some mathematics has always bothered me with pi.
Circumference = C
Diameter = D
Radius = r = (1/2 D) or 2R = D
Pi is a ratio of
C/D
area of a circle:
Area = Pi x r^2 =>> A= (C/D) * r^2 =>> A = (C/(2R)) * (R^2)
Circumference of a circle
C = Pi * D =>> C = C/d * D =>> C = C
how could we solve circumference with out first knowing the circumference ?
a bit of a conundrum.
knowing the circumference we can have an answer for what is PI.
we need a way to solve for circumference that doesn't involve knowing its circumference.
imagine we have a circle with a diameter of 1, which occupies both x and y direction of the circle
x and y directions are both 180 degree lines that intersect at 90 degrees, fold the circle exactly in half on the Y axis of the grid which 1/2s the x axis.
we have x= 1/2 the diameter.
label the edge of the circle as point A on the x axis
label the edge of the circle as point b of the y axis
connect point a and b together with another line label this C
using Pythagorean theory
C^2 = A^2 + B^2
c = sqrt (1/2^2) + 1/2^2)
C = sqrt(1/2)
from here we can also solve the area of the triangle but I'm going to skip that for now.
we can see there is area still uncounted above the triangle, so what is the cord length of the triangle ABC ?
to find that divide the triangle in half
C^2 = A^2 +B^2
(1/2)^2 = (1/2 (sqrt(1/2)) ^2 + B^2
B = sqrt (2) /4
we know the cord length {label it E) = B and the total length of the radius= 1/2,
this tells me
the real question is how many folds(n) does it take for the C length to = 0 distance between points a & B , and E = R
and can they?
the answer is no and its pretty simple to see
we started with 180 degree angle for each fold we are left with 180/(2^n) degrees. this number is infinity increase in smaller scale.
which means E infinity grows by infinity shrinking numbers but never reaches the length of R,
and the space between A & B lines also shrinks infinity but never reaches zero as a & b always have a divergent angle between them
which means Pi is an infinite number as its a area summation of infinity shirking triangles.
we can gain degrees of accuracy the more folds we do and have a
E/R as a % indicator for accuracy.
the best we can do is approximate ratio to the nth decimal place as Pi is an infinite irrational number.
find the area of the circle using some other fun math that allows us to have a high accuracy reading of the pi ratio:
for every fold{n} we do on the circle we make
( 2 * (2^n)) segments {labeled s) with C as its length and has a cord length of E to the center.
Area of a polygon is defined as
A = 1/2 PnR
where:
n = segment count
P = length of the segment
R = cord length of N to the center of the polygon.
translate that to the stuff we solved above
Area of a circle:
A = 1/2 * S * C * E
once we have the area we can solve
pi = area / R^2
paper copy of proving the above method and math
i wrote a short easy program in pascal that does this however, the accuracy is limited by the floating point of pascal real numbers:
{i also included a way to do it with laws of cosine however that is already defined by pi and i solving area and pi..so i cant prove pi while using it lol. }
3.141592653589793238
is the most accurate my program can go do to rounding errors and it terminates on the 34th fold {3.4359738368*10^10 sided polygon} as the length of E reaches the length of R
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