Back to Basics: How to Measure a Circle

[A Gnostic Agnostic]

Greetings everyone,

I am an analyst and have recently concluded an approx. 5-year inquiry (independent) relating to the root cause(s) of human suffering. For the sake of relevance, this inquiry proposed a practical need to look closely at modern-day physics esp. as it relates to any/all discernible laws which govern the physical universe. Though there is indeed a much broader context behind what is to follow (the contents of which will fill a planned future book), only that which relates to the physical universe will be of concern here. What was found has implications for both GR and EU.

contents.
i. Science: Progression (of Knowledge)
ii. Division: the Monumental Blunder of π
iii. Reciprocity: Time and Space
iv. Unity ( Solving e = MC² )
v. Conclusion: Gravity (of Ignorance)
_______________________________________________________________

"In the beginning GOD created the heavens and the earth."
-Creationist

"In the beginning BANG created the heavens and the earth."
-21st Century Scientist

i. Science: Progression (of Knowledge)
Science entails the discipline of challenging our most basic underlying assumptions, should any one or more hitherto taken to be (thus acted upon as) 'true', in reality be 'not necessarily true'. Any true scientist (whose underlying passion is for the truth in/of all matters, no matter how much it challenges the status-quo) must be willing to both acknowledge and admit that not all "beliefs" are true, despite how appealing they may be. The capacity to discern between what is real and what is not (ie. imaginary) is invariably what any faculty of science must endeavor to do.

The 'root' underlying obstacle to the realization of the failure of General Relativity is owing to a much deeper root (and failure): present-day humanity does not know how to properly measure a circle, as the correct value of π clarifies what rationally precedes Einstein's e = MC². That which underlies the latter serves as a viable measure of the ignorance (non-derogatory, non-rhetorical) of present-day humanity as it demarcates what is known (the relation) from what is unknown (the precedent). The precedent is, as it happens, unprecedented.

ii. Division: the Monumental Blunder of π
π is one of the most integral relationships that exists: it relates a line (radius and/or diameter) to a curve (circumference). The nature of the relation between line and curve is indifferent from the nature of the relation between space and time (cyclical movement in characteristic of our kinematic universe). In order to unify these, it is important to understand what Φ (the golden ratio) actually is, and why there is only "1" number in the universe like it.

Φ is the only (positive) number whose reciprocal is precisely one less than itself viz. 1/Φ = Φ - 1. A reciprocal is any number n multiplied by 1/n such to compose 1 (Φ x 1/Φ = 1). Further, Φ is the only irrational which, if/when squared, returns itself plus a rational 1 viz. Φ² = Φ + 1. Observe here Φ in terms of π (base of 2π or "one" full rotation):

(π+π√5)/2π = Φ
(3π+π√5)/2π = Φ + 1
__________________
Difference: 2π and/or 1

If it is possible to express Φ in terms of π,
is it also possible to express π in terms of Φ?

The answer is, "believe" it or not, yes. All that is needed is a unit square s = 1 with corresponding 2r = 1 circle inscribed (r = 1/2)
and a double unit square (2x1) whose diagonal is √5. It is possible to precisely measure π using only these (as they are the ingredients of Φ):

πbywayofΦishax600.jpg

https://i.postimg.cc/d3Kc7ZBY/bywayof-ishax600.jpg
Note: π can also be expressed as √(8√5-8) and/or √(-8+8√5),
implying an "octave" spread out between rational/irrational couplings.

Please see here http://measuringpisquaringphi.com/geome ... ofs-of-pi/ for a website containing several geometric proofs (as well as an actual real-world physical measurement) which corroborates what is illustrated above (if the image is displaying correctly, unable to preview). Another way of seeing this is recognizing that each 1/4 circle is equal to 1/√Φ, thus 1/√Φ + 1/√Φ + 1/√Φ +1/√Φ = 4/√Φ. Note that √Φ and π/4 are reciprocals, a most important (hitherto unrecognized) relationship between space and time (which clarifies the dark matter/energy nonsense).

However, Φ has one important characteristic that is hitherto unrecognized: it naturally conjoins the rational and irrational numbers viz. (1+√5)/2. The integrity of this rational/irrational coupling (complex analysis equivalent: real and imaginary) is preserved in/of 4/√Φ (rational/irrational). What rational is to real/discrete (terminating), irrational is to imaginary/non-discrete (non-terminating). This property emerges by rooting the roots of unity:

1 = Unity
√1 = +1, -1
√+1 = +1 (real)
√-1 = i (imaginary)

Note that 4/√Φ (the correct value of π) is (one of four symmetrical) geometric root(s) of:

f(x) = x⁴ + 16x² - 256
x = ±√9.88854381999...
x = ±i√25.88854381999...
__________________
This crucial "junction function" is practically the spine of the physical universe
and/or real/imaginary (rational/irrational) number system. Note the roots
have an integer difference of exactly '16' viz. 16 = Φπ².

thus π is certainly not "transcendental". The "approximated" number 3.14159... is (only) "transcendental" because it is not π (!) The correct value of π is just as geometric as Φ because π is contained in/as Φ (they are inseparable). The gravity of this oversight can not be under-stated: all major scientific progress in physics is at-a-stand-still until it correctly measures the circle such to reunify space and time. The (re-)unification of π and Φ is practically equivalent to the (re-)unification of space and time (multiplicative reciprocal aspects of motion).

iii. Reciprocity: Time and Space
Where Western science is going wrong (aside from not being able to measure a circle properly) is failing to understand the nature of the relationship between space (s) and time (t) as being multiplicative reciprocal aspects of motion, wherein v is a measurable velocity:

v = s/t (velocity)
e = t/s (energy)
s/t x t/s = 1 (concerns unity)
_________________________
All motion(s) have their own corresponding energy constituency.

