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The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.The decimal representation of phi is 1.6180339887499... .
The
Use of Golden Mean in Ancient Greece
Constructing the Golden Section
We start with a problem in aesthetics. Consider the following line segment
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What is the most "pleasing" division of this line segment into two parts? Some people might say at the halfway point:
_________________.________________
Others might say at the one-quarter
or three-quarters point. The "correct answer" is, however, none of these,
and is found in
Western art from the ancient Greeks
onward (art theorists speak of it as the principle of "dynamic symmetry"):
_____________________.____________
Here, if the left-hand portion is of length u = 1, then the right-hand portion is of length v = 0.618... A line segment partitioned as such is said to be divided in Golden or Divine section. What is the justification for endowing this particular division with such elevated status? The thinking is that the length u, as drawn, is to the whole length u+v, as the length v is to u. In symbols,
The rich geometric connection between the Golden mean and Fibonacci's sequence is seen via the following diagram ([10,11,12,13]). Starting with a single Golden rectangle (of length and width 1), there is a natural sequence of nested Golden rectangles obtained by removing the leftmost square from the first rectangle, the topmost square from the second rectangle, etc.
Construct
IT!
Constructing Golden Section
Golden
Mean ArchitectureNumber Series
If you start with the numbers 0
and 1, and make a list in which each new number is the sum of the previous
two, you get a list like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... to infinity-->
This is
called a 'Fibonacci series'.
If you then take the ratio of any
two sequential numbers in this series, you'll find that it falls into an
increasingly narrow range:
1/1 =
1
2/1 =
2
3/2 =
1.5
5/3 =
1.6666...
8/5 =
1.6
13/8 =
1.625
21/13
= 1.61538...
34/21
= 1.61904...
and so on, with each addition coming
ever closer to multiplying by some as-yet-undetermined number.
The number that this ratio is oscillating
around is phi (1.6180339887499...). It's interesting to note that the ratio
21/13 differs from phi
by less than .003, and 34/21 by only about .001 (less than 1/10 of one
percent!), thus providing our less
technically-advanced ancestors
an easy way to derive phi on a large scale in the real world with a high
degree of precision.
Geometry
If you have a rectangle whose sides
are related by phi (say, for instance, 13 x 8), that rectangle is said
to be a Golden Rectangle.
It has the interesting property that, if you create a new rectangle by
'swinging' the long side around one of its ends
to create a new long side, then that new rectangle is also Golden. In the
case of our 13 x 8 rectangle, the new rectangle
will be (13 + 8 =) 21 x 13. You can see this is the same thing that's going
on in our number list, but when you discover
it geometrically it looks downright magical. If you start with a square
(1 x 1) and start swinging sides to make rectangles,
you wind up with Golden rectangles without even trying. Here's the list,
in case it isn't obvious:
1 x 1
2 x 1
3 x 2
5 x 3
8 x 5
13 x 8
21 x 13
34 x 21
and so on, with, again, each addition
coming ever closer to multiplying by phi.
Ancient architecture is filled
with Golden rectangles.
When you swing the long side of a Golden Rectangle around to create a new rectangle, the line it forms with the short side is made up of two sections having lengths of phi and one, respectively. This division of a straight line into a phi proportion is what is actually meant by the term 'Golden Section'.
Pure Math
Proportion is the relationship
of the size of two things.
Arithmetic proportion exists
when a quantity is changed by adding some amount.
Geometric proportion exists
when a quantity is changed by multiplying by some amount.
Phi possesses both qualities.
If you study the Fibonacci series
and the Golden Rectangle carefully, you will eventually realize that
phi + 1 = phi * phi.
Consider: suppose that you start
with a Golden rectangle having a short side one unit long. Since the long
side of a Golden rectangle equals the short side multiplied
by phi, the long side of our rectangle is one times phi. or simply phi.
Now suppose that you swing the
long side to make a new Golden rectangle. The short side of the new rectangle
is, of course, phi units long, and the long side is
that times phi, or phi * phi. This describes a Geometric proportion.
But we also know from simple geometry that the new long side
equals the sum of the two sides of the original rectangle, or
phi + 1. This describes an Arithmetic proportion. Since
these two expressions describe the same thing, they are equivalent, and
so phi + 1 = phi * phi.
(Now that we know this, we can
discover the exact value of phi. )
The resulting proportion is
both arithmetic and geometric. It is thus perfect proportion; you could
think of it as the place on some imaginary graph where
the curved line of multiplication crosses the straight line of addition.
Nature
In pure mathematics, an increase
in size can be any imaginable number, even one like e or pi. But in the
world of nature, things always grow by adding some
unit, even if the unit is as small as a molecule. So it's not surprising
that phi turns out to be an ideal rate of growth for
things which grow by adding some quantity.
Some examples:
The Nautilus shell (Nautilus pompilius)
grows larger on each spiral by phi. 
