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Thread: squaring the circle with compass/straightedge to better than 5/100,000th's numerical accuracy

  1. #1 squaring the circle with compass/straightedge to better than 5/100,000th's numerical accuracy 
    (6/5*Phi^2)-Pi=.000048132 devin-m's Avatar
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    squaring the circle with only compass and straight edge... the impossible geometry problem.

    i thought i'd share my extremely close approximation of squaring the circle with compass and straight edge (numerically off by 0.000048132910080579383).

    ...& suppose you have a brand new apple macbook with retina display 2560x1600 pixels and you fill your screen top to bottom with a perfectly pi-area circle and square... the error in pi-area-square-height produced by this method would be much less than a single pixel on the screen... roughly 1/50th of a retina-pixel error by my calculations.

    method:

    step 0: construct a circle with radius equal to 1-- area equals pi exactly

    step 1: construct a regular pentagon side length equals 1
    instructions can be found here: https://en.wikipedia.org/wiki/Pentagon
    *please note in my drawing below i erroneously used an approximated rather than exact pentagon construction method, but i've included instructions for the exact method at the bottom

    step 2: create a line between 2 non-adjacent corners of the pentagon -- this line is exactly length Phi=1.6180339887...
    PHI: The Divine Ratio

    step 3: from the Phi line construct a square area=Phi^2=2.6180339887... -- side length Phi=1.6180339887...
    https://mathbitsnotebook.com/Geometr...ionSquare.html

    step 4: divide the square into 5 equal sections
    https://www.mathopenref.com/printdividesegment.html

    step 5: extend the square and create a rectangle beside it equal to one of the 5 sections of the square

    3.141640786499873817846 = rectangle area = 6/5*Phi^2
    3.141592653589793238463 = pi

    approximation error = 0.000048132910080579383 = 3.141640786499873817846 - 3.141592653589793238463

    step 6: square the rectangle
    Squaring A Rectangle

    square area = 3.141640786499873817846
    circle area = 3.141592653589793238463
    approximation error = 0.000048132910080579383

    phi = (1+sqrt(5))/2 = 1.6180339887
    source: https://en.wikipedia.org/wiki/Phi

    (6/5)*((1+sqrt(5))/2)^2 = 3.141640786499873817846

    ---------------------------------
    Please note: the pentagon construction method i used in the first step of the drawing is approximated but at the very bottom I have included instructions for constructing the pentagon with compass and straightedge exactly.






    please note the exact method of pentagon construction is as follows:

    Source: https://en.wikipedia.org/wiki/Pentagon
    "Pentagon at a given side length
    Draw a segment AB whose length is the given side of the pentagon.
    Extend the segment BA from point A about three quarters of the segment BA.
    Draw an arc of a circle, centre point B, with the radius AB.
    Draw an arc of a circle, centre point A, with the radius AB; there arises the intersection F.
    Construct a perpendicular to the segment AB through the point F; there arises the intersection G.
    Draw a line parallel to the segment FG from the point A to the circular arc about point A; there arises the intersection H.
    Draw an arc of a circle, centre point G with the radius GH to the extension of the segment AB; there arises the intersection J.
    Draw an arc of a circle, centre point B with the radius BJ to the perpendicular at point G; there arises the intersection D on the perpendicular, and the intersection E with the circular arc that was created about the point A.
    Draw an arc of a circle, centre point D, with the radius BA until this circular arc cuts the other circular arc about point B; there arises the intersection C.
    Connect the points BCDEA. This results in the Pentagon"

    Source: https://en.wikipedia.org/wiki/Pentagon


    Image Source: https://en.wikipedia.org/wiki/Pentagon


    Last edited by devin-m; December 1st, 2017 at 12:47 PM.
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    (6/5*Phi^2)-Pi=.000048132 devin-m's Avatar
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    https://science.howstuffworks.com/ma...cci-nature.htm

    https://en.wikipedia.org/wiki/Golden_ratio



    Image Source: http://webecoist.momtastic.com/asset...piral-main.jpg


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