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.
