What are the odds?
If we were to begin with a square and then cross the diagonals to find the centre, like this…
…we could use the centre to draw a circle inside the square with a diameter equal to the sides of the square, like this:
If we were then to measure the sides of the square and add them up, we could divide that value–the perimeter of the square–by half, and then by π, in order to arrive at a radius that would describe an outer circle whose perimeter was equal to that of the square, like this:
Having done so, we could then draw another circle, this time centred on the circumference of the outer circle, with its edge just touching the circumference of the original, inner circle, like this:
Going through these steps, we would have an immediate and intuitive understanding that the relationship in size between the smallest circle and the inner circle was not arbitrary and random, but highly specific and tightly constrained. We would have demonstrated to ourselves, in the very act of drawing, that the two are bound together in ratio characterised by the square’s relationship to the circle it perfectly contains.
Now, as it happens, this ratio can be expressed in whole numbers very neatly as the ratio 3 : 11, e.g., if the radius of the smallest circle has a value of 3, the radius of the circle inside the square will have a value of 11.
These numbers are inherent to the proportion of a circle and square of equal perimeter, and can be expressed as a ratio, a fraction or a percentage.
3 : 11 = 3/11 = 27.3%
All signify the same proportion.
Why does this matter? Remarkably, this proportion exactly defines the relative size of the earth and moon. We saw that the smallest circle relates in size to the original circle as 3 does to 11; the same is true for how the size of the moon relates to that of the earth. Let’s do the math:
3 / 11 = 0.2727 = 27.3%
The radius of the moon is 1079.57 miles
The radius of the earth is 3959 miles
1079.57 / 3959 = 0.2727 = 27.3%
That the relative size of the earth and moon should conform so perfectly to the proportions generated by “squaring the circle” is astounding. Making the randomness of this coincidence even harder to believe, the same value of 27.3 defines the moon’s sidereal orbit around the earth. It takes the moon 27.3 days to make one full circle around the earth as measured against the backdrop of the fixed stars. Combine this with the fact that, despite the enormous difference in their sizes and relative distance from earth, the sun and moon are so perfectly proportioned in their size and distance that they both appear to be the same size to us on earth.
What are the odds?