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

• Mathematics

• Geometry

Phi appears in:

• The proportions of the human body

• The proportions of many other animals

• Plants

• DNA

• The solar system

• Art and architecture

• Music

• Population growth

• The stock market

• The Bible and in theology

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.

Phi Related Jewelry by Ka Gold Jewelry

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In the 1202 AD, Leonardo Fibonacci wrote in his book “Liber Abaci” of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi.  This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning “son of (the) Bonacci”.

Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,  233, 377 . . .

This sequence is shown in the right margin of a page in Liber Abaci, where a copy of the book is held by the Biblioteca Nazionale di Firenze. Click to enlarge.

The relationship of the Fibonacci sequence to the golden ratio is this: The ratio of each successive pair of numbers in the sequence approximates Phi (1.618. . .) , as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60. This relationship wasn’t discovered though until about 1600, when Johannes Kepler and others began to write of it.

The table below shows how the ratios of the successive numbers in the Fibonacci sequence quickly converge on Phi.  After the 40th number in the sequence, the ratio is accurate to 15 decimal places.

1.618033988749895 . . .

Compute any number in the Fibonacci Sequence easily!

Here are two ways you can use phi to compute the nth number in the Fibonacci sequence (f_n).

If you consider 0 in the Fibonacci sequence to correspond to n = 0, use this formula:

f_n =  Phi ⁿ / 5^½

Perhaps a better way is to consider 0 in the Fibonacci sequence to correspond to the 1st Fibonacci number where n = 1 for 0.  Then you can use this formula, discovered and contributed by Jordan Malachi Dant in April 2005:

f_n =  Phi ⁿ / (Phi + 2)

Both approaches represent limits which always round to the correct Fibonacci number and approach the actual Fibonacci number as n increases.

The ratio of successive Fibonacci numbers converges on phi

Tawfik Mohammed notes that 13, considered by some to be an unlucky number, is found at position number 7, the lucky number!

The Fibonacci Sequence and Gambling or Lotteries

Some people hope that Fibonacci numbers will provide an edge in picking lottery numbers or bets in gambling. The truth is that the outcomes of games of chance are determined by random outcomes and have no special connection to Fibonacci numbers.

There are, however, betting systems used to manage the way bets are placed, and the Fibonacci system based on the Fibonacci sequence is a variation on the Martingale progression. In this system, often used for casino and online roulette, the pattern of bets placed follows a Fibonacci progression: i.e., each wager should be the sum of the previous two wagers until a win is made. If a number wins, the bet goes back two numbers in the sequence because their sum was equal to the previous bet.

In the Fibonacci system the bets stay lower then a Martingale Progression, which doubles up every time. The downside is that in the Fibonacci roulette system the bet does not cover all of the losses in a bad streak.

An important caution: Betting systems do not alter the fundamental odds of a game, which are always in favor of the casino or the lottery. They may just be useful in making the playing of bets more methodical, as illustrated in the example below:

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GoldenNumber.Net explores the appearance of Phi, 1.618 (also known as the Golden Ratio, Golden Mean, Golden Section or Divine Proportion, in mathematics, geometry, life and the universe and shows you how to apply it, and its applications are limitless:

• Art
• Architecture
• Design of any kind – Graphics, logos, products, fashion, web sites and more
• Photo composition, photo cropping matting and framing
• Personal beauty and facial or dental cosmetic procedures to enhance beauty
• Stock market and FOREX analysis

The Golden Section is a ratio based on a the number Phi, 1.618…

The Golden Section or Ratio is is a ratio or proportion defined by the number Phi (= 1.618033988749895… )

It can be derived with a number of geometric constructions, each of which divides a line segment at the unique point where:

the ratio of the whole line (A) to the large segment (B)

is the same as

the ratio of the large segment (B) to the small segment (C).

In other words, A is to B as B is to C.

This occurs only where A is 1.618 … times B and B is 1.618 … times C.

