The number Five (5) and Phi

The number 5 is intrinsically related to Phi and the Fibonacci series

Phi can be derived from several formulas based on the number 5.  The most traditional, based on the geometric construction of phi is this:

This formula for phi can also be expressed all in fives as:

= 5 ^ .5 * .5 + .5

Another formula for phi based entirely on 5's, an original insight contributed by Erol Karazincir (), is as follows:

And, as pointed out by W. Nathan Saunders, the terms in above representation of phi can be expressed in yet another way that involves four 5's:

(5+√5) x (5-√5) = 5 + 5 + 5 + 5

Phi appears in the geometry of the 5-sided pentagon

Take a pentagon with 5 equal sides and connect all the points to form a 5-pointed star.  The ratios of the lengths of the resulting line segments are all based on phi.

Phi appears in the natural logs and trigonmetric functions:

Phi can be related to e, the base of natural logs,
through the inverse hyperbolic sine function:

Phi = e ^ asinh(.5)

Determining the nth number of the Fibonacci series

You can compute the nth number in the Fibonacci series (f_n) using phi and root 5:

5 is the 5th Fibonacci number

5 is also the 5th of the Fibonacci numbers, including 0, 1, 2, 3, and 5.

5 appears in the human body, which has proportions based on phi

Another interesting aspect of phi and five is in relation to the design of the human body, which in addition to being based on phi relationships in its proportions, has: 5 appendages from the torso, in the two arms, two legs and a head, 5 appendages on each of legs and arms in the five fingers and five toes, 5 openings on the face, and 5 senses in sight, sound, touch, taste and smell.

5 deserves a "high 5" for its role in phi, don't you think!