History, Mathematics, Metaphysics, Philosophy
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Rede-phi-ning π: On Measuring a Circle

π and human suffering

The following investigation is a product of the ongoing scientific inquiry ‘whence human suffering?‘, the same encountering a critical need to call into serious question the long-standing pi (π) “approximation” methodology (ie. of exhaustion) first employed by Archimedes (late, c. 287 – c. 212 BCE), and then by mathematicians and scientists ever since.

To begin, the author draws attention to an important inquiry: ‘ does π ever naturally emerge as a product of a square? ‘ If so, it must be measureably so such to negate any/all need/inclining for “approximation” methodology(s) employing the use of multiple straight-edged polygons. Now consider the quadratic:

x² – x – 1 = 0

and find it to have positive solution x = (1+√5)/2 which, as the reader may recognize, is the so-called golden ratio (hence: Φ). By expressing Φ in/on a base of 2π (thus generally applicable to rotational motion):

Φ = (π+π√5)/2π = 1.618…

and then squaring:

Φ² = (3π+π√5)/2π = 2.618…

we find a numerator difference (being a matter) of a discrete :

Φ² – Φ = /2π

and so we have an answer to the former inquiry: 2π discretely emerges as a natural product of a square (if/when based on the same).

Concerning Φ: there are non-trivial (universally unique) properties it possesses as intrinsic – it is the only positive number (irrational, no less) whose reciprocal is precisely one less than itself:

                                                Φ = (1+√5)/2 = 1.618...
                                                1/Φ = (Φ – 1) = 0.618...

and (as we previously encountered) Φ is the only positive number whose own square is precisely one greater than itself:

                                                Φ = (1+√5)/2 = 1.618...
                                                Φ² = (Φ + 1) = 2.618...

If π is a natural product of a square, we must be able to utilize the geometry implied by Φ such to precisely measure this emergent π and, importantly: do so without the need/inclining for”approximation”

Prior to this endeavor, the author implores the reader to suspend (if even temporarily) any/all hitherto taken-to-be-true notions concerning π: both quantitative and qualitative.

The square is composed of four equal sides whose interior angles are four right angles. The circle is composed of four symmetrical quarters whose axial radii also compose four right angles. By way of inscribing a circle of diameter d = 1 (equiv.: r = 1/2) inside the unit square s = 1, we find four axially situated points (D₁₋₄ shown above) dividing the circumference of the circle into four equal quarters (each c/4 wherein c = π). These four critical points both simultaneously and geometrically correlate the r = 1/2 circle with the unit square s = 1. Further, these same points compose the square whose side lengths are equal to the reciprocal of √2 viz. s = 1/√2, noting:

1/√2 = √2/2

By extending any two opposite sides of the unit square s = 1, we obtain the remaining constituents of Φ: √5 (as the diagonal of the emergent 2×1 rectangle) and (a division by) 2. This extension of the unit square can be performed on both sides wherein the 8 vertices of both 2×1 rectangles can be used to compose another larger circle whose diameter is equal to their own √5 diagonal:

By extending the √5 diameter circle in all directions by one (1) discrete unit, we find the real geometric basis underlying the circumference of the r = 1/2 circle (such to measure):

Upon one full rotation, D (= Φ) incessantly coincides with the full circumference of the r = 1/2 circle while”kissing” each of the four sides of the unit square equidistantly. The real geometric square underlying this relation can be obtained arithmetically via:

wherein the irrational √Φ has an underlying magnitude(s) of ±1.27201964… and whose own reciprocal (normalizing to 1) is:

If/when plotting the first three powers of Φ (as they relate to the geometry we are presently working with):        

the square of the golden ratio can be seen to geometrically coincide with a real diameter (2r) of a real circle in real relation to a real square(s) of equal area – the emphasis on real being as (in) contrast to “transcendental”. A real circumference of a real circle (ie. π) can not possibly be “transcendental” if possessing a real geometric radius. The area of the inscribed square (whose vertices are D₁₋₄ as shown) is equal to the radius of the circle viz. r = 1/2 = s². 

We began by correlating the four right angles of the square to the four axial radii of the circle, the latter dividing π into four symmetrical quarters (each π/4). We observed the four associated axial points to simultaneously correlate the square s = 1 with the circle r = 1/2 and found them vertices of the square s² = 1/2. We also found how the real circumference of the r = 1/2 circle naturally emerges by way of a rotational motion utilizing the real geometry implied by Φ. 