What underlies this is the failure to recognize that the 3 dimensions normally associated with space actually precedes all considerations of space and time. That is: space and time are not "two separate things" (or even "things" at all for that matter: they are abstract) - what applies to one aspect also applies to the other aspect (like a see-saw). In this way, not only does "space" have 3 dimensions, but time also has 3 dimensions:

s³/t x t³/s = (st)²

Recall the "junction function" in light of what follows.

iv. Unity ( Solving e = MC² )
The solving of " Einstein's " e = MC² entails deriving that which rationally precedes (ie. underlies) it. In other words: one must be able to show why e = MC², and this is now possible. Unfortunately for General Relativity, what underlies e = MC² is catastrophic for it:

π ≠ 3.14159... (human approximation error)
π = 4/√Φ
π² = 16/Φ
16 = Φπ²
e = MC²
1 = Φπ²/16
∞ = Φ(π/4)²
_____________________________________________
Wherein 16 = Φπ² rationally precedes e = MC²,
Φ is a universal 1D electric constant
π² is a universal 2D magnetic constant

Einstein's General Theory of Relativity is certainly a failure: the universe is certainly not gravity-centric,
as "gravity" can be expressed in terms of magnetism (an inward motion).
The correction(s) implies: no "big bang", no "black holes", no "dark matter", no "dark energy" etc.
(rather than the universe coming into being with a "BANG", it seems GR is destined to go out with one).
All of these (and much more) are artifacts rooted in the de-coupling of space and time via Φ and π.
If/when allowing the "speed" of light to be '1' (what it should be according to the Φ property(s), as light does not technically have a "speed"),
the expansion of the universe (scalar constant) is counter-acted by gravitation (scalar inverse of expansion, distributed).
The "balance" between the two is what constitutes orbital motion according to the nature/constituency of the body(s) concerned.

v. Conclusion: Gravity (of Ignorance)
The π problem truly stands as a monumental measure of the ignorance of 2000+ years of humanity (dark ages). If any advanced civilization wished to quickly gauge the general development of other living beings, the first thing that civilization would check is the value of π (part. whether or not it is known to be in relation to the integral golden ratio). As it stands: our human civilization is not as "advanced" as it may want to believe, hence the absolute imperative for human beings to incessantly question our most basic underlying assumptions, and this includes such basic ones as our knowing how to properly measure a circle.

To close: the solving for e = MC² is a product of the ongoing inquiry " whence human suffering? " as the deficiency of π was/is found to be underlying several thousand years of suffering, as the disunity of line and curve is tantamount to a dishonoring of a most primordial coupling: that of Φ and π. This single reunification alone has the capacity to spark a global revitalization of science such that truth is made the (only) authority rather than authority made the (only) truth. In parting: in a world wherein in takes a "believer" to somehow "believe" the opposite of what is true, knowing all not to "believe" definitely tends towards all knowledge indefinitely.

[A Gnostic Agnostic]

https://www.youtube.com/watch?v=_j2tq1GsXbI

[paladin17]

(π+π√5)/2π = Φ
(3π+π√5)/2π = Φ + 1
__________________
Difference: 2π and/or 1

If it is possible to express Φ in terms of π,
is it also possible to express π in terms of Φ?

The answer is, "believe" it or not, yes.

It is not hard to see that in your expression pi may be changed to any other number, and the equation would still hold. So you didn't really express anything in terms of anything else.

The "proofs" that you provide simply postulate this hypothetical relation between phi and pi, never actually arriving at it from any considerations. E.g. see here: it is implicitly assumed that you can exactly fit 4 circles of a circumference = 2 inside a 2*Sqrt(phi) diameter circle. This is not proven anywhere (and cannot be proven, because it is wrong), yet this is the basis of the "proof" of phi/pi connection given there.
(Also, in this picture AE = Sqrt(phi+1), not phi, but it doesn't really matter, since only the mentioned implicit introduction of the needed result is necessary for the "proof").

[A Gnostic Agnostic]

It is not hard to see that in your expression pi may be changed to any other number, and the equation would still hold.

This is true, though thoroughly irrelevant. There is only one 'base' which indicates a 'full circle': 2π.
This is the same full circle which is used to correlate the 4 equidistant points via a "one" full rotation.

So you didn't really express anything in terms of anything else.

A base of 2π means 'per full circle'.

(1+√5)/2 is just "baseless" arithmetic.
(π+π√5)/2π implies a real kinematic geometry.

One can thus measure the 2π that emerges as a natural product of the square of Φ
without the need for any "approximation". It is the latter that is catastrophic:
the method of exhaustion involving inscribed/circumscribed polygons
will always miss an entire constituency of the circle. This is reflected
in/as the "approximated" and "transcendental" (erroneous) π of 3.141...
...the correct ratio of π is 4/√Φ which can be expressed in a number of ways:

√(8√5-8) and/or √(-8+8√5)
_____________________________________________________
This is octave-based, as in music. Relates to frequency.

-√¯2(1+√5)¯ + √¯10(1+√5)¯ (and/or vice versa)
_____________________________________________________
This is mechanically how a circle is captured/measured:
-(1/4) rotation +(5/4) rotation = 4/4 = 2π/2π wherein
the first -1/4 rotation "collects" information about 1/4
of the rotation (circle), then constructs a full circle with it.

8/√2¯(1+√5)¯
_____________________________________________________
This is 4x the reciprocal of the square root of √Φ as
1/√Φ = π/4 such that 1/√Φ + 1/√Φ + 1/√Φ + 1/√Φ = 4/√Φ.

The de-coupling of Φ and π is practically equivalent
to the de-coupling of space and time, as neither
are autonomous from one another: they are one
via reciprocity.

s/t = v wherein v is velocity
t/s = e wherein e is energy
________________________
s/t x t/s = 1 = Φ(π/4)²

The "proofs" that you provide simply postulate this hypothetical relation between phi and pi, never actually arriving at it from any considerations.

See above re: reciprocity. Taking the reciprocal of √Φ viz. 1/√Φ recovers π/4.
This is their real (not hypothetical) relation, and is a natural consequence
of ordinary mathematics.

The Giza pyramid was constructed with a height of √Φ
(ie. by way of use of the so-called Kepler triangle)
and was done so for an extremely important reason(s):
1/√Φ is the reciprocal of π/4, and this relationship
is thoroughly concealed by the "approximation"
method(s) which decouples this natural relation
and substitutes it with an "approximated" and
"transcendental" pi.

