## 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.

## 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:

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:

A numerical series discovered by Leonardo Fibonacci

## Phi appears in:

The proportions of the human body

The proportions of many other animals

The solar system

The stock market

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.

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 🙂

DMac01 says

I got it Jonathan…. very good!

Madi says

Is this supposed to be a poem?

Michael says

♡

Justin says

Jonathan I already knew everything on this page but was looking for something more, your post was it!!! Thank you!

Nami says

Thank you!!!! Really helped 🙂

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.

Greg says

“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.”

I guess I extended the line upward, too, but I got a “$”.

Anne Marie Cook says

The .”s” is actually a coil

Leckner says

The Theory of 0

First before myriad things there was 0, or source, or infinite.

0=infinite

If 0 then unit=true & unit.id=1

if unit=true then unit.count=infinite

If unit.count=infinite then

enumeration of unit = numbers.exist & numbers.infinite = true

if numbers.infinite=true then unit.state.varriance=true & potential for unit.state.varriance=true

For unit.id=1to unit.id=infinite Do Until Step 1

enumerate unit.id

calculate potential for unit.state.varrience

Loop

VictorRosez says

Two different infinities are equal in size.

Choim says

You make me wonder, is there any difference between any two infinities?

Dave says

Wow… Your brain sounds pretty unmumbojumbo to me! Pure clarity – well, almost! 🙂

Gary B Meisner says

See also https://www.goldennumber.net/theology/.

truth_explorer says

your description is actually better than i think you realise.

it is a poetic expression of symbol. but actually has more mathematical depth than youve described.

Bobby says

You are smart.

Tim says

I didn’t get it, but apparently I am in the minority

Antonia says

I’m being reminded of a particular book/tom hanks movie here….

Sweta says

Yup, me too. The Da Vinci Code by Dan Brown. That’s what brought me here.

anonymous says

same

Michael says

Hello Johnathan,

I am experimenting with PHI and I have a question that I believe you can shed some light on if you don’t mind.

ovolbc@gmail.com

please reply

LAURA says

Thank you! This helped a lot!

wolfman says

very clever and very cool

Rebecca says

Yes, it makes perfect sense. It’s interesting how the symbols so simply show everything.

winnie says

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

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 …

Andrea says

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

Pano says

Consider the set Φ={1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…,∞}

(Refer to this set when needed.)

How did we get this list?

Begin with 0. The 1st whole number after 0 is 1. We add 0+1 (and 0+1=1) <–Notice the last two numbers here are both 1 so we add them.

Adding them: 1+1=2

This gives us our next sum; so far we just have the set= {1, 2} a subset of Φ (top of page) so we're getting there…

We keep adding the last two numbers: 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, and we keep the sums in a set = {1, 2, 3, 5, 8, 13} another subset of Φ

Let's keep going, last two above are 8 & 13, so: 8+13=21, 13+21=34, 21+34=55, 34+55=89, 55+89=144, 89+144=233,…etc.

Eventually we get the set Φ={1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…,∞}

If we divide the last two numbers in the set, like 233/144 = 1.6180555 so this number, or ratio, "approaches" φ=1.6180339887…

I'll explain what I mean by "approaches."

We're going to start dividing the last two numbers in the set Φ

Look at 1&2 …Nothing special: 2/1= 2

Next 3/2 = 1.5 Ok, it's a little closer

Next 5/3=1.666… better

Let's skip ahead, 233/144= 1.6180555… which is approaching 1.6180339887… =φ

233/144 is the 1st ratio that gets us close to φ (within 1.618)

See how our ratio went from 2, 1.5, 1.666, skipped ahead, 1.6180555… and if we keep going, with bigger and bigger numbers the more exact the decimals become.

And we started at 0, right? 0+1, 1+1, 1+2, etc, so "as we begin at zero and the numbers get larger and larger they approach infinity, but our ratio get's closer and closer as it approaches φ=1.6180339887…."

I hope this helped.

esnart says

this is so interesting.i was so good in math at school.i came across the phi when i read he da vinci code by dan brown and ever since iv become interested.thanks for this explanation.it has made it much easier to understand.

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!

Peter Hedding 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.

Edgewalker says

Pi is defined as the ratio of a circle’s circumference to its diameter, so it’s not possible for any number to be “more accurate” for this purpose. Do you see why?

larry coleman says

“PHI” UNIFIES ALL THING AND SUFFUSES ALL THINGS

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

olivia says

If you divide the distance from the tip of somebody’s head to the floor and the distance from their belly button to the floor, you get Phi. Same with from your shoulder to your fingertips divided by elbow to fingertips, and hip to floor divided by knee to floor. Another thing, take the number of female bees in a hive and divide it by male bees, anywhere in the world and you’ll get Phi.

