Pi is vaguely described as "(perfect) circle's circumference to its diameter".
sqrt(16/(lim_(n->infinity) F_(n+1)/F_n)) = sqrt(16/((sqrt(5)+1)/2)) = 4/sqrt(phi) = approx
3.1446055110296931442782343433718357180924882313508929506596 ..
It is based on the fundamental Fibonacci sequence (the F there is Fibonacci number).
Why do we nowdays still use the Pi's arithmetic value 3.141... instead of the "more natural" Pi value 3.144... even the ancient egyptians used and
which does fulfill the description of (perfect) circles circumference to its diameter?
Proof:
"From here things begin to get really interesting . As can be seen, BC above is equal to one half the length of the pyramid's side. Therefore, the
perimeter of the base equals BC x 8, and in relative terms this equals 0.618034 x 8 = 4.9443. The relative height of the pyramid is 0.78615, and, if
one uses this length as the radius of a circle, then the circumference (perimeter) of that circle will also be 4.9443. Also, perhaps more important
factor, is that the length of side OD (0.78615), when multiplied by 4 yields an amount (3.1446) that is almost exactly equal to Pi (3.1416). This
finding means that the 38010' right triangle offers a unique and most interesting point of intersection between the Pi ratio and the golden ratio
phenomenon. How this unexpected agreement comes to be is that :
As we saw in the 38010' right triangle, 0.618034 ÷ 0.78615 = 0.78615. This means, that 0.618034 = 0.78615 x 0.78615. Therefore, 8 x 0.618034 is the
same as 8 x 0.78615 x 0.78615;
As we also saw, 4 x 0.78615 is a very close approximation for Pi . Therefore,2*Pi can be said to equal 8 x .78615. For the circumference of the circle
using 0.78615 as its radius, we then have C =2*Pi*R = (8 x 0.78615) x 0.78615 .As a result, the Great Pyramid turns out to have the same perimeter
length when measured in a horizontal plane, as a square, and in a vertical plane, as a circle. "
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There are several rumours that even nasa didn't use the ordinary Pi value in some of their lunar missions. Why?
You can solve the maths yourself: 4- 4sin²x = cos x , 0