We may now obtain the exact circumference of the r = 1/2 circle by observing the nature of the relationship between √Φ and π/4:

                                              π ≠ 3.14159265358979...
                                            (human approximation error)
                                             Line and curve are resolutely
                                                  reciprocally related:
                                                         1/√Φ= π/4
                                          ...from Φ's own root is derived π..."

The author wishes to impart that Archimedes’ “approximation” methodology catastrophically misses an entire constituency of the circle (albeit small, non-trivially so). A real, symmetrical 1000mm diameter circle will certainly have a real circumference greater than ~3141.6mm (ie. the latter is too short). Should this ever become a source of dispute, the author suggests a simple experiment such to resolve: actually measure a real 1000mm diameter circle, and should it discretely measure (any) more than c = 3141.6mm, the same would resolutely demonstrate the deficiency of a “transcendental” π of 3.14159… as 4/√Φ is a real root of an integral function:

f(x) = x⁴ + 16x² – 256

It is the opinion of the author that the very notion π is somehow “transcendental” (let alone “proven” to be so) is absurd. A real circle is composed of a real radius relating four discretely real  loci. While the “approximated” number of 3.14159… is indeed “transcendental,” it is so for a simple reason: it is not really π, but an “approximation” of π deficient from the thousandth decimal place. Because Φ is geometric, π follows, as from the root of the former do we derive the latter naturally by way of reciprocity viz. 1/√Φ = π/4.

As for the golden ratio: the author suggests stripping it of any/all exotic and/or esoteric notions, and rather focus on the real underlying mechanics (ie. the practicality of the relation): Φ naturally couples terminating rationals with non-terminating irrationals and perpetually suspends their sum on a rational base of 2, giving rise to the universally unique (properties of the) Φ ratio.

The geometric union of Φ and π is reflected in/as the above integral function: the real/imaginary roots reflect a discrete rational integer difference of ’16’. The real element is imperatively fixed to the ratio of 1/2 as this constitutes the real, scalar constituency of a real circle, the same 1/2 to be found in/of:

                                                     1/2 + √5/2 =
                                        ("real" terminating rational)
                                                           +
                              ("imaginary" non-terminating irrational

In other words: all real circumferences of all real circles resolutely possess a real, scalable base of 1/2 (such to scale from) and only the golden ratio permits/employs such a universal scalability

Thus as it concerns the outstanding Riemann Hypothesis problem; in particular, the underlying question: 

“for which s does ζ(s) = 0?”

the problem (ie. question) is outstanding due to the catastrophically culprit “approximation” (ie. deficieny) of π. In short: Euler’s famous solution to the Basel problem such to derive a ζ(2) involves a sin(x)/x relation, thus implies (radians in terms of) a π of 3.14159… 

While the solution fits a mathematically constructed “reality” upon a “transcendental” π of 3.14159… the real unrecognized problem is the real, physical universe does not employ such an “approximated” (let alone “transcendental”) π. For this reason, the hypothesis itself is not (only) a problem, but in reality a symptom of a much deeper underlying problem (hithertomeasurable over a span of at least ~2200 years): a deficient π as due to a deficient “approximation” methodology.

The underlying magnitude of such a blunder of millenia compels the author to sympathetically hypothesize: the Riemann Hypothesis problem will not be solved until humanity consciously acknowledges the underlying “approximation” deficiency in/of a π of 3.14159… 

Finally, as for the concerned inquiry ‘whence human suffering?‘, though the real underlying root lies beyond the limited scope of this investigation, for the purposes of the latter alone (suffice it to say): as a natural consequence of a more general failure(s) to incessantly challenge basic underlying assumptions (incl. and esp. the bases of any/all oustanding “beliefs”), human beings suffer knowing not how to (properly) measure a circle, as:

π ≠ 3.14159…

                                                        π/4 = 1/√Φ
                                                         π = 4/√Φ
                                                         π2 = 16/Φ
                                                         16 = Φπ2
                                                        (e = MC2)
                                                        1 = Φπ2/16
                                                        1 = Φ(π/4)2

π is ∴ not “transcendental” (!)

such an endeavor rationally provides a real means to discern what is real from what is not (the same needed to discern a real π from an imaginary “transcendental” one). Whereas the latter is a measure of millenia of human ignorance, the former rationally clarifies the universal constancy of both: Φ and π (not as two, but as one).