E.g. see here: it is implicitly assumed that you can exactly fit 4 circles of a circumference = 2 inside a 2*Sqrt(phi) diameter circle.

The website (not mine, though the proofs are valid) already addresses this:

Note: A few readers have asked, “How do you know that the diameters of 4 blue circles fit exactly tangent across the diameter of the Big Yellow Circle in Proof 1 Fig 1?”

Answer:

Blue circle diameter = 2 / Pi, given where Circumf = 2, and diam = C / Pi, Proof 1 Fig 1,

Big Yellow circle diameter = 2 sqrt Phi, from Kepler’s Triangle Proof 1 Fig 1.

In Proof 1 Fig 1, we do not know yet what Pi is and so we ask, What is the value of Pi when the diameter of the Big Yellow Circle = 4 times the diameter of the Blue circle? The answer is:

2 sqrt Phi = 4 (2 / Pi), Solve for Pi

Pi = 4 / sqrt Phi,

Pi = 4 / 1.272019650… = 3.144605511…

Therefore, the diameter of 4 Blue circles fit exactly tangent across the diameter of the Big Yellow Circle in Proof 1 Fig 1 when Pi = 3.144605511… . And since Pi is a universal constant, not a variable, there is no need to look for another value of Pi.

There is no real basis for any outstanding assumption regarding these circles fitting exactly.
The proof that I provide is less intensive to understand if one is able to imagine rotation.
If not, the video linked in the first reply shows exactly how/why this works both
arithmetically and (correspondingly) geometrically via √2/2 being the key, hence
√2¯(1+√5)¯/2 is arithmetically/geometrically the real "square root" of Φ
responsible for the construction of the (any) perfectly symmetrical circle.

https://i.postimg.cc/d3Kc7ZBY/bywayof-ishax600.jpg
By rotating the extended AC exactly one full rotation (ie. 2π) that point D will invariably
"kiss" both the circle and the unit square 4 times equidistantly. This is the '4' in/of 4/√Φ.

Pi can thus not be "transcendental", as it implies a real radius and
requires 4 real equidistant points. The "proof" that "pi" is "transcendental"
is concerning the "approximated" number 3.141... not π, as π = 3.1446055...
which is a precise ratio of 4/√Φ rather than an "approximation".

There is so much more this implies, I wish I could get into it but at this time
I am only making attempt to have others see that the "approximation" of π
is the underlying problem. A curve should never be "approximated" by way of
→∞ straight lines esp. if/when given as a natural product of a square.

This is not proven anywhere (and cannot be proven, because it is wrong), yet this is the basis of the "proof" of phi/pi connection given there.

It is on the same website under 'Geometric Proofs of Pi' and already discusses/addresses this contention.
If you believe it to be "wrong" can you explain exactly how/why?

(Also, in this picture AE = Sqrt(phi+1), not phi, but it doesn't really matter, since only the mentioned implicit introduction of the needed result is necessary for the "proof").

AFE is the Kepler triangle 1/√Φ/Φ.

AF is 1.
FE is √Φ.
AE is Φ.

AE is certainly not Sqrt(phi+1).

The Kepler triangle, as with the golden ratio, as with phi, establishes
a fixed/scalar relationship between rational and irrational numbers.

Φ is irrational/rational viz. 3.236.../2
π is rational/irrational viz 4/1.272...

π ≠ 3.141...
π = 4/√Φ
π² = 16/Φ
16 = Φπ²
e = MC²

1 = 16/Φπ² is rational/irrational
1 = Φπ²/16 is irrational/rational
hence
1 = Φ(π/4)²

Line and curve (and/or space and time) are one by way of reciprocity.
This follows as an obvious natural consequence of properly measuring the circle
such to find that the reciprocal of √Φ viz. 1/√Φ is π/4. This relation is integral
to the structure of the physical universe, as Φ is the only number in the universe
whose reciprocal is exactly 1 less than itself. It is also the only irrational whose
square is exactly one more than itself:

1/Φ = Φ - 1
Φ² = Φ + 1

thus Φ must be contained in/as any true solution for oneness/unity.
What I am doing is showing why e = MC²... it is because 16 = Φπ²
rationally precedes/underlies it. In other words: if not the latter, the former
would not be a valid coupling.

[paladin17]

So you didn't really express anything in terms of anything else.

A base of 2π means 'per full circle'.

(1+√5)/2 is just "baseless" arithmetic.
(π+π√5)/2π implies a real kinematic geometry.

So by the same token:
1 + 2 = 3 is just "baseless" arithmetic;
(e+2e)/h = 3e/h (e is the elemental charge, h is the Planck's constant) implies a real fractal subquantum aetheric interdimentional cymatic physics.

This is true, though thoroughly irrelevant.

One can thus measure the 2π that emerges as a natural product of the square of Φ
without the need for any "approximation".

No, one can't. They are unrelated.

the correct ratio of π is 4/√Φ

No, this is wrong.

1/√Φ recovers π/4.

It doesn't. It's a different number.

Therefore, the diameter of 4 Blue circles fit exactly tangent across the diameter of the Big Yellow Circle in Proof 1 Fig 1 when Pi = 3.144605511

This is very true. Except in reality the circles would not fit, because Pi is smaller than the value given here.

[A Gnostic Agnostic]

So by the same token:
1 + 2 = 3 is just "baseless" arithmetic;

Yes, I agree.

(e+2e)/h = 3e/h (e is the elemental charge, h is the Planck's constant) implies a real fractal subquantum aetheric interdimentional cymatic physics.

This is true, though thoroughly irrelevant.

Irrelevant at best, yes, however I don't know what point (if any) you are attempting to make here.

I admit I do not know what you actually mean by "a real fractal subquantum aetheric interdimentional cymatic physics".
If you mean some/any fixed/scalar integral structure(s) and/or integrity(s) underlying (and/or as preceding)
the strictly physical universe, I know such things do exist however do not adopt any of the modern-day "quantum"
(non-)mechanics (mainly because it is coming from people who do not know how to measure a circle properly)
which brings us full circle to my 2π point: back to basics (how to measure a circle) and we can actually do this
with the (addition of) the 2π that is a natural product of a square (multiplicative) revealing reciprocity
as a natural consequence.

