Since at least the time of Archimedes over 2000 years ago, mathematicians have been approximating the curvature constant. They've done this while/as assuming (without foundation) inscribed (n) & circumscribed (N) polygons of sides n=N indefinitely approaches the circumference of a circle with 0.000... margin of error. This assumption is over 2000 years old & has neither been proven nor challenged (until now.)
In light of the area of a circle being a = πr², two points:
i. Neither polygon ever has an area a equal to the circle they are approximating, and
ii. Neither polygon (therefore) has a radius equal (to the same.)
Conclusion: Archimedes' method is
not a like-for-like comparison & is therefore
invalid.
What's happening is: the circumscribed polygon is invoking an area
greater than the circle while/as the inscribed is invoking
less. This means the equivalent radii associated with each polygon are of a slightly larger & smaller circle(s)... not the one we want. If/as the areas are re-normalized, Archimedes' method only resolves up to a max. of 0.49952094... of a radius of 0.5000... which means 3.14159... is NOT the circumference of a circle whose radius is 0.5000... This also means the lower- and upper-bounds calculated using polygons are wrong: they wildly underestimate the actual radius.
As for proving & verifying π ≠ 3.14159...:
1. Plot two concentric circles x² + y² = 1/4 (minor) and x² + y² = 5/4 (major) to produce an annulus of area equation π((√5/2)² - (1/2)²) = π. Find its uniform width w = -1/2 + √5/2 & note this to be equal to the RECIPROCAL of the so-called "golden ratio".
2. Plot a point anywhere on the minor circle such as (0, 1/2) and square 3.14159...'s quarter off it via. (0, 1/2 + (3.14159.../4)²).
3. Find the square of the approximated pi's quarter catastrophically fails to satisfy the uniform width of the annulus.
For a formal proof, see
https://papers.ssrn.com/sol3/papers.cfm ... id=4307169.
FAQ:
"Why should the square of 3.14159...'s quarter satisfy the width of this π annulus?"
If given a single r = 1/2 circle, if/as a second circle expands from inside it, each π/4 imperatively squares with itself at r = 1/2 & the result is the annulus whose width is imperatively the square of this length: (π/4)² = w = -1/2 + √5/2. By plotting the square of the approximated pi's quarter from the minor, we can try/test/falsify the numerical integrity of 3.14159... by showing there is not enough length in this number to satisfy the annulus. This is owing to Archimedes' crude approximation method & failure to challenge his assumptions re: an indefinite approach.
Any one quadrant sweep of length w = -1/2 + √5/2 contains the exact same area as the square of pi's quarter.
This implies a square-sweep equivalence wherein both produce and/or contain the exact same area.
"If π ≠ 3.14159... what does it equal?"
(π/4)² = w = -1/2 + √5/2
π/4 = √(-1/2 + √5/2)
π = 4√(-1/2 + √5/2)
= √(8√5-8)
≈ 3.144605511029693144...
This implies for p = 1/2 + √5/2 ≈ 1.618...
π = 4/√p
π² = 16/p
16 = pπ²
1 = p(π/4)²
wherein '1' is a product of the so-called golden ratio & square of pi's quarter.
One may also use an
inverse square contained in a unit square/circle ratio 4r²/4ar² for r = 1/2:
(π/4)p = 4r²/4ar² = 1/a
p = 1/(π/4)²
p = 1/(π²/16)
p = 16/π²
π² = 16/p
π = 4/√p
etc.
There are many other ways, but the inverse square is irrefutable because the universe is based in/on it.
According to the inverse square law, the circumference of a circle whose radius is 0.5000... is NOT 3.14159...
How could everyone have had pi wrong for so long & how did nobody notice it?
1. Human ignorance (don't underestimate it - it is the underlying problem.)
2. Failure to challenge basic underlying assumptions (ie. lack of use of scientific method of falsification.)
3. Not a single mathematician has ever checked their approximation vs. reality.
In reality, a real 1000mm diameter circle
certainly has a circumference of
at least 3144.6mm (to within tolerance.)
The sooner humanity fixes this error, the better. If this error is not fixed, humanity is not going to accomplish anything beyond what it has.
