Discovery of the true geometric π and Squaring the circle

08/₃₄	Tracing it with the radius of the circle `CS` as the base, it is possible to obtain directly the value φ or `CA` of an immediately greater radius, from which to draw the new circle. It follows that if we assign value 1 to `CS`, `CA` will be Φ [1,618]. while if we assign value `1` to `CA`, `CS` will be Φ [0,618], and this latter is the case we propose, assigning to the circle with radius `CS` the symbol Φ`⁰`. In fact, sticking to the golden criterion inherent in the rectangle thus outlined, the arc `Ae` with center `S` will establish with the line `db` a golden proportion inside `CS`, equivalent to `SA`, which will permit to draw with that radius concentric circle [`C`] classifiable Φ². With the same procedure [specular `L` in the figure] applied to this new circle, the fourth internal concentric circle will be obtained which can be classified as Φ³; it's just one of many ways, and so far, nothing surprising. We see circles whose sum of the diameters of each contiguous pair is equal to the diameter of the circle that immediately contains it, golden foundation in the reverse path traced – as immediate proof of what Kepler had recognized – in the well-known Fibonacci series, in which each digit is the sum of the two preceding it. It is actually a multiplier / divisor, which is `Φ` so the 1st and 2nd factors, instead of `1 1` for Fibonacci, are in the present case `Φ 1`, to be applied or added, and then proceed by dividing or multiplying, depending on the direction chosen and always by `Φ` the last number obtained. Being born and based on integers (like Frederick II of Swabia's rabbits, before ending up in the casserole dish), of the Fibonacci sequence we can only say that the relationship between two consecutive values of the series – once surpassed the first low numbers that alter the increment in a gross way with respect to the pinpoint accuracy of the golden sequence – progressively approaches and with alternate fate in ± to the φ ratio, but only for the collateral effect deriving from the applied method, certainly not as cause or matrix that can be identified with that ratio; at most a popular version of it for breeding. The Fibonacci series in fact does not constitute a mathematical or natural law, but draws its essence from the implicit parallelism to the golden rationale; nevertheless it is the case to complete Kepler's intuition with the clear statement that this series will never reach the φ ratio. Here is a page of 48 numerical developments compared with 36 decimal digits for sufficient verification, and since all of reality is interwoven with nuances it is difficult to believe that a certain discontinuity could replace the divine ratio in living things (even if it is playing a lot of fun for the promoters of trading):

In fact, sticking to the golden criterion inherent in the rectangle thus outlined, the arc Ae with center S will establish with the line db a golden proportion inside CS, equivalent to SA, which will permit to draw with that radius concentric circle [C] classifiable Φ².

We see circles whose sum of the diameters of each contiguous pair is equal to the diameter of the circle that immediately contains it, golden foundation in the reverse path traced – as immediate proof of what Kepler had recognized – in the well-known Fibonacci series, in which each digit is the sum of the two preceding it. It is actually a multiplier / divisor, which is Φ so the 1st and 2nd factors, instead of 1 1 for Fibonacci, are in the present case Φ 1, to be applied or added, and then proceed by dividing or multiplying, depending on the direction chosen and always by Φ the last number obtained.

Being born and based on integers (like Frederick II of Swabia's rabbits, before ending up in the casserole dish), of the Fibonacci sequence we can only say that the relationship between two consecutive values of the series – once surpassed the first low numbers that alter the increment in a gross way with respect to the pinpoint accuracy of the golden sequence – progressively approaches and with alternate fate in ± to the φ ratio, but only for the collateral effect deriving from the applied method, certainly not as cause or matrix that can be identified with that ratio; at most a popular version of it for breeding.
The Fibonacci series in fact does not constitute a mathematical or natural law, but draws its essence from the implicit parallelism to the golden rationale; nevertheless it is the case to complete Kepler's intuition with the clear statement that this series will never reach the φ ratio.
Here is a page of 48 numerical developments compared with 36 decimal digits for sufficient verification, and since all of reality is interwoven with nuances it is difficult to believe that a certain discontinuity could replace the divine ratio in living things (even if it is playing a lot of fun for the promoters of trading):