of number, shape, geometry and mathematics. The theme of my work: Proportion is everything, must be "ruled out" by the most intensely rigorous mathematical exploration, or we take the risk of missing the path to the ultimate truth and most fundamental truth of the real physical universe.

2) What is the shape of an atom? --->(when printed, p. 25)

In order to meet our desperate need to protect and sustain the life-supporting environment on Earth, we must know exactly how Nature works. In order to do that, we must either prove beyond all doubt or rule out the "ball-bearing" shape of the "planetary" model of the atom. This may mean, of course, revising Quantum Theory.

The problem is obvious. Chaos Theory is a partial answer to the question: How does Nature produce the variety of shapes that we see in the universe? The simplest principle of Chaos Theory is that: Nature repeats the same patterns over and over but on a different scale. This makes sense, but it is also not complete. It also is not a theory of "chaos" only. It is a valid description of all of Nature, that which appears "orderly" as well as that which appears "chaotic."

The obvious problem is: How could Nature begin to construct the universe with atoms that are spheres and produce immediately molecules that are crystalline, polyhedrons, the shapes of polished, faceted gems? And then all of the shapes of Nature that are larger than molecules that again are repetitious patterns of angular or polyhedral shapes? We have to find the explanation for either one of two distinctly different phenomena:

1) Either atoms are spheres and there is some almost magical, fantastic "boundary" event where the spherical shapes make a transition to polyhedral shapes at the molecular level; or

2) Atoms are polyhedrons in the first place, and that is why all of the subsequent shapes of Nature are also polyhedrons, and such polyhedral shapes are often not visible to us because of the very fine precision of the scale on which they are built. I offer my own theory as to how atoms are polyhedrons, which I call the "Living Crystal Electromagnetic Atom," described in Part 3 of Precision of the Ancients, entitled "Medicine and the Atom."

We have to do this. We are scientifically suicidal if we continue to assume spherical atoms. How to do it experimentally is a challenge. The only idea I have come up with is that if atoms were traveling down an inclined plane, spheres would roll; polyhedrons would bounce, tumble and slide. That's not much to go on, but attempts should be made to test this.

3) Sequential shaping and the "evolution" of shapes as the evolution of life:

Civilization asked the question: How does Nature produce the great variety of species that we observe in the universe? Charles Darwin proposed an answer: Natural Selection. He knew, as scientists know today, that his answer makes sense, but is probably not complete. The most obvious missing information is that having a bird develop a different beak, or grow larger claws, or become an excellent swimmer, all evolution of new "survival traits," does not in itself explain exactly what occurs that creates a "bio-chemical firewall" that prevents the bird with new traits from successfully reproducing by mating with an "old bird" with old traits. In other words, we can see the logic of natural selection, but the actual complete creation of a new, separate, species is clearly more than just the development of new survival traits.

Let's ask another question: How does Nature produce the great variety of shapes that we observe in the universe? First, the initial scientific response is that the variety of shapes is not necessarily so great. From ancient times up through the present, scientists, especially biologists, have observed the proportion of Phi and the spiral and other common shapes repeated in Nature over a broad range of natural occurrences and living organisms. Vast millions of ordinary individuals have observed the similarities among veins and arteries, the channels of a river delta, the branches of a tree. So that is the initial perception. There are a variety of shapes, but perhaps also some shapes repeated everywhere or almost everywhere.

The second response comes to us from "Chaos Theory." Nature repeats the same patterns over and over but on a different scale.

I propose that the complete answer, and valid answer, is "Proportion." To me, proportion is the mechanism through which all the shapes of Nature, from atoms to the galaxies and all that exists in the universe, are made. To me, proportion is the "number system" of Nature which is in reality the source of number though not number in itself. Number is a way of naming proportions, because there are so many proportions, a number system is required to name them.

