What makes a single number so interesting that ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it? It's a number that goes by many names. This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section. It was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in "De Divina Proportione" in 1509, by Johannes Kepler around 1600 and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.” With the movie release of the “The Da Vinci Code”, the quest to know Phi was brought even more into the mainstream of pop culture. The allure of “The Da Vinci Code” was that it creatively integrated fiction with both fact and myth from art, history, theology and mathematics, leaving the reader never really knowing what was truth and … More on Math, Myth and Truth

# The Golden Ratio: Phi, 1.618

This site is dedicated to sharing the best information on Phi, the number 1.618, with insights from dozens of Contributors on a broad range of topics (See Site Map). We're here to help you understand its unique properties in mathematics and geometry, appreciate its many appearances in life and the universe, and benefit by applying it in art, design, beauty, financial markets and more. Share your thoughts on the Discussion Page or on any of the article posts. Enjoy the "phinomenon!"#### Featured Articles: Math or Myth? An overview and the evidence

## Golden Ratio Myth, Fact and Misunderstanding: The missing evidence

There are many misconceptions and misrepresentations about the golden ratio. Some look too fervently for patterns and say it exists where it really doesn't. Some whose goal is to debunk golden ratio myth say it doesn't exist where it really does, missing the obvious and often not stating what proportions appear instead. People on both sides often just repeat what they've heard rather than personally performing the analysis required to support their conclusions. Intelligence and education are not always factors in getting to the truth, as even Ph.D.'s in mathematics sometimes get it wrong. As the author of this site since 1997, I've changed my views and the information on this site as well. Let's look at some of the common points of confusion and debate, covering beauty, the Parthenon, the UN Secretariat Building, the Great Pyramid, Nautilus shell, use by famous artists (Da Vinci, Botticelli, Seurat, etc.) and other topics. I'll provide objective answers, with additional evidence in the … More on Math, Myth and Truth

#### Latest blog articles

## Another FastOne by FastCoDesign on the Golden Ratio?

Was it routine site maintenance or censorship at FastCoDesign? Why delete reader comments? On April 13, 2015, FastCoDesign.com took a stand on design with an article by John Brownlee titled “The Golden Ratio: Design’s Biggest Myth – The Golden Ratio is Total Nonsense in Design. Here’s Why.” (Find article here.) By the end of June 2015, the article had hundreds of comments from readers, almost entirely negative, as readers expressed their anger, disappointment and frustration at the ignorance, inaccuracies and bias in the poorly researched and lamely written article. And then all the comments just disappeared! Descriptions of the article used by readers included "appalling, sensationalist, dangerous, stupid, under-informed nonsense, an exercise in ignorance and conceit, seething with misinformation, misleading, utter nonsense, profound ignorance, lame, entirely incorrect, click bait, what a troll, simplistic, naive, puerile, opinionated, unsophisticated, boring, fallacious, … More on this article

## Fast Company Design, John Brownlee and the Golden Ratio “Myth” in Design

Is D for Design or Deception? In April 2015, FastCoDesign.com published an article by John Brownlee titled "The Golden Ratio: Design's Biggest Myth - The Golden Ratio is Total Nonsense in Design. Here's Why." (Find article here.) The article, unfortunately, is filled with bias and errors, but immediately ranked on the top page of Google results for searches on "Golden Ratio." Why? Because Google places a high value on the assumed credibility of a site. Fastcodesign ranks #2,295 of all US sites on Alexa while GoldenNumber.net ranks #93,094. Unfortunately, "credibility" is not always the same as reliability, accuracy and truthfulness. Here is my review of the many inaccuracies in the article. Start with an upside-down Golden Spiral on the Parthenon First, the top of the article shows the Parthenon with the caption "Is the Parthenon designed after the Golden Ratio? NOPE!" As proof, we're shown a golden spiral that is drawn upside-down. It's also oversized, as shown by the red … More on this article

