Squaring the Circle--and Earth-Moon Ratio

Squaring the Circle — Earth and Moon Ratio

The Ancient Challenge

"Squaring the circle" is the classic problem of constructing a square with the same area as a given circle
using only compass and straightedge.

It is mathematically impossible to do exactly because π is irrational — but very close approximations
are possible.

Earth and Moon Approximation

Earth's diameter: 7920 mi
Moon's diameter: 2160 mi

7920 / 2160 = 3.6666

Pi Ratio diameter = 3.6694

This squaring the circle technique = 3.6693, has an accuracy that exceeds 99.9%

Answers are the same with negligible error‼

Thank you Michael Joyce for putting this into proper mathematical format for me.

"Here follows a mathematical proof of Gertjan’s discovery.

If the radius of each of the small 8 circles is 1, then the square will have sides of length 18.
D4 represents the diameter of a circle, R1, R2, R3 the radii of other circles.

1- To calculate angle “a”
The diagram in the lower right quadrant shows an enlargement of the second small circle
cosine a           = 0.5 divided by 1 
                        = 1/2
Therefore a     = 60 degrees.

2- To calculate the length R1
cosine a           = 2 divided by (R1 - 1)
therefore R1    = 5

3- To calculate the length R2 using Pythagoras’ theorem
R2²        = GH² + 9²
GH         = 18 – DG
GH         = 9 - FG

sin 60°  = EF divided by 4
EF         = 4 x sin 60°           = 3.4641016151
FG         = EG(R1) - EF          = 1.5358983848
GH        = 9 - 1.535898384    = 7.4641016151

R2²        =  GH² + 9²
             = (7.4641016151)² + 81
             = 136.7128129211
R2         = square root (136.7128129211)
             = 11.6924254507

4- To calculate the length R3
R3         = R2 - FG
             = 11.6924254507 - 1.5358983848
             = 10.1565270658

Squaring the Circle:
Area of square = 18 x 18                                                       = 324
Area of circle   = pi x R3² = 3.141… x 10.1565270658 ²           = 324.071122253... (Q.E.D)

5- To calculate the length D4
D4         = DF - 4 x sin 60°
             = 9 - 3.4641016151
             = 5.5358983848

Earth : Moon diameters
R3 (10.1565270658 x 2) : D4 (5.5358983848) or (20.3130541317 : 5.5358983848)
Or 3.66933 : 1

Open Wonder

Nature seems to "square the circle" with Earth and Moon — a cosmic coincidence or deeper harmony?

These ratios invite contemplation of geometry in the heavens.

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