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 :)

    • Kinson says

      Wow… That was almost a complete paradigm shift. I have just recently started researching numbers again, as I had a thought years ago before I knew any if this. I wasn’t bad at math I just didn’t really try as hard as I could, but nonetheless; I had thought there were obvious shapes that seem to pop up all around the world and universe and quickly began thinking of geometry and math and discovered Phi, then my journey stopped as life got in the way…

      Something has brought me back and I can’t help but feel I was meant to learn more and expand my horizon further. As any knowledge I can learn about the fundamentals, the absolute truths, and all that is seemingly ingrained into not only our surroundings and cosmos, but in the very vessels that allow us, the universe, the chance to experience itself. Your post did an absolute fantastic push in my new journey.

      I think of how it screams volumes next to E=mc^2. Great mind.

  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

  10. jim carlin says

    I read-the male brain is a line
    the female brain is a wave
    viva la difference
    which is language of the heart :)

    • says

      Phi is much more than amazing. Nobody in my year knows what it is either. I’m five decades older than you. Just to show how amazing, I’ll give one example from my life that is happening right this second.. I’m a retired high school Social Studies teacher. I did my student teaching with a guy who is a Vietnam vet with 3 Purple Hearts, a three decade devotee of Parmahansa Yogananda and the Self-Realization Church, and my best friend. The city we live in, where he has spent practically his whole life, has a Central Ave. If you drop a location pin from a Google map app on a cell phone for 6180 N. Central it lands in the middle of avenue itself, right in front of my friend’s Self-Realization Church. The place he has been learning to be enlightened for most of his adult life. He was married there, and even renewed his vows there. But I can guarantee that he was never taught that his International Church, started almost 100 ears ago, has locations worldwide, with addresses, area codes, zip codes, phone number, etc., aligned as much as possible, whenever and wherever possible, with the Golden Ratio, the DIVINE PROPORTION, 1:1.618. How do I tell him? Do I tell him? Hell, I’ve been trying to tell the Administrator of this very website that his belief system is just as ensnared with the Golden Mean as my friend’s Organized Religion, as they all are. The belief system responsible is the oldest of them all; the Master Architect of the Universe, the ancient Egyptian god PTAH, Patron of Memphis, and of artisans and craftsmen, who built the Temple ‘Aeigyptos’ in its’ honor, and practically everything else ever since.

  11. jim carlin says

    Occam’s razor=be brief

    over lapping ellipses=symbol for empathy

    there is a good orderly direction=a safe universe


  12. Leo Darnell says

    Excellent article about Phi. Me too, I find it interesting to know that its not only for Mathematics, it can be linked to numerous useful factors.


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