## Common Core Curriculum Math Standards for the Golden Ratio

Welcome educators, teachers, parents and students to the Golden Ratio Academy page. The Golden Ratio appears not only in nature and the arts, but also in the Common Core Curriculum mathematics standards. This page is dedicated to educators, parents and students seeking the best, most accurate information on the golden ratio. You can return to this page by simply going to the easily remembered link of goldenratio.academy in your browser. Separating Math and Myth: Teaching Golden Ratio Fact, not Fiction. Much information is available on the golden ratio from a variety of sources, but it's amazing how much of it is incorrect. It perpetuates like a myth or urban legend. Even content produced by Ph.D's in mathematics and educational sources such as the Discovery Channel contains errors. My goal is to provide you with the resources to provide our students with the best information available on this fascinating topic. As background, I've been researching and writing on this topic … More on this article

#### Applications for Better Art, Design and Composition

## Google Logo and the Golden Ratio in Design

New Google logo design finds visual harmony using the Golden Ratio. Google's design follows in the footsteps of Leonardo da Vinci and other masters When Luca Pacioli published "The Divine Proportion" in 1509 (with illustrations by Leonardo da Vinci), he described his work on this "golden ratio" of 1.618 as a "very delicate, subtle and admirable teaching" that would "delight in diverse questions touching on a very secret science." Johannes Kepler later called it "a precious jewel" of geometry. The designers at Google have apparently found its value too, as we see when we study and appreciate the underlying design of Google's new logo, iconic G, the microphone icon and even the layout of the Google search page. This is the kind of thoughtful design work that follows in the footsteps of Leonardo, Michelangelo, Raphael, Botticelli, Seurat, Le Corbusier and other masters of design, and that would make Pacioli proud. Here's a version without the arrows for a clearer … More on Art and Design

## Raphael and the Golden Ratio in Renaissance Art

Raphael was one of three Master artists of the Renaissance Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance and lived from 1483 to 1520. He is recognized as one of the three great masters of that period, accompanied by Michelangelo and Leonardo da Vinci. His work is admired for its form, composition, and visual achievement of the ideal of human grandeur. One of his most famous works is The School of Athens, a fresco in the Apostolic Palace in the Vatican. It captures the spirit of the Renaissance, and is revered as his masterpiece. It was painted between 1509 and 1511. The School of Athens: Inspired by a union of art and mathematics It was also in 1509 that Luca Pacioli published the book De Divina Proportione (The Divine Proportion), with illustrations by Leonardo da Vinci. MonaLisa.org reports that The School of Athens "incorporates many of the mathematical theories of Luca and Leonardo." "Civilisation" author Kenneth … More on Art and Design

## Leonardo Da Vinci, Salvator Mundi and the Divine Proportion

In 2011, the discovery of a lost painting by Leonardo da Vinci was announced to the world. This painting, Salvator Mundi, had been in the art collection of King Charles I of England in 1649, was auctioned in 1763 and then lost for many years. Its recovery was led by Robert Simon, an art historian and private art dealer, and it was restored by Dianne Dwyer Modestini. Many unique qualities of this painting led experts to confirm that it is indeed an original work of Leonardo da Vinci, one of only fifteen now in existence. Da Vinci and the Golden Ratio in Art Composition. A lesson for all artists and designers. This article reveals new insights into da Vinci’s genius that can be found in this painting. Da Vinci created the illustrations for the book “The Divine Proportion” written by his contemporary, Luca Pacioli. He used the Divine proportion, also known as the golden ratio, in his composition of earlier paintings, including The Annunciation in about 1473 and The Last Supper in 1495. … More on Art and Design

## Da Vinci and the Divine Proportion in Art Composition

Leonardo Da Vinci has long been associated with the golden ratio. This association was reinforced in popular culture in 2003 by Dan Brown's best selling book "The Da Vinci Code." The plot has pivotal clues involving the golden ratio and Fibonacci series. In 2006, the public awareness of the association grew when the book was turned into a movie starring veteran actor Tom Hanks. Da Vinci's association with the golden ratio, known in his time as the Divine proportion, runs much longer and deeper. Da Vinci's illustrations appear in Pacioli's book "The Divine Proportion" Da Vinci created the illustrations for the book "De Divina Proportione" (The Divine Proportion) by Luca Pacioli. It was written in about 1497 and first published in 1509. Pacioli was a contemporary of Da Vinci's, and the book contains dozens of beautiful illustrations of three-dimensional geometric solids and templates for script letters in calligraphy. The original manuscript can be viewed online … More on Art and Design