J.F. Meyer is a pseudonym for the author of this paper who wishes to remain anonymous. The inquiry ‘whence human suffering?’ began as a personal inquiry into the suffering of someone the author refers to as “Isha”. She and her family were displaced from Iraq due to ongoing wars in that region. By circumstance, the author happened to meet Isha, and she eventually imparted to the author her own childhood experiences as involving: being sexually exploited (ie. raped) as a very young child; being forced into an “arranged” marriage wherein she was physically abused to the point she had to flee; being disowned for her choice to marry someone her family did not approve of due to “religious” reasons; being used as added “incentive” for a family business transaction (as a wife/concubine) etc. There is much more to this underlying reality precipitating the inquiry, but suffice it to say: as the author looked deeper for the real root(s) of the suffering of Isha, they found not only what underlies her own suffering, but the (same) suffering of many others. It was for this reason the inquiry naturally evolved from a personalized ‘whence her suffering?’ to a more generalized ‘whence human suffering?’ and is examined according to a scientific methodology(s), the results of which are contained in/as a practical theorem to be made publicly available on the ThinkSpot platform in the near future. The author concludes by stating that now the reader can “perhaps…better understand my wish to abstain from biographical notes: there is nothing of myself that merits any attention before those like Isha, as the latter is the reason for the endeavor entirely.”

4 Comments

  1. Interesting Mr. Meyer.

    I was just reading on this question of the quadrature of the circle by Nicholas of Cusa, who also challenges Archimedes’ approach, although from a fundamentally different standpoint.

    Cusa’s essential argument is that a circle has no sides, it is of a fundamentally different species than a polygon i.e. a many sided shape. Even if one were to add sides ad infinitum to the polygon, Cusa argues that while it may appear to approach identity with the circle, in reality, it is not, quite the opposite: by increasing the sides infinitely, the polygon becomes infinitely further from being resolved into an identity with the circle. Even a difference of virtual zero still implies a discontinuity between a circle and a polygon, a boundary condition. Moreover, one cannot create a circle from a polygon, but one can create a polygon from a circle, simply be folding the sides over i.e. circular action.

    So there is a fundamental question of measurement: that which has no sides cannot be measured by that which has sides; linearity cannot measure circularity.

    Circular action is not linear action. In terms of causality, this has many implications, especially when considered in terms of the very small and very large.

    Cusa makes the point that the circle is of a higher power than the polygon. Circular action can generate linear action, but linear action cannot generate circular action, however close it may be approximated. From the standpoint of causality, circular action is of a higher power or “cardinality” than linear action.

    Do you accept the idea of a higher cardinality?

    The tendency today in science seems to pave over these questions with statistical description i.e. linear thinking, but who is to say the interaction at the level of the very small is linear? In fact, as Riemann stated in his Habilitation Dissertation, linear extension, the assumptions of Euclidean geometry, work fairly well when it comes to explaining/describing phenomena from the standpoint of the sense-perceptual world, but once things are extended into the domains of the very small and very large, beyond certain boundary conditions, these same assumptions of linear space and time don’t hold anymore. Hence, Riemann concludes his dissertation:

    “This path leads out into the domain of another science, into the realm of physics, into which the nature of this present occasion forbids us to penetrate.”

    Riemann makes the point that these questions of the very large and very small cannot be understood through mathematics alone, but require us to enter into the realm of physical science, to experiment and discover new boundary conditions i.e. discontinuities. It is knowledge and discovery of these discontinuities which then force the creation of the appropriate language, mathematical or otherwise, to describe physical reality i.e. physical reality must dictate the mathematics, not the other way around.

    So what would you say to someone who argues that circular action (which generates a circle) is a fundamentally different kind of action than linear action (which cannot generate a circle)? A discontinuity exists, however small. Such boundary conditions and discontinuities in the world of the very small, atomic physics, cannot simply be paved over by statistical explanations–statistics is not causality.

    Cusa’s argument about the quadrature of the circle seems relevant in this respect because there are processes in the universe which are fundamentally non-linear, and cannot be explained in linear terms. This is the essential crux of Cusa’s argument concerning the quadrature of the circle. We must rise above linear thinking, the universe is fundamentally non-linear.

    David Gosselin

  2. Mr_Mayhem says:

    This might be used as a new paradigm for interpreting how the continuum of space and time, or perhaps cause and effect, unfolds.

    The fun seems to happen when feedback occurs, and the new actual unfold in a continuous manner from an increased or decreased range of possibilities, setting the stage for further recursions and complexity.

    A related center of inquiry is the boundary between randomness and self-organization.

    The sought after idea relates to an organizing principle which organizes other organizing principles in some kind of super-universal principle.

  3. J. M. Meyer says:

    Hi David,

    Thanks for your reading & interest.

    Cusa is correct that a circle has no sides, however: it is important not to therefor discount a circle from having right angles. As a square has four right angles on the outside-facing-in, a circle has the same four on the inside-facing-out. This relation of 4 right angles is what it being reflected in/as the ‘4’ in/of 4/√Φ.

