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

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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