## The UN Secretariat Building, Le Corbusier and the Golden Ratio

Some claim that the design of the United Nations headquarters building in New York City exemplifies the application of the golden ratio in architecture. Debunkers of the golden ratio say no, that this is just another groundless myth to be dispelled. Let's look at the history of its design, the sources of the claims and mathematics of the dimensions. Perhaps we can come to a solid conclusion that can be agreed upon by all. A lead architect of the UN Building, Le Corbusier, created a system of design based on the golden ratio The building, known as the UN Secretariat Building, was started in 1947 and completed in 1952. The architects for the building were Oscar Niemeyer of Brazil and and the Swiss/French architect Le Corbusier. On Corbusier, Wikipedia states: Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. Explaining the Modulor, Wikipedia states: Le Corbusier developed the Modulor in the long tradition of Vitruvius, … More on Art and Design

## Logo Design using the Golden Ratio

Phi is often applied in product logos. From Renaissance artists of the 1500's to graphic artists of today, phi is recognized for its ability to give a sense of aesthetic appeal in balance and harmony of design. Product logos represent an image that must make a positive and memorable impact on the conscious and subconscious minds of consumers, so it is no surprise to find phi proportions in many logo design of major companies. The Phi grid proportions are provided by PhiMatrix software. See other examples of logo design at the PhiMatrix site. Note: The logos presented on this page are for illustration purposes only of principles of graphic design and are do not imply in any way an endorsement of, or affiliation with, this web site or its affiliate web sites by the companies shown. Note how every dimensions of each letter of this logo is apparently based on proportions of phi (first golden ratio) or phi squared (second golden … More on Art and Design

## Product Design and Marketing applications of the Golden Mean

Phi is used in the design of many consumer products. Phi has been used to bring beauty, balance and harmony to some of the world's greatest art and architecture. It is also used to add style and appeal in the marketing and design of everyday consumer products. The applications are endless, but are illustrated by a few of the products below. Click on the image for a larger version. The gauge shown here was developed by Dr. Eddy Levin and is offered for sale on this site. Photos with the gauges are also courtesy of Dr. Levin. Photos with golden ratio grid lines were created with PhiMatrix software. For another example, note the dimensions of the classic Hewlett Packard HP12C Financial calculator. The official dimensions on the HP site are 5” x 3.1” which has a ratio of 1.6129: 5” x 3” would have been the simplest, most obvious English measure dimensions. That would have resulted in a width to height ratio of 1.6667. Dimensions of 5” x 3.2” would have resulted in a … More on Art and Design

## Aston Martin, James Bond and the Golden Ratio

What do James Bond, Aston Martin and the Golden Ratio have in common? James Bond, also known as 007, drove an Aston Martin DB5 in the movies GoldFinger and GoldenEye, and Aston Martin is now boasting its application of the Golden Ratio in the design of its latest DB9 and Rapide S automobiles. The Aston Martin Rapide S is described as: "Breathtaking Proportions - The ‘Golden Ratio’ sits at the heart of every Aston Martin. Balanced from any angle, each exterior line of Rapide S works in concert and every proportion is precisely measured to create a lithe, pure form. Our engineering follows the same principle. A near perfect weight distribution ensures Rapide S is balanced in form and balanced in function." The Aston Martin DB9 is described as: "Perfectly Proportioned - Every inch of DB9's form is designed for elegance and balance. The simple beauty of nature guides the design of DB9, with the 'golden ratio' setting all proportions. The result is a profile where every line, dimension … More on Art and Design

#### Stock Market and FOREX Trading Analysis

## Stock Market Analysis, Phi and the Fibonacci Sequence

Human expectations occur in a ratio that approaches Phi. Changes in stock prices largely reflect human opinions, valuations and expectations. A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that approaches phi, with 61.8% positive and 38.2% negative. Phi and Fibonacci numbers are used to predict stocks Phi (1.618), the Golden Mean and the numbers of the Fibonacci series (0, 1, 1, 2, 3, 5, 8, ...) have been used with great success to analyze and predict stock market moves, known as retracements. Forbes ASAP featured a story on the work of scientist Stephen Wolfram in cellular … More on Trading