If it is true that proportion is the cause of Nature's shapes, then it is the cause of the shapes of molecules, and the cause of the capacity of molecules to assemble themselves and disassemble themselves and build living tissue and living organisms. This means that there is a direct succession of cause an effect. The causal chain of reality in the universe is that proportion causes shapes, shapes cause molecules, and molecules possess the ability to organism themselves into the process that we call "life." Therefore, once again, if squaring the circle reveals proportion, then it is true that squaring the circle reveals the Secret of Life in the Universe, that "secret" being that proportion is the sufficient cause of the self-organization of matter.

I believe the process of "proportion" begins with sequential shaping, described under Values and Constructions under the Construction of 7.778, Construction #10. Sequential halving is the proportional function that appears to us in Nature as the "binary" number system in genetics and electronics and electro-chemistry. Simply stated, the binary system is actually the most readily observed aspect of what is actually proportion in action. The observed "binary system" is far from sufficient in itself to produce the products of Nature. Proportion is the cause that is prior to "binary" sequential halving. This all needs to be explored further.

4) The Economy of Nature:

Scientists love to argue "Occam's Razor," saying that if there are two possible solutions, or answers, to a question about the natural universe, then the simpler solution is more likely to be correct. Then they choose the more complex answer and claim that it is the simpler.

Another way of identifying the economy of Nature is to state that if a simple method or phenomenon evolves that addresses a particular function or need, it is never followed by a second, more complex, method to address the same or a similar function or need. In simpler words: If Nature needs to dig a hole, and a stick is sufficient to dig a hole, Nature will not invent a shovel, just more sticks.

Therefore, if Proportion is sufficient for Nature to create shapes out of matter, then it will not invent anything further, no "number system" and no "mathematics" of Nature will be necessary or evolved. If proportion is sufficient, Nature will just invent more proportion. Thinking about proportion (which I sometimes capitalize out of both reverence and emphasis that I see Proportion as a subject of study, a field of knowledge), I can suggest how "proportion" is most likely Nature's form of "number." All we have to do is visualize a simple right triangle and assign that a value of 1. A slightly different right triangle could be 2. A square 3, although we see it as 4. A pentagon is a symbol of 5 for us, but Nature might recognize it as some other number. However, if proportion is the coded language of Nature, number values would most likely arise from the several different characteristics of any polygonal or polyhedral shape: not only the number of vertices, but the precise angles, the lengths of each side, the proportions. This attempted visualization brings us right back to our own number system. The numbers I used here, to define angles and side lengths, are in themselves an acceptable way to describe proportion. We don't need to invent a new number system, but we do need to see the number system we use differently when we use it to describe proportion. This would make us just as economical as Nature, use our own number system to describe proportion, even though proportion is not the same thing as our number system, and not the same thing as our mathematics. It is proportion. It is like anything else. Do it a while, and then you will come to know what it is. If you don't do it, you will just continue to see it as a "coincidence," which means, really, that you don't see it and it does not exist for you.

If proportion is Nature's coded language, then proportion is the true basis for all number systems. This means that there are many possible number systems, but only one reality that is the reality of proportion. Genes must be Nature's only memory. Genes are a record of an event that occurred in the past and can be repeated. Genes are, like shapes, patterns that can be duplicated, or copied. Genes are a record of instruction on how to make a copy of something that has been produced before. The printed photocopy of your birth certificate is like the "printed" copy of a newborn turtle. Some changes occur in the record over time, and yet sometimes the changes, if any, are very small over hundreds of millions of years. Genes that have been in the record a very long time are found in many different types of organisms, some the simpler organisms, others more complex. The genes that have been in the record a very long time work together with the new genes that produce new traits. This is why, I believe, Jesus said (Matthew 13:52): "...the kingdom of heaven is like a householder who brings forth from his storeroom things new and old." Every living organism is made by genes, records, that are both new and old.

Lastly, most obviously, proportion is the common quality of any and all existing matter. Everything that exists is "in proportion" to all other things. Therefore, Nature uses that, and does not need to invent anything further in order to be what it is or do what it does. If this is correct, we must understand Proportion in order to be the caretakers of Nature.

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