The Egytians used phi in the design of the Great Pyramids (c. 2500 BC)

Euclid (365 BC - 300 BC) referred to dividing a line at the 0.6180399... point as dividing a line in the extreme and mean ratio. This influenced the use of the term mean in the golden mean.

Phidias (500 BC - 432 BC), a Greek sculptor and mathematician, studied phi and applied it to the design of sculptures for the Parthenon.

Plato (c. 428 BC - 347 BC) considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos.

Leonardo Fibonacci (1170 AD - 1250 AD) discovered the unusual properties of the numerical series that now bears his name, but it's not certain that he even realized its connection to phi and the Golden Mean.

In 1509, Leonardo Da Vinci provided illustrations for a dissertation entitled "De Divina Proportione,” perhaps the earliest reference to the title "Divine Proportion." It was probably Da Vinci who first called it the "sectio aurea” (golden section).

Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of "The Last Supper," from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background.

Johannes Kepler (1571-1630), discoverer of the elliptical nature of the orbits of the planets around the sun, also made mention of the "Divine Proportion," saying this about it:

"Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."

The term “Phi” was not used until the 1900’s

American mathematician Mark Barr was the first person to use the Greek letter Phi to represent the golden ratio.

= 1.61803398874989...

Φ is defined a few different ways:

or1 1 1 1 1 ...

Φ2 = Φ + 11

1

or

Phi is the ratio, if "the whole is to the larger as the larger is to the smaller"

Phi can be found using algebra:

Phi is the ratio, if "the whole is to the larger as the larger is to the smaller"

a b

a b b

b a

b a b b b

a b a a

2a b b

a a

2

1b b

a a

Phi can be found using the quadratic formula:

21 4 1 1

2

Phi can be found using algebra and the quadratic formula:

11

11

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Index Number Fibonacci Number Previous Number Ratio1 0 -- --2 1 0 --3 1 1 14 2 1 25 3 2 1.56 5 3 1.6666666677 8 5 1.68 13 8 1.6259 21 13 1.61538461510 34 21 1.61904761911 55 34 1.61764705912 89 55 1.61818181813 144 89 1.61797752814 233 144 1.61805555615 377 233 1.61802575116 610 377 1.61803713517 987 610 1.618032787

Proceeding from fingertips to your elbow, the fibonacci series emerges, and with it, so does the golden ratio.

A normal human heart beats in a phi rhythm, with the T point of a normal electrocardiogram (ECG or EKG) falling at the phi point of the heart's rhythmic cycle.