## Foreign Exchange Markets (FOREX) and Trading

Phi relationships appear in foreign currency price movements. It has long been known that phi and Fibonacci relationships appear in the stock markets. The foreign exchange market, or Forex, is the largest market in the world. The simplest definition of foreign exchange is the changing of one currency to another. Since there are no touchable commodities affected, this is the most prominent and most liquid financial exchange anywhere. In comparison to the daily trading volume averages of $300 billion in the U.S. Treasury Bond market and the less than $10 billion exchanged in the U.S. stock markets, the Forex market often averages $3.5 trillion exchanged daily. The Forex Market is open 24 … More on Trading

#### What determines our perceptions of beauty?

## The Golden Ratio, Beauty and Design: It’s time to face the facts.

Another attempt to debunk the Golden Ratio is fundamentally wrong. In October 2015, Dr. Keith Devlin was interviewed by science writer Julia Calderone in an article titled "The one formula that's supposed to 'prove beauty' is fundamentally wrong." Professor Devlin is a accomplished Ph.D. in mathematics at Stanford University and NPR's Math Guy. His comments have reach and impact, but his views on the golden ratio are not always accurate, complete, consistent, objective or supported by evidence. Earlier this year, he was featured by a leading online design magazine in an article titled "The Golden Ratio: Design’s Biggest Myth – The Golden Ratio is Total Nonsense in Design. Here’s Why." The article drew hundreds of negative responses (which were later censored and deleted by the magazine) because of its inaccuracies and misrepresentations. A highly regarded NYC designer, Darrin Crescenzi, described the article as "dangerous" and "under-informed nonsense," motivating him to write … More on Beauty

## Celebrity Faces and the Golden Ratio: The real story.

Do these celebrities owe their good looks to the Golden Ratio? Click on images for full resolution. When a joke becomes science, then science becomes a joke. A designer in Russia morphed several celebrity faces as a joke, but it somehow morphed into "scientific evidence" in yet another attempt to debunk the golden ratio. This time it was on Wired.com, a major online magazine site. (Find article here.) Igor Kochmala created some fun images of celebrity faces that he morphed to fit nonsensical patterns of golden ratio spirals. His blog page, called "Yet-another-Fibonacci-homework," humorously featured his "Dr. Fibonazi's Plastic Surgery Clinic." There he transformed five celebrity faces and the Mona Lisa to goofy looking images to supposedly "bring natural harmony" to their faces, as shown below. Click on any image for full resolution. The blog post was picked up by dozens of online magazine sites. Smash.com described the images as "hilarious distortions." … More on Beauty

## The World’s Most Perfect Face: Joan Smalls? Elle says Yes! Golden Ratio says …

In 2014, Elle Magazine announced “Meet the very real woman behind the planet’s most perfect face: It’s Joan Smalls. Supermodels, over? As if. Joan Smalls wields the kind of flawless, unequivocal beauty that is both a throwback to the past and a sign that fashion is finally facing the future.” The Elle article describes Joan's background and rise to supermodel status, saying, "The hair is long and straight, the bone structure aquiline, the smile dazzling. No wonder Smalls is kicking up the kind of fashion-industry excitement that comes along only a handful of times each decade. Over the past year, she has snagged the number-one spot on Models.com, having pulled in a rumored $3.5 million—making her the eighth-highest-paid model in the world, according to Forbes." (Follow her on Facebook, Instagram and Twitter.) Is "flawless, unequivocal beauty" marketing hype or mathematical fact? LA Times reporter Rene Lynch was intrigued, but wanted to do some investigative reporting on this … More on Beauty

