Phi (Φ) is the 21st letter of the Greek alphabet, chosen to represent the golden ratio to recognize the Greek sculptor Phidias, who lived from about 480 – 430 BC. He is regarded as one of the greatest sculptors of classical Greece. His statue of Zeus at Olympia was one of the Seven Wonders of the Ancient World. He also designed the statues of the goddess Athena on the Athenian Acropolis. 

Phi (Φ = 1.6180339…) is an irrational number like pi (π), and was known to the Greeks as the “dividing a line in the extreme and mean ratio” and to Renaissance artists as the “Divine Proportion”. It is also called the Golden Section, Golden Ratio and the Golden Mean. It comes with so many mathematical properties that it would take a whole book to explore this one number. Here, we have tried to include some of the cool characteristics of this number to give you an insight. 

The Fibonacci Sequence is a crazy relative of Phi. It starts with the numbers 1 and 1 and then progresses as follows:

1, 1, 2, 3, 5, 8, 13…

To give another example of Phi in use: a golden rectangle, which can be found in many architectural projects, has a height of 1 and a length of 1.618. The human face has at least 30 different representations of the golden ratio in it. Sunflower seeds grow in a Fibonacci Spiral pattern from the center of the flower. The examples are endless!

The spiral growth of seashells provide a simple, but beautiful, example –

Phi has two properties that make it unique among all numbers. 

  1. If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or     Φ² = Φ + 1. 
  2. If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or  1 / Φ = Φ – 1.

Phi can be expressed as a limit:

Phi, 1.618, the golden ratio, expressed as a limit
or

These relationships are derived from dividing a line at its golden section point, the point at which the ratio of the line (A) to the larger section (B) is the same as the ratio of the larger section (B) to the smaller section (C).

This relationship is expressed mathematically as:

A = B + C, and

A / B = B / C.

Solving for A, which on both sides give us this:

B + C = B²/C

Let’s say that C is 1 so we can determine the relative dimensions of the line segments. Now we simply have this:

B + 1  = B²

This can be rearranged as:

B² – B – 1 = 0

When we put the values and solve the quadratic equation, we get – 

( 1 + √5 ) / 2 = 1.6180339 (Φ)

The significance of the equation, ( 1 + √5 ) / 2 = 1.6180339… = Φ, is that this gives the value of the relationship that appears when dividing a line such that the ratio of the entire line to the larger segment is the same as the larger segment to the smaller segment. This dividing is the golden ratio that appears extensively in mathematics, geometry and nature. It has thus also been applied to the design arts to achieve a natural appearance in composition and to enhance aesthetics.

To celebrate this special number, we even have a day called the Phi Day, which was last celebrated on 6th Jan 2018 (1/6/18). The next date that aligns is 6 Jan 2118.