This ratio has been used by mankind for centuries

The Greeks recognized it as “dividing a line in the extreme and mean ratio”	The Renaissance artists knew it as the Divine Proportion
and used it for beauty and balance in the design of architecture, perhaps as early as the Parthenon	and used it for beauty and balance in the design of art

It appears in the design of Notre Dame in Paris

and continues today in many examples of art, architecture and design.

It also appears in the physical proportions of the human body, movements in the stock market and many other aspects of life and the universe.

Dr. Stephen Marquardt has discovered a template for human beauty using the Golden Section, with obvious relevance in the plastic surgery industry. Marquardt’s analysis takes ethnicity into account, and illustrates variations that are are both numerous and subtle.

See the other sections and pages of this site listed above and to the right for a broad sampling of the many appearances and applications of the golden ratio.

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“The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.”
–Einstein, Albert (1879-1955), What I Believe.

“When one sees eternity in things that pass away, then one has pure knowledge.”
–BHAGAVAD GITA

“Without mathematics there is no art.”
–Luca Pacioli

“Like God, the Divine Proportion is always similar to itself.”
–Fra Luca Pacioli

“The good, of course, is always beautiful, and the beautiful never lacks proportion.”
–Plato

“Measure what is measurable, and make measurable what is not so.”
–Galilei, Galileo (1564 – 1642), Quoted in H. Weyl “Mathematics and the Laws of Nature” in I Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.

“[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.”
–Galilei, Galileo (1564 – 1642), Opere Il Saggiatore p. 171.

“The human mind has first to construct forms, independently, before we can find them in things.”
–Einstein, Albert (1879-1955)

“Nature hides her secrets because of her essential loftiness, but not by means of ruse.”
–Einstein, Albert (1879-1955)

“Where there is matter, there is geometry.”
–Kepler, Johannes (1571-1630), (Ubi materia, ibi geometria.) J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990

“Mathematics seems to endow one with something like a new sense.”
–Darwin, Charles, In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

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There are millions of places to find phi, but here’s some help in finding phi to a million places.  You can download the PhiCalculator, a program provided free of charge by Alireza Shafaei.  It will compute phi to the number of decimal places you specify, to 1 million and maybe more, and output the results to a text file.  He’s also provided the C++ Source Code and a text file of Phi to 100,000 places.

Note:  PhiCalculator was scanned with McAfee as of March 29, 2008 and found to be virus free but the user assumes all risks of use.

For those of you with smaller appetites, here it is to 20,000 places:

Did you find what you were looking for?

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Φ, the Greek symbol for phi, represents the number 1.61803398874989…, which is intrinsically related through math and geometry to the number 5.

V is the Roman symbol which represents the number 5.

So when does Φ + V = 5?

All you mathematicians are probably rushing to your calculators, knowing that the answer must be when V = 5 – Φ, or 3.38196601125011…

But this is a riddle, not an equation.

The answer is:  “When you express it phonetically rather than mathematically.”

Just say “phi” and add the “v” sound to get “phive,” or five!

Curious that these numbers should all be related phonetically and mathematically, isn’t it?

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The generally accepted pronunciation of phi is fi, like fly.

Most people know phi as “fi,” to rhyme with fly, as its pronounced in “Phi Beta Kappa.”  In Dan Brown’s best selling book “The Da Vinci Code,” however, phi is said to be pronounced fe, like fee.

The following is offered in response to the questions received on phi’s correct pronunciation:

Dictionairies either list fi as the only pronunciation for phi or, if both fi and fe are listed, as the primary pronunciation.  See listings at Merriam-Webster and Dictionary.com.

Leading authors on the subject of phi offered the following comments:

• Two in the USA and UK confirmed that fi is the preferred pronunciation.
• One noted that in the UK “phi” was always pronounced to rhyme with “pie” but that some Americans at conferences pronounced it “fee”.
• Another noted that in Greek the letter PHI is indeed pronounced PHEE.  However, in Greek the letter we call PI is also pronounced PEE.  Consequently, depending on whether you want to adopt the Greek or American pronunciation you can pronounce it as PHEE or PHI.  In mathematical circles, the letter used for the Golden Ratio is normally TAU.