This is significant if/when considered in light of 1 = Φ(π/4)² as "one"
can not and does not (in reality) rest on anything but (a scalar base of)
reciprocity, like a see-saw with space and time as reciprocating ends.
In effect, this is the yang and yin that underlies/governs any/all causality.

I do know space and time are naturally discrete units (ie. "quantized") implying (granting) discretion.
I refer to Hamlet: "to be, or not to be", that is the
(definition of) any binary (+/-) discretion:

All (that which is) viz. + (presence of)
(that which is) Not viz. - (absence of)
________________
viz. conjugation All/Not (+/-).

Roots of Unity:
√1 = +1, -1
√+1 = 1 (real) <-rational, terminating numbers (line)
√-1 = i (imaginary) <-irrational, non-termination numbers (curve)
+1 = unity
-1 = not unity
-(-1) = not not unity = unity = +1

This conjugation is reflected in/as space and time which are in discrete units, thus is "subquantum" in the sense that conjugation precedes any/all quantitative considerations, as the universe is not only quantitative, but qualitative. This applies to numbers, such as √5, the latter being absolutely integral to the unification of space and time as one (as they always have been, are, and always will be).

I don't know what your meaning/understanding of "aetheric" is, however this is likely the domain that Western science tries to ignore because of their separation of space and time (catastrophic). Because space and time are not two separate things (or "things" at all) the 3 dimensions that are normally associated only to "space" actually precedes all considerations of space and time entirely. This means that both space and time have 3D indiscriminately associated with them:

s³/t = real, physical
t³/s = ethereal, metaphysical

All universal motion(s) (measured in/as a velocity, v) have in/as themselves
their own corresponding energy constituency:

s / t = v (velocity)
t / s = e (energy)

The "datum" of the physical universe is (discrete units of) motion (there is no other constituency)
as all of the (sub)atomic particles are particularly discrete configurations of motions which can be expressed
in as an array of periodicity(s) of/on a common base of 2π.

This is why it is important to know how a circle is produced naturally by a square.
It is especially important as it relates to the inverse square law which is resolutely uncontested.

As to "interdimensional", space and time are each bound to 3D (there is no 4th "dimension" of time).
The powers of phi: Φ, Φ², Φ³ relate respectively to expansion/equilibrium/gravitation:

Φ = (1+√5)/2 <-expansion
Φ² = Φ + 1 <-discretion (local)
Φ³ = √5 + 2 <-gravitation

wherein the physical universe is only doing one operation: simultaneously expanding/contracting wherein
any being has degrees of freedom (time) to exercise discretion, as rooted in/as space and time.

No, one can't. They are unrelated.

They may appear unrelated to you, and that is a/the problem - they are, in reality, not.

The addition of 2π as a product of a square is certainly not unrelated to a circle: it is the circle.
The circle does not come from anywhere but the incessant squaring of Φ viz. Φ→Φ²=∞=(Φ + 2π/2π).
This 2π is related directly to light, and why the golden ratio is contained/expressed in unity 1 = Φ(π/4)².
However this unity really can only be recognized if/when the integrity of √5 is consciously acknowledged.
The universe would not exist as-is if not for the quality(s) that √5 has. It underlies the basis of all that is physical,
including the pentagram which is constructed by the powers of phi, as example:

http://ksuweb.kennesaw.edu/~sedwar77/ti ... ithphi.gif

we can observe how important the golden ratio is (only) by actually knowing what it actually is/does.
It unites the rational/irrational numbers (ie. line and curve) as incessantly inter-related.
This is contained/captured in/as π as 4/√Φ.

The "approximated" π of 3.141... effectively de-couples Φ from π, as they are not actually autonomous, they were/are always one.

No, this is wrong.

From the previous post:

If you believe it to be "wrong" can you explain exactly how/why?

I'm sorry if you do not like it, but if you state something is "wrong",
if you truly know this to be the case, you would be capable of making it?
Simply stating "wrong" is not science, it is something else entirely.

"Approximating" a circumference with straight-edge polygons is actually practically "wrong"
because an entire constituency of the circle is missed by way of (if even unbeknownst).

It doesn't. It's a different number.

There is a reason it is a different number.

It is a different number because π = 3.141... is not π, but an "approximation"
of 4/√Φ and not the actual precise ratio (value) of π, hence a discrepancy.

"Approximated" π/4 = 0.78539816...
√(8√5-8)/4 = π/4 = 1/√Φ = 0.78615137...

Things tend to be not so precise if/when "approximating", hence
eliminating the need for "approximation" will give a more accurate answer.

This is very true. Except in reality the circles would not fit, because Pi is smaller than the value given here.

"Approximated" pi is smaller because the approximation method misses an entire constituency of the circle.

In reality, the ratio of any 90-degree angle to corresponding equidistant arc is 1/√Φ = π/4. This is why a diameter is equivalent to 2r:
each axis of the 90-degree angle is each its own r, thus the "diameter" is actually also a "right angle" of the same length.

This reciprocal relation can be seen clearly as a natural consequence only if/when properly measuring (not "approximating") a circle.
This is why a 1000mm diameter circle was physically measured and shows that a circumference of 3142mm is clearly short/deficient.
It was further measured to be at least 3144+mm and certainly short of 3145mm.

Denying real-world physical measurements is not science, it is something else entirely
and motivated by something else entirely - certainly not any altruistic pursuit of science.

I understand many people have a psychological/emotional attachment to the "approximated" and "transcendental" pi value of 3.141...
and it is naturally hard for many people to challenge such basic underlying assumptions (despite this being what science is all about)
however the inability to consider/face the possibility "we were wrong" is the essence of destructive hubris underlying present-day "civilization" with "approximated" pi being a 2000+ years long-standing measure of the same. It truly does "measure" the magnitude of the ignorance associated with any/all suffering(s) associated with anything in relation to division viz. "us vs. them". Such problems are not (now) outside the scope of science, as a rational solution to unity exists by way of reciprocity as it naturally includes both rational and irrational aspects/elements/numbers.

