Properties of the Golden Ratio:

Properties of the Golden Ratio:
The Golden Ratio can be expressed as 1.618 and 0.618 and is known as Phi and phi, respectively; phi being the reciprocal of Phi… This is a very unique property that only the Golden Ratio possesses:

1 / Phi = phi  (1 / 1.618 = 0.618)
and…
1 / phi = Phi  (1 / 0.618 = 1.618)

Also, Phi Squared = Phi + 1  (1.618 ^2 = 1.618 + 1)
…and Phi multiplied by phi = 1  (1.618 * 0.618 = 1)

Phi is not a fraction: In other words, there is no way to express Phi as using two integers, e.g. (2/3)

Deriving Phi:
Phi = Square root of 5 + 1 / 2…  or  (Sqr(5)+1)/2

Phi to 31 decimal places: 1.6180339887498948482045868343656

Geometry in Nature and the natural world:
Our reality is very structured, and indeed Life is even more structured.  This is reflected though Nature in form of geometry. Geometry is the very basis of our reality, and hence we live in a coherent world governed by unseen laws.  These are always manifested in the natural world.  The Golden Mean governs the proportion of our world and it can be found even in the most seemingly proportion-less living forms.

Clear examples of geometry (and Golden Mean geometry) in Nature and matter:
All types of crystals, natural and cultured.
The hexagonal geometry of snowflakes.
Creatures exhibiting logarithmic spiral patterns: e.g. snails and various shell fish.
Birds and flying insects, exhibiting clear Golden Mean proportions in bodies & wings.
The way in which lightning forms branches.
The way in which rivers branch.
The geometric molecular and atomic patterns that all solid metals exhibit.
Another, less obvious, example of this special ratio can be found in Deoxyribonucleic Acid (DNA) – the foundation and guiding mechanism of all living organisms:

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Title: Properties of the Golden Ratio:
Date Posted: September 17, 2006
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Category: Uncategorized