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Excuse Me While I Drill A Hole My Head

  1. Apr 19, 2003 #1
    Maybe I'm a little slow, but the true meaning of pi just occured to me.

    tan(pi/4) = 1

    arctan(x) = [sum]n=0[oo] (-1)n * [ x2n+1 / (2n + 1) ]

    pi = 4 * arctan(1) = 4 * [sum]n=0[oo] (-1)n * [ 1 / (2n + 1) ]
    = 4 * ( 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... )
    = 4 * ( 2/3 + 2/35 + 2/99 + ... )
    = 4 * [(1 * 2/3) + (4 * 1/70) + (8 * 1/396) + ...]

    Remember the equation for the area of a circle.

    A = pi * R2
    = (2*R)2 * [(1 * 2/3) + (4 * 1/70) + (8 * 1/396) + ...]
    = D2 * [(1 * 2/3) + (4 * 1/70) + (8 * 1/396) + ...]

    Now, draw a circle with diameter D. In that circle, place a square with area (2/3)*D2. Notice that there is still some empty space left on the sides square (where the square and the circle do not both exist). Place four squares with area 1/70*D2 in each of these four empty spaces. Continue to fill up the unoccupied space of the circle with progressively smaller squares ad infinitum.

    Pi is simply an infinite series sum of scaling factors needed for filling a circle with squares in order to perform a numerical integration method.

  2. jcsd
  3. Apr 20, 2003 #2

    Its a bit more complicated, or simple really., glad to see someone else has found the Antiqity Keys to the Universe!

    There is an obvious scaling factor, and when you invert the 'outside'square into the 'inside' circle, then the dimensional scale is maintained!


    The movement, or transition from a Square to a Circle in '2-dimensions
    ', involves the inversion of area external to internal.

    Below is representative of such an invertion:


    And the hidden relationships of interconnective Pi and geometric shapes can be clearly seen, as the shape expands, it expands to precise geometric formulas.

    Click on the image for a larger pic.

  4. Apr 21, 2003 #3
    Well, yeah that's a neat way to get pi, but excuse me if I am wrong isnt the true meaning of pi the ratio of a circles circumference to it's diameter? I think the most simple and elegant way to find pi is to just inscribe a polygon inside a unit circle, we know how to find the length of the sum of the sides of the polygon so pi is the limit as the number of sides goes to infinity. This is what makes the most sense to me. But there are many various other ways to get it. The two others that I like best are Wallis forumula and a monte carlo method where you have a unit circle inscribed inside a unit box and then you start randomly dropping points inside the box and can get pi as a ratio of the number of points inside the circle versus the total number of points (this ones fun to see a computer program do, the applet I saw do it also had a timeline above it which showed what the current value of pi was and how it changed in time as you added dots, it was neat to see it converge very quickly and accurately to pi!)
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