It should be clear the implications: reciprocity underlies the "golden rule"
and there is a real, rational basis for it that which is resolutely scientific.

[paladin17]

Irrelevant at best, yes, however I don't know what point (if any) you are attempting to make here.

The point is that if you multiply and divide your expression by Pi, it doesn't add anything to it. You might as well take any other number.

1 = Φ(π/4)²

This is an incorrect expression. The right side is not equal to 1.

π = 3.141... is not π, but an "approximation" of 4/√Φ

Quite the contrary, 4/Sqrt(phi) is the approximation of Pi (which also lacks any mathematical basis, I would claim).
A much better numerical approximation may be acquired e.g. through Leibnitz's series formula (which is mathematically justified).

In reality, the ratio of any 90-degree angle to corresponding equidistant arc is 1/√Φ = π/4.

No. The arc length for a 90-degree angle is R*Pi/2, where R is the radius of the arc.
Phi doesn't appear anywhere here. It is a different, unrelated number.

[A Gnostic Agnostic]

The point is that if you multiply and divide your expression by Pi, it doesn't add anything to it. You might as well take any other number.

It does add: by establishing a normalizing base of 2π (full rotation, 360-degrees) - this applies both quantitatively and qualitatively.
Quantitatively, the expression is the same unique scalar ratio 1.618... regardless of (though including) the use/value of pi.
Qualitatively, the expression establishes a common denominator 2π thus implicitly/explicitly concerns any/all universal motions via. s/t
wherein 1/√Φ = π/4 relates rational/irrational numbers (real and imaginary) by way of (multiplicative) reciprocity.

https://postimg.cc/rRLRYNx2

Space and time are reciprocal aspects of motion - nothing more or less - practically a numerator and a denominator.
The integrity of the natural relation between rational/irrational (real and imaginary) numbers is contained in both
the golden ratio and the correct ratio of π as 4√Φ.

This is an incorrect expression. The right side is not equal to 1.

It does if/when π = 4/√Φ viz. 3.1446055..., as it does (as the site linked shows).
http://measuringpisquaringphi.com/geome ... ofs-of-pi/

A physical measurement was made. The result shows that 3.141... is certainly deficient.

The right side will appear not to equal 1 if using the "approximated" π of 3.14159...
...hence the discrepancy.

Quite the contrary, 4/Sqrt(phi) is the approximation of Pi (which also lacks any mathematical basis, I would claim).

4/√Φ is not an "approximation" at all, it is a discrete ratio.

3.141... is an "approximation" and not a discrete ratio.
It is not even a ratio. It is an "approximated" number.
3.1446055... is also not a ratio. It is also an "approximated" number.
4/√Φ is a discrete ratio. The '4' is extremely important.

Miles Mathis wrote a paper(s) claiming π = 4. He is half-correct:

http://milesmathis.com/pi2.html

Abstract: I show that in all kinematic situations, π is 4. For all those going ballistic over my title, I repeat and stress that this paper applies to kinematic situations, not to static or geometric situations. I am analyzing the equivalent of an orbit, which is caused by motion and includes the time variable. In that situation, π becomes 4.

Miles did not apply what he found to his original premise: all kinematic situations.
We live in a universe of motion (always kinematic) thus "π becomes 4" on the numerator.
What Miles did not do is place this '4' on a base of motion. This is what √Φ is, as
because it is a root, there are two solutions (+/-) which form a symmetry (conjugation).

Take a circle. Inscribe a cross such that there are 4 radii forming right angles.
Each quarter is π/4. A diameter is also 2r, the same as the length of two right-angled radii.

π defined as the ratio of a circumference to a diameter is incorrect.
The correct relation is 2r/c/4 = 8r/c (see Miles' paper with an integral
showing C = 8r). Pi must be in a real, rational (geometric) relation to '4'
owing to the four equidistant (symmetrically situated) points that relate
the square to the circle. If these points are not captured, there is no
valid coupling between the unit square (line) and 2r = 1 circle (curve).

The s=√2/2 square captures these four equidistant points. This is why √Φ = √2¯(1+√5)¯/2
as these 4 points touch the unit square/circle (space and time are discrete units).

The golden ratio incessantly coincides with the entire circumference of the 2r = 1 circle
"kissing" the unit square at 4 equidistant (symmetrical) points.
This real relation is how/why π can not possibly be "transcendental", it is geometric.

π = 4/√Φ is one of four roots of f(x) = x⁴ + 16x² - 256
The other three roots form the same symmetry which couples
rational and irrational numbers (and is the function through which
all real/imaginary magnitudes transit). Their numerical difference
is a factor of 16:

±√9.88854381999... <-real
±i√25.88854381999... <-imaginary
_____________________________
integer difference: 16 (=Φπ²).

I can hypothesize the following:

The Riemann Hypothesis will never be proved true/not
until humanity learns how to properly measure a circle.

as the proof entails a correction to π from an "approximated" 3.141... to 4/√Φ.
The truth of the Riemann Hypothesis will never come without the conscious acknowledgement
of the error in/of the "approximation" of 3.141...

No. The arc length for a 90-degree angle is R*Pi/2, where R is the radius of the arc.
Phi doesn't appear anywhere here. It is a different, unrelated number.

It is the same, thus "yes".

0.5√¯(8√5-8)¯/2 = 0.78615137... = 1/√Φ

Phi appears when you set C² to the square of π viz. (8√5-8) in/of e = MC²:

e = M(8√5-8)
M = e(√5+1)/32

M (mass) is (equal to) energy multiplied by 1/16 of Φ.
Set e to 16 (reciprocal).

M = 16(√5+1)/32
M = Φ

Phi appears in both e = MC² as 16 = Φπ² and unity as 1 = Φ(π/4)²
and the golden ratio is a (the) universal scalar constant (as it is)
as it concerns the rate ("speed") of light (causality).

Gravity can be expressed in terms of magnetism.
This is thus catastrophic for General Relativity
and rather embarrassing (Einstein's own equation proves he was wrong).
But then again, even worse is the whole "approximation" of π problem
that underlies the last 2000+ years of human history
as well as the present-day nonsense of "us vs. them".

