The constant value phi (Φ) is a wondrous mystery of nature.

What is the Golden Ratio?

The value phi (Φ), often referred to as the Golden Ratio is a mathematical constant that is found extensively
in nature. Unlike most of the mathematical constants, however, phi has no reason to exist other than "because" (or
at least, we haven't yet discovered its reason to exist — it just does.) And this number is everywhere. For example,
The ratio of the length of the large intestine to the length of the small intestine is the golden ratio. The average
person's total height compared to their waist height is the golden ratio.

Make a rectangle that is not a square and has the properties of not being too long or too wide, but is "just right"
to be pleasing to the eye; the ratio of its length to its width will be very close to the rolden ratio. Artists often
use the golden ratio because it produces pleasing shapes. Buildings such as the Greek Parthenon were designed using
this concept. The Greeks knew the the golden ratio as The Section.

Golden Ratio \(\Phi=\frac{\textrm{length}}{\textrm{width}}\) \(=\frac{1+\sqrt{5}}{2}\) ≅ \(1.618033989\)

The golden ratio, Φ, also appears in connection with the Fibinacci sequence. As each number in the Fibinacci sequence
is divided by the number preceeding it, the quotient gets closer to Φ.

The Fibinacci Sequence

Leonardo Fibinacci, an Italian mathematician who lived from around 1180 to 1250 AD, discovered this number pattern, and
mathematicians are still finding new and interesting ways in which this number series describes nature.

In the Fibinacci sequence, each number in the sequence is the sum of the two numbers before it:

Sunflowers, pineapples, and pinecones (as well as many other plants, especially flowers) all show two spiral patterns —
one clockwise and one counter-clockwise. The number of spirals in each pattern is always a number in the Fibinacci sequence.
The Fibinacci sequence is also present in objects such as snail shells and various other (usually bioligical) formations.

The Fibinacci Spiral

Each square added to the rectangle follows the Fibinacci sequence. Here, values 1, 1, 2, 3, and 5 are shown.
The rest of the numbers would be 8, 13, 21, and so on. Radial curves drawn in each square create the Fibinacci
spiral. Many objects in nature follow this curve almost exactly.

Golden Ratio of Pentagram Inscribed in a Pentagon

For a pentagram inscribed within a regular pentagon ABCDE, intersects A', B', C' D', E' all divide diagonals
into segments of the golden ratio.

Phi as a Continued Fraction

The golden ratio can be written as a continued fraction:

In fact, there are an infinite number of similar equations involving \(\Phi\) that will result in either \(\sqrt{5}\) or 1.
These are just a few examples. As far as I know, this is the only constant in the universe that can do this so simply. Perhaps
that's why it's everywhere.

Another interesting fact about Φ, is that it is one of the roots of the quadratic equations with values a=1, b=-1, c=-1, and
a=-1, b=1, c=1. The other root of both of these quadratics is Φ-1.