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