As from Adam's own "side" was derived Eve, from Φ's own "side" is derived π.
They were not made as two autonomous things - they were made to be as one.
The (re-)unification (of space and time) happens only by way of reciprocity.

[paladin17]

It does add: by establishing a normalizing base of 2π (full rotation, 360-degrees) - this applies both quantitatively and qualitatively.

No, it doesn't.
2 and 2e/e is the same number without any qualitative difference.

1/√Φ = π/4 relates rational/irrational numbers

Perhaps, if this expression was correct. However, it is not.

A physical measurement was made. The result shows that 3.141... is certainly deficient.

I have no doubt that it is not hard to come up with a bad measurement.

4/√Φ is not an "approximation" at all, it is a discrete ratio.

However you wish to call it, it has nothing to do with Pi.

Miles Mathis wrote a paper(s) claiming π = 4. He is half-correct:

I'd say he is 0.785375-correct.

[A Gnostic Agnostic]

No, it doesn't.
2 and 2e/e is the same number without any qualitative difference.

2e/e has nothing to do with the circle (as concerned), nor 360-degrees of rotation (periodicity), nor four symmetrical quarters etc.
(π+π√5)/2π → (3π+π√5)/2π entails a numerator addition of 2π, on a base of 2π, which has everything to do with a circle
both quantitatively and qualitatively.

Do you understand that both a base of 2π and an addition of 2π (as a product of a square) implies a circle(s)?
Denying a/the reality because one doesn't like the result is not scientific, it is something else entirely.

Perhaps, if this expression was correct. However, it is not.

Are you able to give even one reason/rationale as to why you believe this?
I myself don't "believe" 1/√Φ = π/4, it can be known the equality is true.

All you keep saying is "no it is not" and/or "wrong" without any substance whatsoever. I'm still waiting for the "because..."
You earlier tried to contest a proof, but mistook a length of Φ as Sqrt(phi+1) which was/is incorrect.
It leads me to believe you don't actually have a substantive refutation which is fine.
However repeating nos and wrongs with no substance is certainly not science.

I have no doubt that it is not hard to come up with a bad measurement.

Correct - this is exactly what "approximating" π with straight-edge polygons is, a bad measurement.

Physically measuring the circle is better, as it is a direct observation, thus more scientific.
Observing how/why Φ geometrically produces the 2r = 1 circle as natural product of a square is even better, as
reciprocity becomes a natural consequence of the relation between line and curve (space and time).

Denying/ignoring real-world measurements/observations is not science, it is the opposite of science.

Science would entail trying/testing/falsifying the measurement(s) taken and/or assertions made.
If the measurement taken is "bad" science would entail methodically deducing specifically where and why
with sound reason/rationale.

However you wish to call it, it has nothing to do with Pi.

Because you said so? Despite the evidence to the contrary?
And here I thought the Relativists were upside-down.

The opposite is true: it has everything to do with π.
This is obscured by any/all "approximation" methods
which use →∞ straight lines to measure a curve (?).

I'd say he is 0.785375-correct.

In your approximation, duly granted.

The problem with "approximations" is they are not precise.
Measuring π as the natural product of a square is precise.

Here it is again:
https://postimg.cc/9R1nKMK8

Upon rotation of AC about the origin, point D both precisely and incessantly coincides with the
circumference of the 2r = 1 circle while/as inscribed in the unit square s = 1. This relation is fixed/scalar.
Further: point D precisely coincides with 4 equidistant/symmetrical points falling on both the circle and the square (4 is thus integral to π).
The 2r = 1 circle is thus a natural product of the (square of the) golden ratio and why π = 4/√Φ (4 is integral to π).
The 8x directions of the originally situated double-unit-squares containing the central unit square is implied by √(8√5-8).

I am not interested in a back-and-forth with anyone who does not have the willingness/capacity
to challenge basic underlying assumptions and/or are psychologically/emotionally attached
to such things as an approximated transcendental pi god of 3.141... it is such attachment(s)
that underlie the lack of progress in not only physics, but human civilization as a whole.

Thus if you wish to continue to engage, please provide something of real substance
that can be approached scientifically. An isolated "no" and/or "wrong" is not scientific.

[paladin17]

2e/e has nothing to do with the circle (as concerned), nor 360-degrees of rotation (periodicity), nor four symmetrical quarters etc.
(π+π√5)/2π → (3π+π√5)/2π entails a numerator addition of 2π, on a base of 2π, which has everything to do with a circle
both quantitatively and qualitatively.

2e/e = 2 is equivalent in qualitative sense to your ((1+Sqrt(5))/2)*Pi/Pi = (1+Sqrt(5))/2. They are both unrelated to any circles or squares or anything else like that. They are just bare numbers.

Do you understand that both a base of 2π and an addition of 2π (as a product of a square) implies a circle(s)?

The number alone (without, say, Pi) does not imply a circle. Your number ((1+Sqrt(5))/2)*Pi/Pi = (1+Sqrt(5))/2 - where Pi, as you can see (can you?..), is not present, isn't any different in that regard.

Are you able to give even one reason/rationale as to why you believe this?

A good reason is that it's not (and cannot be) proven to be so.

However you wish to call it, it has nothing to do with Pi.

Because you said so? Despite the evidence to the contrary?

Not despite the evidence, but due to the absence of evidence.

Here it is again:
https://postimg.cc/9R1nKMK8

Upon rotation of AC about the origin, point D both precisely and incessantly coincides with the
circumference of the 2r = 1 circle while/as inscribed in the unit square s = 1. This relation is fixed/scalar.
Further: point D precisely coincides with 4 equidistant/symmetrical points falling on both the circle and the square (4 is thus integral to π).
The 2r = 1 circle is thus a natural product of the (square of the) golden ratio and why π = 4/√Φ (4 is integral to π).

What you say here has nothing to do with the number phi.
If you inscribe a circle in a square, they'd touch in 4 points anyways.

