The Golden Mean compared to PI

In reviewing some ancient notes of mine, I came across an interesting comparison between the Golden Mean (Phi) and PI. They are more interesting in reverse:

A phi square (area: 2.618, side: 1.618) has grown in area relative to a unit square by the amount (area: 0.618) plus the rectangle (area:1 ). This reveals the role of phi’s reciprocal square (area: 0.384) in being the reciprocal of the reciprocal so that in product they return the unity (area: 1).

On the right, the phi squared square showing how the reciprocal of phi and its square uniquely sum to unity (area: 1), a property that is scale invariant between structures who share the same units and grow according to the Golden Mean.

The Golden rectangle (area 1 plus 0.618) can be seen as the link to the square (area: 2.618) which is a new manifestation of the original square (area: 1). In the opposite direction, of scaling down, the same pattern subsists and the area on the top right (area: 0.618 squared) has become the new manifestation of the original square (area:1) on the lesser scale. This highlights the uniqueness of the Golden Mean within the domain of real numbers as a singularity within the world of geometrical space (proportionality) as with PI and circular areas: PI times their radius squared. A circle’s in-square and out-square, have areas that are doubled (or halved) relative to each other.

The doubling in area of successive out-circles to in-circles due to squaring of the square root of two

The Fibonacci series {1 1 2 3 5 8 13 21 34 55 89 144 233 …} is interesting since, as it grows numerically, adjacent terms are, in ratio, approximations to the golden mean which improve. The terms of the Fibonnaci set are the smallest integers able to do this (the Lewis series being similar but are more generalised). They have been called sacred numbers, found in Nature and the human body.

If one starts with a square of 34 as “one”, adding the previous 21 can shift to a new square side length 55. Higher Fibonacci numbers can then repeat this using convenient Fibonacci numbers, maintain contact with the number field itself whilst avoid real numbers whose fractional parts are actually part of the transfinite continuum, available geometrically but problematic metrologically – working to quantifiable units of measure, had to be the case for the megalithic astronomers, for example, without arithmetic methods.

The successive approximation of Fibonacci series to Phi improves with later pairs, expressing also the magic of the Golden Section in an algorithmic way