## On Use of the Inverse Square Law to Find π

Discussion concerning the first major re-evaluation of Dewey B. Larson's Reciprocal System of theory, updated to include counterspace (Etheric spaces), projective geometry, and the non-local aspects of time/space.
ckiit
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Joined: Sat Dec 21, 2019 7:54 am

### On Use of the Inverse Square Law to Find π

The inverse square law:
invsqrlw.png (133.56 KiB) Viewed 35058 times
...can be used to properly reconcile the area and/or circumference of the circle contained in (x² - x) = 1 = 2r.

The radial application of the positive solution to (x² - x) = 1 as x = (√5 + 1) / 2
geometrically contains the area of the r = 1/2 circle:
radialphi.gif (358.71 KiB) Viewed 35058 times
By simply setting the positive solution to (x² - x) = 1 as equal to 1/(πr²)²
one can directly solve for π without "approximating" it with n > 4 polygons:
pisqsq.jpg (51.55 KiB) Viewed 35058 times
Use of the inverse square law completely circumvents Archimedes' n > 4-sided polygon "approximation" deficiency.
A 360° circle can only be reconciled by a/the figure whose internal angles sum the very same 360° (that is: a perfect square).

To my knowledge, this is the first mathematical proof using such a law in agreement with observation that π ≠ 3.14159...

Let the area of a circle a be described as:
a = πr²
& let r = 1/2
a = π(1/2)²
a = π(1/4)
a = π/4
∴ π = 4a
To calculate the circumference of the r = 1/2 circle,
use 1/(πr²)² = x:
1/(πr²)² = x
1/(π(1/2)²)² = Φ
1/(π(1/4))² = Φ
1/(π/4)² = Φ
16/π² = Φ
π² = 16/Φ
π = 4/√Φ
∴ π = 2√(2(√5-1)))
≈ 3.1446055...
To calculate the area of the r = 1/2 circle,
use 1/(4ar²)² = x:
1/(4ar²)² = x
1/(4a(1/2)²)² = Φ
1/(4a(1/4))² = Φ
1/(4a/4)² = Φ
1/a² = Φ
a² = 1/Φ
a = 1/√Φ
∴ a = √(2(√5-1)))/2
≈ 0.78615137...
The approximation of π using n > 4-sided polygons whose internal angles sum greater than 360° is the "Blunder of Millennia" as:
only the polygon whose internal angles sum exactly 360° can be used to properly reconcile the circle.