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
(See also the quest for
the Holy Grail in Dan Brown's book, The Da Vinci Code
of numbers 1-1-2-3-5-8-13-21-34-55-89-144 on out to
after the start, these numbers, from one to the next, are
ratios of 1 : 1.618)
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
"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
Golden Section or Ratio is 1 :
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
The remaining rectangle,
BEFC is also a Golden Section.
Square out DF then the
resultant larger figure is also 1 :
OB2 + BC2 = OC2
0.25 + 1 = 1.25
The square root of 1.25 =
Therefore 1.118 + 0.5 =
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.