[A Gnostic Agnostic]

2e/e = 2 is equivalent in qualitative sense to your ((1+Sqrt(5))/2)*Pi/Pi = (1+Sqrt(5))/2.

2e/e has nothing to do with a circle, thus not qualitatively equivalent to any expression on a base of 2π.
Anything on a base of 2π is not "unrelated" to a circle. It is a circle.

This is not mine, this is yours trying to call it mine. Here is mine:

π+π√5/2π = Φ
3π+π√5/2π = Φ²
_________________
Φ→Φ² (square) produces an addition of 2π

The base must be per 2π. If it is not, we are not talking about geometry.
Reducing back into arithmetic removes the geometry implied.
We don't want meaningless "numbers", we want a real geometry
which is composed of ratios, like space/time.

They are both unrelated to any circles or squares or anything else like that. They are just bare numbers.

Wow. Who would argue π is unrelated to a circle? Mind-boggling.
Pi is a ratio before it is a number. Pi expressed as a "number"
will always be an approximation of pi, not pi. Pi is a ratio.

The number alone (without, say, Pi) does not imply a circle.

Correct.

Your number ((1+Sqrt(5))/2)*Pi/Pi = (1+Sqrt(5))/2

This is not my number, this is your number.
I am not multiplying (1+√5)/2 by π/π, you are doing that.
(1+√5)/2 is just arithmetic, and implies no circle.
(π+π√5)/2π is not just arithmetic, it is geometric as it implies a circle.

- where Pi, as you can see (can you?..), is not present, isn't any different in that regard.

I can see what you are trying to do: remove the base of 2π
because if/when it is there, it implies a real circle-based geometry
which can be used to precisely measure the emergent circle
that is a natural product of a simple square operation.

It may be because you don't like the result esp. if it doesn't "fit" with your "belief"
in an "approximated" π of 3.141... and/or you can't bring yourself to face the possibility (reality)
that human beings are not nearly as intelligent as they "believe" for not knowing how to measure a circle.

Whatever it is... it is not science.

A good reason is that it's not (and cannot be) proven to be so.

This is not a reason at all, let alone a "good" one.

I agree it can not be proven to you,
for a number of reasons I need not go into.

If you deny the proof(s) that π = 4/√Φ
(of which several proofs are provided)
it can not be proven "to you" because
you deny the reality as it is anyways.

Nobody can "prove" anything to anyone who denies the reality
for substituting with their own "belief"-based one. It is possible
to know that a π of 3.141... is deficient, and the same is demonstrated.

π = 4/√Φ
π² = 16/Φ
16 = Φπ²
1 = Φ(π/4)²

is a natural consequence of π = 4/√Φ, thus the only concern here is π.

Not despite the evidence, but due to the absence of evidence.

Now you are trying to invert the reality - state the opposite of what is true.

There are at least 7 proofs on the site provided:
http://measuringpisquaringphi.com/geome ... ofs-of-pi/
as well as:
https://postimg.cc/9R1nKMK8
as well as a ~2min. video:
https://www.youtube.com/watch?v=_j2tq1GsXbI

The "absence of evidence" you speak of is another kind of absence entirely.

What you say here has nothing to do with the number phi.

Again, the opposite is true: what I say here has everything to do with *"the number" phi.
_____________________________________________
*Phi is not a number, it is first-and-foremost a ratio, like pi.
Phi may be approximated by a "number", but this number is not a ratio, thus imprecise.

If you inscribe a circle in a square, they'd touch in 4 points anyways.

Correct! Thank you for conceding the most crucial point(s).

https://postimg.cc/9R1nKMK8
(√5+1) = AC
(√5+1)/2 = D

Point D. There are four of them as AC rotates 2π (full rotation) about the origin.
These 4 points are the outcome of (√5+1) being divided by two on each perpendicular axis.
x = ±0.5 are two points
y = ±0.5 are two points

...therefor π must be in some real geometric relation to '4', and we find this in/as 4/√Φ.

Further, the 4 equidistant points on the unit square s = 1 relating a 2r = 1 circle are described by the square s = √2/2.
If when √2/2 is subject to the same four D points, the result is √Φ wherein the reciprocal of the latter 1/√Φ = π/4.
Line and curve are thus reciprocally related (as are space and time as multiplicative reciprocal aspects of motion).

https://postimg.cc/rRLRYNx2

This reciprocal relation underlies the nature of the relationship between space and time, wherein:

s/t = v (velocity)
t/s = e (energy)
______________
s/t x t/s = 1

All motion(s) have a corresponding energy constituency, thus all is accounted for by way of reciprocity
wherein the product of any velocity s/t with its own energy t/s is always normalized to unity/one.

If for whatever reason you reject π = 4/√Φ, this will always be the impasse
and further discussion is practically pointless less to prove the point of the OP:
that human beings are so unwilling to challenge basic underlying assumptions
is practically the principle barrier of ALL scientific progression.

π is believed to be 3.14159... because it is assumed there is no means to measure a circle
without "approximating" it (with straight lines, no less). This assumption is false, as are
any/all beliefs associated with π incl. the "transcendental" status of π.
It is believed π was/is "proven" transcendental because 3.14159... was proven to be.
The problem is: 3.14159... is not π, but a deficient approximation of π and transcendental thus.

What is true may not be pleasant.
What is pleasant may not be true.
It brings me no pleasure to admit
this humanity suffers knowing not
how to measure a circle, but
it just happens to be true
regardless of substance of BELIEF(s).

By the way, the equality 1 = Φ(π/4)² can be used to solve for any root(s), such as Nazism/fascism/socialism
and I have already done this. I can thus prove the equality has a practical utility and demonstrate the apparatus
used to derive such roots. It can be tried/tested by anyone and they can report back their results over time.

[paladin17]

This is not mine, this is yours trying to call it mine. Here is mine:
π+π√5/2π = Φ
3π+π√5/2π = Φ²

I hope you can see that Pi can be factored out from the numerator, which would make it obvious that this expression is equivalent to the one I've written.

(1+√5)/2 is just arithmetic, and implies no circle.
(π+π√5)/2π is not just arithmetic, it is geometric as it implies a circle.

These two expressions are equal to each other.

