What is Phi? (The Basics of the Golden Ratio)

Phi for “Neo-Phi-tes:”

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, 3.14 Phi, 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 www.goldennumber.net and developer of PhiMatrix golden ratio design software.


  1. jonathan says

    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 :)

  2. ferfo says

    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 …

  3. Richard Warren says

    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!

    • says

      Finally, someone said it. We’re about to celebrate Pi Day again in schools all across the country, the real one this time, 3/14/15, which everyone and their brother is familiar with, but as just stated it’s rendered obsolete by the Golden Mean, Phi which no one or their whole family ever heard of! Way, way beyond mystifying.

  4. Gayatri says

    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

  5. Bucky says

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

  6. Gregory Weigle says

    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.

    • says

      “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.”

  7. Jim says

    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.

  8. Noor Zahra says

    If i have words consisting numbers of 30, 40, 1 then how will i calculate that according to this golden ratio?
    And what is the process of calculating this, im not getting this

    • says

      To find the golden ratios of any number, just multiply it by 1.618 and divide it by 1.618. Golden ratio relationships with 30 are thus found at 48.54 and 18.54. You can round that to 49 and 19 if needed.

  9. Albino Giraffe says

    This mind-blew me so good. I love the orgasmic effect it instigates in my mind. Truly a large phenomena, and will always be on my mind


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