Measuring people might be less reliable because clothes are a variable and you might want to keep those on for your project 😛

Source: The Da Vinci Code by Dan Brown

Dude man says

Same the only reason I’m on this site is because of da Vinci code I wanted to find more out about PHI

Edgewalker says

No you don’t. Try it. The ratio of navel distance to height isn’t phi. Try not to look for important truths about the natural world in fiction bestsellers.

Gary B Meisner says

Check your data. This isn’t based on a fictional bestseller. The navel to height ratio is very close to phi for many people, including myself. Even in his classic debunking article, Ph.D., Dr. George Markowsky measured his own family and found an average ratio of 1.63. Not everyone will have this ratio of course, but it’s not uncommon.

Dan Havemann says

Why phi,,, now I get it

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.

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.

Nishant Kapoor says

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

Theo says

PHI is the universal number for All life.

Gary Meisner 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.”

'the da vinci code'brought me here. says

totally fascinated!!!!

lammii says

same here

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.

Kaleb says

nice!

Jason says

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

Jan Cloosterman says

.03 ft? That precise a measurement on an ancient crumbling pyramid is nonsensical. It would seem these numbers are especially doctored to fit the phi number. That isn’t too say that the ancient Egyptians didn’t know of it, but rather insufficient proof of it on its own.

Alex says

Is it true that PHI is Gods building block.?

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

Gary Meisner 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.

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

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 🙂

Lakrtzmth says

Phi is amazing.But nobody in my year knows what it is.I’m 10.

Peter Hedding 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.

Robert elvis Barkley says

Very intriguing. I appreciate you sharing

Alan says

Mr Hedding i suggest you 1. Read and learn the word of God YHWH and 2. Study history a bit better there are many deceptions and lies. Suggestion 1 is your best bet because if you understood Phi you would see it proves intelligent design.

Bethany Brandon says

Mr. Alan I suggest you 1. read and investigate just where your god YHWH comes from and 2. study what those originating people knew. Suggestion 1 is your best bet because if you understood where your beliefs originate, you would see they all begin with human beings who were in touch with the transcendent long before the dualism of your patriarchal mythology narrowed everything down to a “daddy-God” who commands you to love him. (btw: when has COMMANDING love ever worked?)

Nami says

The ‘Da Vinci Code’ does wonders!!! Thank you @jonathan, it helped a lot!!

jim carlin says

Occam’s razor=be brief

over lapping ellipses=symbol for empathy

there is a good orderly direction=a safe universe

🙂

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.

Andrea says

We named my son Phi after this equation.. No we’re not math nerds or philosophers, just appreciate things for what they are and knew my son deserved a unique name, however I wanted something with meaning behind it. A friend brought this to our attention after my husband wanted to name him Fi, after a Foo Fighters song (????), but i couldnt get down with the spelling. After studying up on this and discovering that not only is phi present in almost everything throughout the universe, we also came across the part that stated that Phi is hard wired in our DNA to influence how we percieve beauty, and furthermore, that doctors use these phi proportions during cosmetic surgeries BC it is such, and knew that this was the name for him, BC in essence, Phi = beauty to the human mind.. I have only come across one other person who knew where his name came from and what Phi is (a Dr at the hospital where he was born, go figure) but I enjoy explaining it in lamens terms every time we’re asked where it came from.

Andrea says

I also thought about the irony that he was also born in March 2014.. 3/14, pi. It was definitely fate for him to receive this name.

Erik Swain says

Pi is not phi……

Cal Johnson says

awesome!!! 67 years old and learning things everyday. PHI is new and exciting for me.

j carlin says

kewl-numbers 🙂

jim carlin says

formula for the spiral

the cochlea

the chambered nautilus

a spiral staircase in New Mexico-

sea shells

the spiral modulus

we find what we know to look for–and lots more:)

Sue says

I have always wondered if Phi in nature would occur outside of our own planet, or is it a force that occurs due to earth gravitational pull, orbit.

Gary B Meisner says

It’s not limited to the Earth. The rings of Saturn are reasonably close to a phi proportion to the planet itself. It has been found in quantum solid state matter and perhaps even time. See the articles in the Cosmology section of this site at https://www.goldennumber.net/category/cosmology/.

The Wizard of π says

I found it, Mark!

https://symbiosissite.wordpress.com/2016/03/05/i-found-gods-intended-message-encoded-within-the-implied-values-of-our-numbers/

Truth says

I know all about the the Fibinacci Sequence but can anyone define how it relates to numerology if it does. I don’t know how it couldnt being everything in existence was created using numbers. Thanks ………..

jim carlin says

how creation occurs to me is a mystery

Gary B Meisner says

Yes, there are many unknowns. It seems though that we can start with a couple of logical possibilities of how creation has occurred. We’re here because of:

1) A freak chance with no design or purpose, or

2) A life force of some kind that has driven the creation of the universe and the life within it.