    Cusa is also correct re: multi-sided polygons not actually approaching the circumference, but just the opposite – they regress from the real circumference & instead approach the deficiency introduced by the very methodology itself.

    Indeed there is a fundamental question of measurement: one must use the line to measure another line and the curve to measure another curve. What line is to rational, curve is to irrational. The equality 1/√Φ = π/4 states the reciprocal of the square of the golden ratio is equivalent to the curve between two perpendicular radii of r = 1/2. As such, the golden ratio provides the geometry (π+π√5)/2π whereas the “square” viz. Φ² = (Φ + 2π/2π) provides the 360-degree rotation / motion. This rotation/motion is naturally normalized to the “speed” of light c, recalling 16 = Φπ² being the reason e = MC². Western scientists can not “see” this because they are ignorant of the naturally-occurring reciprocal relationship between space and time as multiplicative reciprocal aspects of motion.

    Regarding “higher cardinality”, I accept the idea however attribute it as owing to “lower” dimensions (not “higher” ones). The 3 dimensions normally associated with “space” actually precedes any/all considerations of space & time entirely. That is: time, like space, is also subject to/of (the same) 3 dimensions. If all three degrees of freedom are occupied by a discrete magnitude(s), the matter is discrete. However, if only 1-of-3 degrees of freedom are occupied, the otherwise free two effectively creates “potential” which is energy in the form of a binary (+/-). As such, the “higher dimension” behaviour scientists keep talking about is actually not owing to “higher” dimensions, but lower ones. Because only 4 right angles are needed to relate a circle to a square, using only those ‘4’ will ensure a/the highest “resolution” possible thus circumventing any/all comparatively deficient methodologies. For some reason, Western scientists/mathematicians don’t question why this integral quadrature property of the circle is not reflected in/as π, for surely it is via. 4/√Φ.

    Riemann was also correct: the universe actually does have two primary motions: linear and rotational. Western science(s) is very much linear-thinking, as the “measure” of their failure to relate line-and-curve such to normalize both to the “speed” of light is obviously reflected in/as their deficient π. Riemann thus correctly stated this problem collapses on the doorstep of physics & the key to that door is π. His real element of (r = 1 / 2) is merely a real radius of a real circle, the same any/all circles/cycles/waves/particles scale to/from. In other words: (r = 1 / 2) is the “ground” between number & physics. The problem with Western science is their π of 3.14159… and real physics’ π of 3.1446055… do not match. This measures the severance Western science has with reality.

    I would accept that circular motion is a fundamentally different “kind” of motion than linear motion, however would also qualify it with the stipulation that linear motion has an “equivalent” rotational motions and/or vice verse. That is: given a magnitude m, this same magnitude can be applied to a linear motion and a circular motion. The difference is not “quantitative” but qualitative: the line terminates “somewhere else” however the circle returns to whence it started. As such, that which goes around in circles is bound to return to the initial cause(s) of the circular motion. This has implications for both physics and metaphysics because any/all physical matter is measurable as some discrete displacement(s) from unity viz. 1 = Φ(π/4)² with the “speed” of light therein.

    It is very true: Western science is stuck in linear thinking & π being deficient is but a natural reflection of their unconsciousness of the naturally-existing relation between the line & curve, the rational & irrational, the terminating and non-, the real & imaginary etc.

    Fundamentally the underlying problem collapses on the doorstep of human consciousness itself, hence my initial inquiry “whence human suffering?” having lead to the resolution of e = MC² as 16 = Φπ² implying unity as 1 = Φ(π/4)². This can be used to solve for all that unity is “not” and unity is not a consequence of being unconscious, but conscious. If/when Western science recognizes the naturally occurring relation between space and time themselves as multiplicative reciprocal aspects of motion, human beings will naturally become more & more conscious of the underlying reciprocity mechanic afforded by the golden ratio/rule.

    Hi Mr_Mayhem,

    The “new paradigm” you mentioned occurs only if/when Western science recognizes reciprocity as the key underlying mechanic of the physical universe (such to discover it also applies metaphysically). This mechanic is a natural consequence of ordinary mathematics/logic as per the correct measure of π – line and curve (as with space and time) are naturally reciprocally related.

    This will allow any/all (s / t) motions’ corresponding energy constituency (t / s) to be extracted/interpreted such to better-understand root cause(s) & branch effect(s). All that is displaced from unity has a definite corresponding “cause” giving rise to circular/cyclical motion(s) with an associated gravity.

  4. Milan Kroupa says:

    There are a number of errors so the result is wrong.

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