To complicate matters, when used in connection with fraternities and sororities, the usage varies as well and it is pronounced PHEE when it comes after a vowel, as in Alpha Phi.

My Greek phriend Tassos Spiliotopoulos offers the following:  The letters of the Greek alphabet are written as words and not as single letters, for example the first letter A is written AΛΦA and sounds like Alpha.  When it comes to letters like Π, Χ, Φ (written ΠI, ΧI and ΦI respectively), the misunderstanding comes from the pronunciation of the letter ‘I’ which in English rhymes with fly but in Greek is pronounced EE. The letter Φ is always pronounced PHEE in Greek, and it does not differ if followed by a vowel or a consonant.

So there you have it.  While a linguistic purist might opt for the original Greek fee, most mathematicians know phi as fi.  Either is correct, but if we want to be consistent with the common usage of pronouncing pi as pie, we would then pronounce phi as fi.

Or as the lyrics of the song say, “PotAto, potAHto, tomAto, tomAHto, let’s call the whole thing off.”

Many thanks to Dr. Mario Livio (author of The Golden Ratio), Dr. Ron Knott (author of Fibonacci Numbers and the Golden Section), Steve McIntosh (author of The Golden Mean and President of Now & Zen) and Dr. Eddy Levin (inventor of the Golden Mean Gauge) for their input, and to Geni Flowers for inspiring me to get the answer.

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In the texts of ancient Greece, the letter phi looked like this:

Φ

When you see the Greek letter Phi on a fraternity or sorority house, it usually looks like this:

Φ

When you see Phi on a web site, it often looks like this:

Ø

What’s the slant on this?

Is Phi no longer the upright character it once was?  Has Phi become an empty shell of its former self? (A little set humor for you mathematicians.) Is Phi leaning to the right in its political orientation?  To keep Phi from suffering from an “identity” crisis (a little more math humor), here’s an explanation of what’s going and what you can do to be sure that Phi remains in good standing.

Just like golf, it’s all in the stroke

The simple truth is that the basic Western character set on computers does not include a character for the Greek letter phi.  The only basic ASCII character that comes close in appearance to Phi Φ is the letter O with a stroke through it, or Ø.  As a result, Ø has been masquerading as Φ since the early days of computer usage.

Type a real Φ on your keyboard with Alt-1000

Now that extended character sets are available on most PC’s and in most browsers, it’s possible once again to let Phi be Phi.  All you have to do is hold the Alt key and then enter 1000 on the number pad.

If your PC doesn’t have the necessary character sets installed to do this, you can use Windows’s Character Map program.  To open Character Map, click Start, point to All Programs, point to Accessories, point to System Tools, and then click Character Map.  Scroll down to find the phi symbol, click on select, then copy and paste it into your application.

Letting Φgønes be Φgønes

And while change is always bound to cause some dispute, in the end it’s better to let Φgønes be Φgønes.

Symbol font and Phi:  Alt-618 gives … j, which is phi, 0.618!

On a PC using Symbol font, you can generate a phi symbol in the following ways:

Appropriately enough, a lower case phi, or 0.618, and the reciprocal of Phi, 1.618, can be created with Alt-618: j

Other phi symbols can be created with:

Alt-70:  F

Alt-102:  f

Alt-106 or Alt-618:  j

Alt-232: Φ

Alt-237: φ

Note:  Alt-618 means hold down the Alt key, enter 618 on the numeric pad and then release.  This insight was contributed by W. Nathan Saunders.

Running a character check on Phi

On Windows keyboards, you can

This page is dedicated to Katie (a.k.a. Princess Kate), a high school senior who wrote to question the usage of Ø and who inspired me to dig deeper into the reasons that the Φ symbol wasn’t being used much on web sites … until now.  (3/15/2003, The “Phides” of March, a date made from the Fibonacci series numbers of  0, 1, 2, 3 and 5)

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Uses in architecture potentially date to the ancient Egyptians and Greeks

It appears that the Egyptians may have used both pi and phi in the design of the Great Pyramids. The Greeks are thought by some to have based the design of the Parthenon on this proportion, but this is subject to some conjecture.