If you deny the proof(s) that π = 4/√Φ
(of which several proofs are provided)

No proofs have been provided. I've shown what's the deal with "proofs" that were.

There are at least 7 proofs on the site provided:
http://measuringpisquaringphi.com/geome ... ofs-of-pi/
as well as:
https://postimg.cc/9R1nKMK8
as well as a ~2min. video:
https://www.youtube.com/watch?v=_j2tq1GsXbI

None of which are correct. I've already shown that for the first one.
You may choose the one you think is the most correct, and I'll show you why it is not, if your own skill at basic geometry/algebra is lacking.

https://postimg.cc/9R1nKMK8
(√5+1) = AC
(√5+1)/2 = D

Point D. There are four of them as AC rotates 2π (full rotation) about the origin.
These 4 points are the outcome of (√5+1) being divided by two on each perpendicular axis.
x = ±0.5 are two points
y = ±0.5 are two points

...therefor π must be in some real geometric relation to '4', and we find this in/as 4/√Φ.

Fantastic logic.
Now let us inscribe a circle in a hexagon. They would touch in 6 points.
Therefore, Pi must be in some real geometric relation to 6.
What is it?

[A Gnostic Agnostic]

I hope you can see that Pi can be factored out from the numerator, which would make it obvious that this expression is equivalent to the one I've written.

I can. I hope you can see there is no need/imperative to do this
esp. if/when describing a rotating base (cycles of time, yea?) of 2π.
This is why I reminded numbers are both quantitative and qualitative.

2 ≠ 2π either quantitatively or qualitatively.

These two expressions are equal to each other.

Quantitatively, yes
but not qualitatively:
the latter has a circular base.

2 ≠ 2π

No proofs have been provided. I've shown what's the deal with "proofs" that were.

You're stating the opposite of what is true again - here they are again:
http://measuringpisquaringphi.com/geome ... ofs-of-pi/
https://postimg.cc/9R1nKMK8
https://www.youtube.com/watch?v=_j2tq1GsXbI

The proofs were/are provided.

Contrary to your claim, you've not shown "what's the deal" with any proof:
You once made a contention that was already addressed directly on the site itself (you obviously didn't read it).
You then incorrectly identified a line segment, and that was it.

None of which are correct. I've already shown that for the first one.

You haven't shown anything.
You keep stating the opposite of what is true.
If you believe you have, please quote it.

Here is what I see:

The "proofs" that you provide simply postulate this hypothetical relation between phi and pi, never actually arriving at it from any considerations. E.g. see here: it is implicitly assumed that you can exactly fit 4 circles of a circumference = 2 inside a 2*Sqrt(phi) diameter circle. This is not proven anywhere (and cannot be proven, because it is wrong), yet this is the basis of the "proof" of phi/pi connection given there.
(Also, in this picture AE = Sqrt(phi+1), not phi, but it doesn't really matter, since only the mentioned implicit introduction of the needed result is necessary for the "proof").

The first is addressed on the site (you didn't read), the second is incomprehensible to me as given Φ² = (Φ + 1), √Φ² = Φ.
What I gather you are stating is AE = √Φ² (which equals Φ) then saying not Φ, which is obviously absurd.

You may choose the one you think is the most correct, and I'll show you why it is not, if your own skill at basic geometry/algebra is lacking.

It is supposed to be the other way around: you may choose the one you think is not correct,
and I'll show you why it is, as I know how to measure a circle without "approximating" it.

Fantastic logic.
Now let us inscribe a circle in a hexagon. They would touch in 6 points.
Therefore, Pi must be in some real geometric relation to 6.

Pi is a natural product of a square, not a hexagon.
A square has four right angles. A hexagon does not.
We are measuring the 2π that emerges from the square
(of the golden ratio) without "approximating" anything.

I admit I do not know what the logic behind performing a hexagon operation (whatever that is) would be.

What is it?

The 4 (yes, the same '4' in/of 4/√Φ) right angles of the square is the same constituency of any axes (ratio),
including one which may be inscribed in a unit circle such to correlate the four critical √2/2 points:

https://postimg.cc/9R1nKMK8

this is from the proofs that were already provided. From the image above, point D coincides with:
i. the entire circumference of the circle (as point D manufactures the 2r = 1 circle upon one full rotation of AC about origin)
ii. four critical √2/2 points correlating the unit square s = 1 to/with the associated 2r = 1 circle
iii. the golden ratio (1+√5)/2 viz. AC/D = 1.618...

When you look at any x/y axis, you see four right angles formed outward from the origin.
The unit square itself relates rational/irrational via s = 1 whose diagonal is √2.
The double-unit square itself relates the square of each side/diagonal of the unit square: 1x2 whose diagonal is √5.
The square of (π+π√5)/2π viz. (3π+π√5)/2π is one greater than itself due to an addition of 2π to the numerator.
The addition of 2π is what we measure without the need for any "approximation" that misses an entire constituency
of the circle. To be precise to 130 decimal places, the constituency missed by the "approximation" method is:

0.0030128574398999
058155909600923328
338953188319757871
296846632880969117
842030346562196167
312919154742773679
624632961385764079
968722...

4/√Φ is geometric thus relates to anything/everything inscribed inside a circle... because...

What is it?

it IS a circle.

What is it not?
"transcendental" and/or "approximated".

[paladin17]

2 ≠ 2π either quantitatively or qualitatively.

Yes, but 2 = 2*Pi/Pi, which is what you have in your expression.

You're stating the opposite of what is true again - here they are again:
http://measuringpisquaringphi.com/geome ... ofs-of-pi/
https://postimg.cc/9R1nKMK8
https://www.youtube.com/watch?v=_j2tq1GsXbI

So which one is the most definitive proof? Let's analyze it.

It is supposed to be the other way around: you may choose the one you think is not correct,

None of them are correct.

Pi is a natural product of a square, not a hexagon.

Pi is a number that relates the circumference of a circle to its diameter. There are no squares here. Anywhere.
Neither there are triangles, pentagons, hexagons, octagons or anything else you can inscribe a circle into.
So your argument about squares is invalid.