Noel Reekie's Proof of the Golden Section

 

What really fascinated my brother, Noel, in the realm of mathematics was the ratio of numbers, 1 : 1.618 that is the multiplier as to the Golden Section or Rectangular Ratio, the Divine Proportion. 

 

(See also the quest for the Holy Grail in Dan Brown's book, The Da Vinci Code and the

 Fibonacci sequence of numbers   1-1-2-3-5-8-13-21-34-55-89-144 on out to infinity.

Shortly after the start these numbers, from one to the next, are ratios of 1 : 1.618)

Oddly enough, the square root of 5 plus 1, all divided by 2 also equals 1.618 

 

Noel's wording from here on.

 

Although I've come across dozens of web sites with elaborate proofs as to the Golden Section I've found NONE as simplistic as mine.

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Noel Reekie's Proof that the

Golden Section or Ratio is 1 : 1.618

From square ABCD describe Radius OC from Point O, which is 0.5 of AB, to Point E.

Complete Rectangle AEFD which is the Golden Section.

Remove the original Square.

 

The remaining rectangle, BEFC is also a Golden Section.

Square out DF then the resultant larger figure is also 1 : 1.618

 

OB² + BC² = OC²  

0.25 + 1 = 1.25

The square root of 1.25 = 1.118

Therefore 1.118 + 0.5 = 1.618

 

If you square out the long side of a Golden Rectangle, the Resultant Rectangle is a Golden Rectangle, ad infinitum.

 

If you radius each square, the resultant curve is a Golden Curve (more or less) as in sea shells (Nautilus) etc. on up to the curvature of the whirling galaxies.

 

Although in theory this sequence of curves could go on indefinitely I don't think that nature or modern engineering structures could sustain a sequence of the ratio 1 : 1.618 beyond about 7 or 8 fold.