Welcome to Aquarius, Volume 9 (September 17, 2007)
The Re-Discovery of Proportion, Part 5
Proof that we can square the circle (Proof p. 4-6).
Link to: (Welcome) or (Geometry Alpha Index).
The Construction of 1 and Pi Exactly (from Volume 8) [Squaring-the-circle exactly]
Included in Volume 8, the Values and Constructions required, are descriptions and drawings for the following constructions: (Link back on last page.)
Begin the constructions with [the five-sixths constructions] Constructions #1, #2, and #3.
[The nine-elevenths construction] Construction #4
[The point fifty-five (0.55) constructions] Constructions #5 and #6
[The (secant 18)^4 constructions] Constructions #7 and #8.
[My specific claim of discovery] and the key to the theory and the solution.
Then Construction #9.
[The Key to the Theory and Solution] Between Constructions #9-2 and #9-3
[Construction of 0.7778], as well as 7.778 and 0.2222, 2.222, and 1.222 and 8.778.
Construction #10, a set of constructions.
[The Definition of M and MQ] (MQ = M^2)
[The Construction of TN and NT] Construction #11
[The Construction of MQ and M] Constructions #12 through #16
[The Get Pi constructions] Constructions #17 through #20
[The Pi and 1 constructions] (improved from Volume 1 of Welcome to Aquarius)
[The Pi and MQ Constructions] (alternative square = circle area)
and the Similar Right Triangle (SRT) Construction Table:
Notation and construction of similar triangles (re-construction of a right triangle):
S=sine CS=cosecant C=cosine SC=secant T=tangent CT=cotangent
The reality of re-constructing right triangle angles and sides:
Construct A, and: A=A B=A*cotangent H=A*cosecant
Construct B, and: A=B*tangent B=B H=B*secant
Construct H, and: A=H*sine B=H*cosine H=H
(Below, G = angle of interest, OPP = opposite angle in a right triangle.)
sin = A/H = 1/csc csc=H/A = 1/sin sin G = cos OPP csc G = sec OPP
cos= B/H = 1/sec sec=H/B = 1/cos cos G = sin OPP sec G = csc OPP
tan= A/B = 1/cot cot=B/A = 1/tan tan G = cot OPP cot G = tan OPP
If A=X H=X * CS H= X/S H=B * SC H=B/C
B=X * CT B=X/T B= H * C B=H/SC
If B=X, H=X * SC H=X/C H= B * SC H= B/C
A=X * T A=X/CT A=H * S A=H/CS
If H=X, A=X * S A=X/CS A=H * S A=H/CS
B=X * C B=X/SC B=H * C B=H/SC
and X * sin = X/csc X * csc = X/sin
X * cos = X/sec X * sec = X/cos
X * tan = X/cot X * cot = X/tan
and the following list of values of interest, pertinent to squaring the circle exactly:
Important values and constructions to know (using degrees as angular measure),
(ratios can be converted to line lengths and line lengths can be converted to ratios):
Cosine 36 = 0.809016994374947424102293417182819... times 2 = Phi
Phi = 1.61803398874989484820458683436564... and inverse is
phi = 0.618033988749894848204586834365638... or "little phi" or 1/Phi
and = (Sine 18) * 2, Sine 18 = 0.309016994374947424102293417182819...
also, (Sine 18) = (Cosine 36) minus 0.5
(phi)^2 = 0.381966011250105151795413165634362... <------- * * *
phi + (phi)^2 = 1 (known to all mathematicians and geometers)
(Phi)^2 = 2.61803398874989484820458683436564... = (1.618033988...)^2
and (Phi)^2 / 2 = (sine 18) + 1 = 1.30901699437494742410229341718282...
and note that (sine 18) + 0.5 = 0.809016994374947424102293417182819...
Also, Phi is the limit of the Fibonacci series: 1+1, 2+1, 3+2, 5+3, 8+5, 13+8, and so on, where the last sum divided by the previous sum = 1.618033988... as the sums grow larger. Previous sum divided by last is inverse (0.618033988...).
