The Golden Ratio also known as PHI

Math could be the foundation of reality both being intangible and having substance, not needing anything to exist, not needing creation. It does not require even time to exist. If so then maybe it is time for a closer look at the numbers. If numbers are just symbols what do they really represent?

The Golden ratio is also known as the Divine proportion. Nature uses it, in it's beauty and perfect design as do those that design with genius.

The Golden ratio is one of those rare irrational infinite numbers. It's like Pi goes on forever which is much like infinite being.

It is also called the Golden Section, Golden Ratio and the Golden Mean.

Phi, like Pi, is a ratio defined by a geometric construction

Just as pi (p) is the ratio of the circumference of a circle to its diameter, phi () is simply the ratio of the line segments that result when a line is divided in one very special and unique way.

Divide a line so that:

Sectioning a line to form the Golden Section

the ratio of the length of the entire line (A)
to the length of larger line segment (B)

is the same as

the ratio of the length of the larger line segment (B)
to the length of the smaller line segment (C).

This happens only at the point where:

A is 1.618 ... times B and B is 1.618 ... times C.

Alternatively, C is 0.618... of B and B is 0.618... of A.

Phi with an upper case "P" is 1.618 0339 887 ..., while phi with a lower case "p" is 0.6180339887, the reciprocal of Phi and also Phi minus 1.

What makes phi even more unusual is that it can be derived in many ways and shows up in relationships throughout the universe.

One unique point exists that divides a line into two unequal segments so that the whole is to the greater as the greater is to the lesser. there is only one way to divide a line so that its parts are in proportion to, or in the image of, the whole or in the image of, the whole: Almost sounds like the relationship between the divine being and men.

Every nth Fibonacci number is a multiple of Phi(n),
where Phi(n) is the nth number of the Fibonacci sequence.

Given 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

(Every 4th number, e.g., 3, 21, 144 and 987, are all multiples of Phi(4), which is 3)

(Every 5th number, e.g., 5, 55, 610, and 6765, are all multiples of Phi(5), which is 5)

And another:

The first perfect square in the Fibonacci series, 144,

is number 12 in the series and its square root is 12!

0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

or, if not starting with 0:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

What does Phi look like?

whole greater	=	greater lesser

where:

AB AC	=	AC BC

This odd, obscure ratio equals PHI.

The value of this isn't immediately obvious, but becomes much more apparent when we divide the long segment by the short segment to locate a new point D. The simple way to accomplish this is to set a compass to line segment BC, then rotate around point C to mark a new point D on segment AC.

AC?the old long segment?is now the whole.
CD is the new long segment.
AD is the new short segment.

AC CD	=	CD AD	= 1.618 = PHI

All three lines are in the same PHI ratio relationship.

Repeat this operation again. Set the compass to line AD and rotate around point D to locate new point E on the long segment CD. Line CD is thus divided into two unequal segments CE and DE. The three new line segments are again in PHI ratio:

CD DE	=	DE CE	= 1.618 = PHI

Repeatedly dividing each long segment by the short segment will create new line segments that are also in PHI ratio at each smaller scale, no matter how many times this division is repeated.

DE EF	=	EF DF	= 1.618 = PHI

Thus, in Extreme & Mean Proportion, PHI recurs again and again. Every time the lesser is used to divide the greater into two shorter, unequal segments, the resulting line segments are in PHI ratio.

From these simple operations on a line, you can see that PHI is a resonant ratio? a geometric echo. Once established, this ratio recreates itself through successive operations.

Recursion is a very useful property, and a key characteristic in the design of forms and flow systems that are stable at multiple levels of scale. Recursion is also a mechanism to pass information as ratio through successive layers of scale. PHI is the recursion required for regeneration the resonant ratio to govern and guide growth.