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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 number is the multiplier as to the Golden Section or the Rectangular Ratio that has also become known as 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. Somehow I think there's more to this than just coincidence.


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."


Noel Reekie's Proof that the

Golden Section or Ratio is 1 : 1.618

Starting 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 generated curve is the Golden Curve that is found in nature as in sea shells (Nautilus) etc. etc. on up through plant life to the very tremendous spiral curvature of the whirling galaxies in the Heavens.


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.