series, Golden rectangles, phi=1.6180339887499
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
The decimal representation of phi is 1.6180339887499...
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
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:
= Whoa! That one doesn't count.
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.
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:
and so on, with, again, each addition coming ever closer to multiplying
Ancient architecture is filled with Golden rectangles.
the Golden Rectangle Text-only
version (character-based graphics)
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'.
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,
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.
phi graphically See
the 'imaginary' graph.
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.
The Nautilus shell (Nautilus pompilius) grows larger on each spiral
The sunflower has 55 (see number list) clockwise spirals overlaid
on either 34 or 89 (see number list) counterclockwise spirals, a