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 above)
and find √5 as any diagonal between two opposite corners (AB as shown)
(this diagonal thus being equal to the diameter of the circle).
Add 1 unit distance to this diagonal (√5 + 1) and find the mid-point (/2).
Note: this midpoint coincides with the circle whose diameter is 2r = 1
by way of rotating the √5 diagonal about the origin of the circle we began with,
and thus "kisses" the square four times equidistantly, thus a more precise π
can be begotten directly from Φ expressed as a ratio of Φ.
This ratio is 4/√Φ = π and/or 16/Φ = π².
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)=x4 + 16x2 - 256
Φπ² = 16
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).