Begin with phi = 0.618033988749894848204586834365638...
(Pi*Ho) = 3.14460551102969314427823434337184... = 4 times sqrt(phi)
sqd = 9.88854381999831757127338934985021... = 16 * phi
(Pi*Ho)/ 4 = sqrt(phi) = 0.786151377757423286069558585842959...
FX = (5/6)=0.833333333333333333333333333333333... <------- * * *
sqrt(5/6)=0.912870929175276855761616304668004...
SX = (6/5)=1.2
sqrt(6/5)=1.0954451150103322269139395656016...
Phi^2=2.61803398874989484820458683436564... (also = Phi +1) <------- * * *
divided by Pi =0.833346100984274536545032628319445...
which = (5/6) * MQ, where MQ = 1.00001532118112944385403915398333...
and sqrt(MQ) = M= 1.00000766056122262280424348031297...
or M = 1.000007661...
QM = inverse of MQ = 0.999984679053605550929814794430672...
which = Pi * VP, or Pi * 0.318305009375087626496177638028635... <------- * * *
and VP = (5/6) * (phi)^2 or (5/6) * 0.381966011250105151795413165634362...,
therefore VP = 0.318305009375087626496177638028635...
and (phi)^2 = (0.618033988749894848204586834365638...)^2
= 1/ 2.61803398874989484820458683436564... = (phi)^2 because I use upper case and
lower case to distinguish phi [0.618033988...] from Phi [1.618033988...]
Note that a hand-held scientific calculator may show Phi as 1.618033989...
PF=0.55 (label for this value derived from "Point Five") <-------* * *
ES= (1/0.55)=1.81818181818181818181818181818182...
(label for this value derived from "Eighteen Series")
*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
LN= (11/9) = 1.22222222222222222222222222222222... (eLeven Nines)
NL = 0.818181818181818181818181818181818... (Nine eLevenths)
* 1.2 = 0.981818181818181818181818181818182...
SF = (secant 18)^4 = 1.22229123600033648574532213002996...
(label for this value derived from Secant 18 to Fourth power)
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... <------- * * *
NT, inverse of TN = 0.999943537369751504522585249236876...
(labels "TN" and "NT" from mind of author)
The value 0.2222 (stopped there at 4th decimal place)
The value 0.7778, = 1 - 0.2222
Also, 1.222 and 8.778 and 0.1222 and 0.8778
These values, from LN to 0.7778, in particular TN and NT, possess unique properties that reveal the meaning of proportion.
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
Proof that we can square the circle: Phi^2 = [Pi * (5/6) * MQ]
.(Detailed construction procedures with drawings appear in Volume 8)
Look again at the beginning of the "Eighteen Series"
PF=0.55 (label for this value derived from "Point Five")
ES= (1/0.55)=1.81818181818181818181818181818182...
(label for this value derived from "Eighteen Series")
*1.2 = 2.18181818181818181818181818181818...
*1.2 = 2.61818181818181818181818181818181...
*1.2 = 3.14181818181818181818181818181818...
Obviously, the third value, 2.618181818... is close in value to Phi^2.
Obviously, the fourth value, 3.141818181... is close in value to Pi.
This close: 2.618181818... = Phi * MQ * TN
or 2.618033988... * 1.000015321... * 1.000056466..., or 2.618033988... * 1.000071788...
and 3.141818181... = Pi * MQ * TN or Pi * 1.000071788...
With the construction of MQ and TN, we can then construct the value of 1.000071788...
and then construct a right triangle with Altitude A = 3.141818181... and Base B equal to our new value 1.000071788..., and the tangent of that right triangle will be Pi exactly.
You are already buried in drawings and construction procedures from Volume 8. Here I present only the calculations that can be performed by construction.
