This thread culminated into the derivation of a more precise calculation of π as shown here:

and elaborated on page 3.Beginning with a circle whose diameter is √5,

place two unit squares inside of the circle side-by-side

(either horizontally or vertically, the latter is shown)

and find √5 as any diagonal between two opposite corners (AB shown).

Note: this diagonal isequalto the diameter of the circle as √5.

Add 1 unit distance (BC shown) to this diagonal and find the midpoint (D shown)

such to satisfy (√5+1)/2 as (AB+BC)/D wherein D effectively halves AC.

By rotating the √5 diagonal about the origin of the circle we began with,

the midpoint (D) incessantly coincides with the circle whose diameter is 2r = 1

and"kisses"the unit square four timesequidistantly, thus a more precise π

can be measured directly by way of Φexpressed as a ratio of Φ.

This ratio is 4/√Φ and/or (8√5-8)^{1/2}

Concerning:

https://en.wikipedia.org/wiki/Kepler_triangle

I am unsure as to the motive for the above, but the construction of the geometry

is conceptually unsound: concerning the same Kepler triangle, a circle that circumscribes it

completely ignores the imperative need for the diameter of the circle to be √5, the same to which

the Kepler triangle must apply. If the circle is not √5, there is no valid coupling and π remains 'transcendental'

however if coupled to the √5 diameter circle (as shown), the coupled relation permits a geometry

whose co-operation between Φ and π are respectively reflected in the following:

Φ solves for

f(x) = x² - x - 1

π as 4/√Φ solves for

f(x)=x

^{4}+ 16x

^{2}- 256

thus

**16=Φπ²**

and General Relativity states:

**e=MC²**

thus must concern the herein derived:

**1=Φπ²/16**

The former f(x) graphed with an overlay of the same geometry:

__________________________________________________________________________

This is important because it confirms the bi-rotation model (mandated by the geometry as intrinsic)

and implies a bi-

*orientation compliment*, the properties of which can be derived inductively -

the implications of which amounts to the capacity to calculate universal roots using universal geometries

that satisfy cubed proportionality (as time and space reciprocally do) viz.

**(s³/t ∞ t³/s)**.