Since at least the time of Archimedes over 2000 years ago, mathematicians have been approximating the curvature constant. They've done this while/as assuming (without foundation) inscribed (n) & circumscribed (N) polygons of sides n=N indefinitely approaches the circumference of a circle with 0.000... margin of error. This assumption is over 2000 years old & has neither been proven nor challenged (until now.)
In light of the area of a circle being a = πr², two points:
i. Neither polygon ever has an area a equal to the circle they are approximating, and
ii. Neither polygon (therefore) has a radius equal (to the same.)
Conclusion: Archimedes' method is [i]not[/i] a like-for-like comparison & is therefore [i]invalid[/i].
What's happening is: the circumscribed polygon is invoking an area [i]greater[/i] than the circle while/as the inscribed is invoking [i]less[/i]. This means the equivalent radii associated with each polygon are of a slightly larger & smaller circle(s)... not the one we want. If/as the areas are re-normalized, Archimedes' method only resolves up to a max. of 0.49952094... of a radius of 0.5000... which means 3.14159... is NOT the circumference of a circle whose radius is 0.5000... This also means the lower- and upper-bounds calculated using polygons are wrong: they wildly underestimate the actual radius.
As for proving & verifying π ≠ 3.14159...:
1. Plot two concentric circles x² + y² = 1/4 (minor) and x² + y² = 5/4 (major) to produce an annulus of area equation π((√5/2)² - (1/2)²) = π. Find its uniform width w = -1/2 + √5/2 & note this to be equal to the RECIPROCAL of the so-called "golden ratio".
2. Plot a point anywhere on the minor circle such as (0, 1/2) and square 3.14159...'s quarter off it via. (0, 1/2 + (3.14159.../4)²).
3. Find the square of the approximated pi's quarter catastrophically fails to satisfy the uniform width of the annulus.
For a formal proof, see [url]https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4307169[/url].
FAQ:
[b]"Why should the square of 3.14159...'s quarter satisfy the width of this π annulus?"[/b]
If given a single r = 1/2 circle, if/as a second circle expands from inside it, each π/4 imperatively squares with itself at r = 1/2 & the result is the annulus whose width is imperatively the square of this length: (π/4)² = w = -1/2 + √5/2. By plotting the square of the approximated pi's quarter from the minor, we can try/test/falsify the numerical integrity of 3.14159... by showing there is not enough length in this number to satisfy the annulus. This is owing to Archimedes' crude approximation method & failure to challenge his assumptions re: an indefinite approach.
Any one quadrant sweep of length w = -1/2 + √5/2 contains the exact same area as the square of pi's quarter.
This implies a square-sweep equivalence wherein both produce and/or contain the exact same area.
[b]"If π ≠ 3.14159... what does it equal?"[/b]
(π/4)² = w = -1/2 + √5/2
π/4 = √(-1/2 + √5/2)
π = 4√(-1/2 + √5/2)
= √(8√5-8)
≈ 3.144605511029693144...
This implies for p = 1/2 + √5/2 ≈ 1.618...
π = 4/√p
π² = 16/p
16 = pπ²
1 = p(π/4)²
wherein '1' is a product of the so-called golden ratio & square of pi's quarter.
One may also use an [i][b]inverse square[/b][/i] contained in a unit square/circle ratio 4r²/4ar² for r = 1/2:
(π/4)p = 4r²/4ar² = 1/a
p = 1/(π/4)²
p = 1/(π²/16)
p = 16/π²
π² = 16/p
π = 4/√p
etc.
There are many other ways, but the inverse square is irrefutable because the universe is based in/on it.
According to the inverse square law, the circumference of a circle whose radius is 0.5000... is NOT 3.14159...
[b]How could everyone have had pi wrong for so long & how did nobody notice it?[/b]
1. Human ignorance (don't underestimate it - it is the underlying problem.)
2. Failure to challenge basic underlying assumptions (ie. lack of use of scientific method of falsification.)
3. Not a single mathematician has ever checked their approximation vs. reality.
In reality, a real 1000mm diameter circle [i]certainly[/i] has a circumference of [i]at least[/i] 3144.6mm (to within tolerance.)
The sooner humanity fixes this error, the better. If this error is not fixed, humanity is not going to accomplish anything beyond what it has.