Phi Basics

May 16, 2012

What is Phi?

(or Phi for “Neo-Phi-tes”)


What is Phi?

Phi ( Φ 1.618033988749895… ), most often pronounced fi like “fly,” is simply an irrational number like pi ( p = 3.14159265358979… ), but one with many unusual mathematical properties.  Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation.

Phi is the basis for the Golden Ratio, Section or Mean

The ratio, or proportion, determined by Phi (1.618 …) 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.

Circle with a diameter of 1 and circumference of pi, 3.14Phi, like Pi, is a ratio defined by a geometric construction

Just as pi (p) is the ratio of the circumference of a circle to its diameter, phi () is simply the ratio of the line segments that result when a line is divided in one very special and unique way.

Divide a line so that:

Sectioning a line to form the golden section or golden ratio based on phi

the ratio of the length of the entire line (A)
to the length of larger line segment (B)

is the same as

the ratio of the length of the larger line segment (B)
to the length of the smaller line segment (C).

This happens only at the point where:

A is 1.618 … times B and B is 1.618 … times C.

Alternatively, C is 0.618… of B and B is 0.618… of A.

Phi with an upper case “P” is 1.618 0339 887 …, while phi with a lower case “p” is 0.6180339887, the reciprocal of Phi and also Phi minus 1.

What makes phi even more unusual is that it can be derived in many ways and shows up in relationships throughout the universe.

Phi can be derived through:

Phi appears in:

For an overview of key content of this site, read the article, Phi: The Golden Number by Gary Meisner, author of and developer of PhiMatrix golden ratio design software.

Be Sociable. Share the Phi-nomenon!

Leave a Comment

{ 18 comments… read them below or add one }

jonathan October 30, 2012 at 1:47 pm

The Theory of 0

First before all was thought.
There was nothing. = 0
If 0 then infinite
if infinite then all numbers

If numbers then separation or difference
if separation then equations

if numbers then lines = I
if pi then circles = O circle and line = Φ

it should be noted also that if you move the left half of the circle up touching on the line where the right half of the circle touches you get a wave. It is commonly believed that waves make up the sub-atomic particles. Just some mumbo jumbo in my brain. Hope this helps someone :)


DMac01 January 14, 2013 at 11:36 pm

I got it Jonathan…. very good!


winnie January 7, 2013 at 3:52 pm

Joe, really nice. Got lost at a point though.


ferfo January 24, 2013 at 4:37 pm

Si la luz se propaga en medio de ondas…+ y dios es Luz… Las partículas sub atómicas son ondas… y todo crece a razón de Phi… ntonces….
Translation: If light propagates waves amid … + God is light and … Sub atomic particles are waves … and everything is growing by Phi …


Andrea January 26, 2013 at 2:58 am

Cant Understand Joe…..! Can anyone EXPLAIN….!


Richard Warren March 2, 2013 at 6:28 am

You can also use phi to calculate the circumference of a circle to within .04 of pi.
It’s probably more accurate than pi. Which pretty much makes pi obsolete.
Go the golden mean!


larry coleman March 12, 2013 at 2:15 pm



Gayatri July 1, 2013 at 12:35 am

Hi everyone
I’m in 10th grade and am doing my math project on Phi. I’d like to know if there are any applications of Phi that can be easily demonstrated to an audience that has never heard of it before. Also, is there anything I can make a scaled model of to show? The only materials I’ve got are thermocol and modelling clay. Please reply to this by 6th July


Bucky July 13, 2013 at 7:50 am

I thought Phi is only one Mathematics constant.
I can’t believe it is associated with so many interesting things!
Thanks for your share.


Gregory Weigle August 7, 2013 at 8:03 pm

Go to YouTube and look up, “The Revelation of the Pyramids”. That’s where I ran into this ratio and was dumbfounded. Great documentary and informative.


Nishant Kapoor August 15, 2013 at 1:48 pm

thanks didn’t know that…phi….is associated with so many things


Theo October 16, 2013 at 10:35 pm

PHI is the universal number for All life.


Gary Meisner October 18, 2013 at 4:41 am

“Universal” means that it applies without exception to all cases, which to my knowledge isn’t true. More accurate adjectives would be “pervasive,” which means “spread throughout” or “dominant,” which means “main, major or chief.”


'the da vinci code'brought me here. November 5, 2013 at 2:58 am

totally fascinated!!!!


lammii December 9, 2013 at 9:37 pm

same here


Jim December 18, 2013 at 2:47 pm

I have been very focused on sacred geometry this year. I have received much guidance regarding the creation of geometric forms and their uses. I love how numbers pull together so many different aspects of life. Like Phi and the Pythagorean musical scale. They are both about 5th’s. The relationship between 3 and 2.


Kaleb January 5, 2014 at 4:17 am



Jason January 19, 2014 at 11:15 am

True Dimensions of The Great Pyramid of Khufu… Base=777.03 ft, Height=480.24 ft… Calculated through Phi=1.618


{ 1 trackback }

Previous post:

Next post: