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 = (**3****π**+π√5)/2π = **2**.618…

we find a numerator difference (being a matter) of a *discrete* **2π**:

Φ² – Φ = **2π**/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 resolutelyreciprocally related:1/√Φ= π/4...from Φ's ownrootis 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 1000*mm* diameter circle will *certainly* have a *real* circumference *greater *than ~3141.6*mm* (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 = 3141.6*mm*, 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 n*on-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 bas*es* 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 π^{2}= 16/Φ 16 = Φπ^{2}(e = MC^{2}) 1 = Φπ^{2}/161 = 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.”

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