Look again at the value labeled VP:
VP = 0.318305009375087626496177638028635...
or VP = [(5/6) * (phi)^2] or (5/6) * 0.381966011250105151795413165634362... .
1/VP = 3.14164078649987381784550420123877... or Pi times MQ.
This value of 3.14164078649987381784550420123877... (Pi*MQ) or a fraction or multiple of it is found in many trigonometric ratios. The two easiest to describe are:
1) Right triangle with tangent = square root of 0.8:
The right triangle with tangent = sqrt(0.8), has a cosine = 0.745355992... and the secant has a value of 1.341640786..., and that plus 1.8 (the secant squared) = 3.141640786... .
2) Phi raised to the fourth power, or Phi squared squared:
Phi^2 squared, or Phi^4 (6.854101966...) plus 1 = 7.854101966..., divided by 10
equals 0.7854101966..., times 4 = 3.141640786...
Therefore, we know that we can construct the value Pi*MQ (3.141640786...) and that means of course that a right triangle with A=3.141640786... and B=1.000015321... has a tangent = Pi.
Proof
:MQ = 1.000015321..., and 1/MQ = QM = 0.999984679..., and the value of QM is equal to Pi * (5/6) * (phi)^2. Therefore, Pi * MQ = Pi / [Pi * (5/6) * (phi)^2]
Or, Pi / [Pi * (5/6) * (phi)^2] = 1/ [(5/6)*(phi)^2] and this is redundant due to the fixed definition of MQ and QM. That definition is, sqrt(MQ) = M, and M is that number such that M^2 plus (1/M^2) = 2 exactly. Therefore, the only way the proposition that Phi^2 = Pi*(5/6)*MQ can be proven wrong is to prove that the factor MQ given as a factor of Phi^2 is not M^2. However, the definition is circular (or reciprocal, or proportional) because in reality, MQ is the inverse of a value, [Pi * (5/6) * (phi)^2], which has Pi as a factor multiplied by two other known defined factors. That means that the values of M and MQ must always be at least as precise as the value we assign to Pi. This reality of proportion therefore means that we cannot prove that we have the wrong value for MQ without also concluding, by definition, that we also have the wrong value for Pi. In conclusion, if we accept a value for Pi, however precise and however computed or constructed, we must accept the proportional values for M and MQ by the same standard. Otherwise, we are only saying that we cannot be certain of M because we are not certain of Pi, but this is not a valid dismissal of the proof because we accept Pi as a defined value even though we may claim that we cannot write it precisely. Likewise, M and MQ are defined values that we may claim that we cannot write precisely for the same reason. The proof should not be discredited for the reason that neither Pi nor M nor MQ can be written exactly in our decimal digital number notation. This is a limitation of the notation, not a limitation of the numbers.
Thus, we see that Phi^2 = Pi*(5/6)*MQ, and Pi*MQ = secant + secant^2 of the right triangle with tangent = sqrt(5) * (2/5) or sqrt(0.8).
Facts for tangent=sqrt(0.8) right triangle:
G= 41.81031489..., OPP= 48.18968511...
Sine = 0.666666666..., Cosecant = 1.5
Cosine = 0.745355992..., Secant = 1.341640786...
Tangent = 0.894427191..., Cotangent = 1.118033989...
Therefore this proof for squaring the circle, and the facts of the tangent = sqrt(0.8) right triangle, are an introduction to "Proportion," a reality separate from any number system discovered or invented by humans, and which I believe is the true "number system" of Nature, sufficient to enable matter to be self-organizing. And this is what I believe was the original purpose of the ancient geometric riddle. I see the binary number system, found in genetics and electronics, as a special case of "Proportion," which I call "sequential halving," described in these web pages under the title The Pyramid Mark (Two): the square root of two. To square the circle, as described in detail in Volume 8, we employ "sequential shaping" which is probably a more fundamental reality of Proportion than sequential halving.
Link to: (Welcome) or (Geometry Alpha Index).