If there is a life force, there are two more possibilities:

1) It has somehow created life forms (e.g., you) whose intelligence, emotions and awareness surpass its own, or

2) It has created intelligent beings whose human qualities are a reflection of its own greater yet capacities.

Which choices strikes you as the most rational?

jim carlin says

#2 -G Meisner (Occam’s razor)

I finally constructed a reply-

like my experience

my truth is wordy

C S Elliot said “experience is a brutal teacher

but we learn-God do we learn”

Vinyasi says

http://goldenfed.blogspot.com/2016/06/the-euclidean-algorithm-in-expanded.html

Describes how the Golden Ratio comes in an Infinite Variety beyond the mere quadratic 1.618 which we are familiar with. The Silver Ratio is similar as it can be geometrically implied. More on this latter distinction, later….

Ada says

look up the golden ratio of the muslim kaaba or quran. you will be surprized at what you find.

Gary B Meisner says

You can read about it here: https://www.goldennumber.net/golden-ratio-of-earth/

buck Owens says

What keeps most of us in the dark when we talk about Pi, Phi, or any other equations; yet we don’t rightly secure balance our own speech. Live is temporary and subject to death & birth and Life is endless and exempt from death and birth.

It’s important first that we hearsee. Then we ahaaa moment, which we can seesee. That leads to speaksee and for some who are compelled to be impaled to teachsee. Ok, I know you like it better all cleaned up. So here it is:

Hear one, See one, Speak one, Teach one, Be one, Are one. All roads may lead to Rome, but all mathematically equations lead to “Existence is One”. And Philosphy, Science, Religion, Art, Music, Poetry,

and Knowledge to its unity leads to the same three words: “Existence is One.” So if you want it all, there it Is.

There is nothing else.

There is a lie in every belief, If you can proceed past the lie in your own BE-LIE-F, not only will you not be prejudice, but you will know who we, thee, she and he Are; The I am . . . am I. This is

immortality, welcome to “It’. If this wasn’t so, I would tell you something else.

buckowens67@gmail.com

davronbek says

Hi everyone!

Sorry,I can’t understand how phi could be central part of all things in universe?

If it is possible who can be prove by picture or sth like that please

Thanks in advance!

abdullayev-davronbek@mail.ru

Gary B Meisner says

No one who really understands the topic would claim that phi is the central part of all things in the universe. It’s not. It is, however, a number with some very unique properties, and it appears in a number of surprising places. For a clearer understanding, see my articles at https://www.goldennumber.net/golden-ratio/ and https://www.goldennumber.net/golden-ratio-myth/.

Louis Charles Morelli says

There is something deeper about Phi and pi, nobody told it yet. The number 1,618 is merely a point that points out a location in the universal formula for all natural systems, called Matrix/DNA. It happens that this location is just where the force that is responsible by replication, or reproduction of things in the system, begins to reproduce the left face of the spherical formula, building the right face. Since the reproduction results into two equal copies, it produces de bilateral symmetry. It happens for example, when the DNA separates into two strands and each one strand replicates itself, resulting two DNA’s.

For understanding it you need see the universal formula at http://theuniversalmatrix.com . This formula was found when I was making comparative anatomy between living and non-living system. Extracting the internal energy-circuity of all natural systems I observed that it was resulting all times the same pattern. Big surprise was when founding that the formula is the template behind the building blocks of galaxies and the DNA, the fundamental unit of information.

Now we will understand Phi. This formula built all natural systems and then, we found it encoded into the general light wave resulting from all seven radiations in the electromagnetic spectrum. Here I found why there is the Fibonacci sequence. When the light wave propagates creating the seven layers, each new layer is longer than the anterior, just 0,,618 longer. So, each next layer will be the size of the anterior, which is the unit 1, plus 0618. Since that each layer has its specific size, frequency and vibration, we can see how the natural forces applies different functions, dynamics, to organisms, like the different sizes and functions of the different fingers in a hand. I am discovering lots of other things now, about Phi, registering it as articles in my website. ( if you see the Matrix/DNA formula, Phi is over F5, the reproduction function.

Paul Christy says

Maybe I missed it, but I didn’t see it. I found it odd this page doesn’t have the “exact” value of phi, which is (5^1/2 + 1) / 2.

I also didn’t see any mention of phi’s reciprocal, which I find rather interesting because it has the exact same decimal value as itself (truncating the leading 1 or 0, obviously):

phi = 1.618033988749894842….

1/phi = 0.618033988749894842….

Anyway, just wanted to point out some more phi fun facts 🙂

Gary B Meisner says

This site has over 100 pages devoted to the golden ratio, so this page is just an introductory index of topics. If you click on the Mathematics link on this page it will take you to the information you mentioned. See https://www.goldennumber.net/math/