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

Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden ratio to be the most binding of all mathematical relationships and the key to the physics of the cosmos. It was not known as the golden ratio in his time, but he describes it with his first reference to proportion:

“Now it is not possible for two things to be combined well on their own without a third, for some bond is required between the two to draw them together. The very best bond is that which, as much as possible, makes itself and the conjoined entities, one; and it is proportion that by nature best accomplishes this. So  whenever the middle item of three numbers or volumes or powers is such that the first is to the middle as the middle is to the last, and again, that the last is to the middle as the middle is to the first, then the middle becomes first and last, and the last and first for their part both become middles. Accordingly it follows, of necessity, that they all turn out to be the same, and since they have all become the same as one another, they will all be one.” Translation © 2021 by David Horan

This is analogous to what Euclid later wrote.

Euclid (365 BC – 300 BC), in “Elements,” referred to dividing a line at the 0.6180399… point as “dividing a line in the extreme and mean ratio.” This later gave rise to the use of the term mean in the golden mean.  He also linked this number to the construction of a pentagram.

The Fibonacci Sequence was written of in India in about 200-300 BC and brought to the Western world around 1200 AD

What we now as the Fibonacci sequence is named after Leonardo Pisano Bonacci (aka Bigollo) of Pisa, an Italian born in 1175 AD, who later became known as Leonardo Fibonacci. His book Liber Abaci, published in 1202, introduced this sequence to Western European mathematics in the form of a math problem on the breeding of rabbits. He learned of it though while being educated in North Africa with an Arab master, where he was exposed to the much earlier knowledge of Indian mathematicians. Liber Abaci became a pivotal influence in adoption by the Europeans of the Arabic decimal system of counting over Roman numerals. (3)

The sequence itself though had been described as early as the 2nd or 3rd century BC in the works of Acharya Pingala, an Indian mathematician who enumerated the possible patterns of Sanskrit poetry that could be formed from syllables of two lengths.

The contributions of Pingala and Fibonacci are important, but it’s not apparent that anyone even realized its connection to the Golden Ratio until the 1600’s by Johannes Kepler and others.

It was first called the “Divine Proportion” in the 1500’s

Leonardo Da Vinci provided illustrations for a dissertation published by Luca Pacioli in 1509 entitled “De Divina Proportione” (1), perhaps the earliest reference in literature to another of its names, the “Divine Proportion.”  This book contains drawings made by Leonardo da Vinci of the five Platonic solids.

The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty. 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 “Golden Ratio” was coined in the 1800’s

It is believed that Martin Ohm (1792–1872) was the first person to use the term “golden” to describe the golden ratio. to use the term. In 1815, he published “Die reine Elementar-Mathematik” (The Pure Elementary Mathematics). This book is famed for containing the first known usage of the term “goldener schnitt” (golden section).

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

It wasn’t until the 1900’s that American mathematician Mark Barr used the Greek letter phi (Φ) to designate this proportion. This appeared in the “The Curves of Life” (page 420) in 1914 by Theodore Andrea Cook . By this time this ubiquitous proportion was known as the golden mean, golden section and golden ratio as well as the Divine proportion.  Phi is the first letter of Phidias (1), who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter “F,” the first letter of Fibonacci.  Phi is also the 21st letter of the Greek alphabet, and 21 is one of numbers in the Fibonacci series.  The character for phi also has some interesting theological implications.

Recent appearances of Phi in math and physics

Phi continues to open new doors in our understanding of life and the universe.  It appeared in Roger Penrose’s discovery in the 1970’s of “Penrose Tiles,” which first allowed surfaces to be tiled in five-fold symmetry.  It appeared again in the 1980’s in quasi-crystals, a newly discovered form of matter.

Phi as a door to understanding life

The description of this proportion as Golden and Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life.  That’s an incredible role for a single number to play, but then again this one number has played an incredible role in human history and in the universe at large.

Source – The Divine Proportion : A Study in Mathematical Beauty by H. E. Huntley

(1) Page 25
(2) Page 157
(3) Page 158

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