## Facial Analysis and the Beauty Mask

"Beauty is in the phi of the beholder." It has long been said that beauty is in the eye of the beholder and thought that beauty varies by race, culture or era. The evidence, however, shows that our perception of physical beauty is hard wired into our being and based on how closely the features of one's face reflect phi in their proportions. The Golden Ratio appears extensively in the human face, as demonstrated in a 2009 university study on attractiveness and as illustrated by the video below of Florence Colgate, Britain's "Most Perfect Face" of 2012, : http://www.youtube.com/watch?v=kKWV-uU_SoI The image analysis shown in the video was done with PhiMatrix Golden Ratio Design and Analysis Software But let's take a deeper look yet at beauty based on analysis of the evidence. A template for human beauty is found in phi and the pentagon Dr. Stephen Marquardt has studied human beauty for years in his practice of oral and maxillofacial surgery. Dr. Marquardt performed … More on Beauty

## Beauty, the perfect face and the Golden Ratio, featuring Florence Colgate

What determines the beauty of a perfect face? We may never answer the question "how many angels can dance on the point of a needle," but how about this one: "How many Divine proportions ratios are there in the face of an angel?" The YouTube video and information in this article may give some insight into the answer, with details on the contest, a recap of explanations of beauty that fall short and new insights and a video illustrating the impact of the golden ratio in beauty. The British contest in 2012 to find "Britain's Perfect Face" draws over 8,000 entries, and finds Florence Colgate as the winner British company Lorraine Cosmetics sponsored a contest in 2012 to find “Britain’s Perfect Face.” Billed as the "Naked Competition," and seeking the most perfect makeup-free face, contestants were required to have completely natural faces, without make-up, botox or cosmetic surgery. The competition drew entries from 8,045 contestants, who were initially screened by a panel that … More on Beauty

#### What is Phi?

## 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. 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 … More on Phi

## What is the Fibonacci Sequence (aka Fibonacci Series)?

Leonardo Fibonacci discovered the sequence which converges on phi. In the 12th century, Leonardo Fibonacci wrote in 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. 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, . . . 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. 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 … More on Phi

## The Golden Section / Golden Ratio

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 … More on Phi

#### Appearances in Life and Nature

## Is the Nautilus shell spiral a golden spiral?

Nautilus shell spirals may have phi proportions, but not as you may have heard. The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. Some say yes, but offer no proof at all. Some show examples of spirals, but incorrectly assume that every equi-angular spiral in nature is a golden spiral. One university math professor says no, but only compared the nautilus spiral to the spiral created from a golden rectangle. Another university professor says no, but only measured height and width of the entire shell. Let's look at this objectively and solve this mystery and debate. The Golden Spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral. A traditional Golden Spiral is formed by the nesting of Golden Rectangles with a Golden Rectangle. This resulting Golden Spiral is often associated with the … More on Nature

#### Appearances in the Cosmos

## Stars that Pulsate to the Golden Ratio

A research article called "Strange Nonchaotic Stars" published at the American Physical Society reported the discovery of a class of white-blue variable stars that pulsate in a fractal pattern at frequencies close to the golden ratio. Download the paper in PDF format here: strange-nonchaotic-stars-pulsate-golden-ratio-1501.01747v2 Pulsations described as "strange nonchaotic behavior." The discovery was reported by John F. Lindner, Vivek Kohar, Behnam Kia, Michael Hippke, John G. Learned, and William L. Ditto. The analysis was perfomed by John Lindner at the University of Hawaii and his colleagues. The stars may be the first example in nature of what chaos theorists call a “strange nonchaotic attractor,” a system that has fractal structure, but not the sensitivity to initial conditions of a chaotic system like the weather." The fractal pattern was described as “strange.” "Nonchaotic" means the pattern is orderly rather than random. Most fractal patterns in nature, such as weather, are … More on the Cosmos

## Phi and the Solar System

The dimensions of the Earth and Moon are in Phi relationship, forming a Triangle based on 1.618. The illustration shows the relative sizes of the Earth and the Moon to scale. Draw a radius of the Earth (1). Draw a line from the center point of the Earth to the center point of the Moon (square root of Phi). Draw a line to connect the two lines to form a Golden Triangle (Phi). Using dimensions from Wikipedia and geometry's classic Pythagorean Theorem, this is expressed mathematically as follows: Dimension (km) Proportion (Earth=1) Mathematical Expression Radius of Earth 6,378.10 1.000 A Radius of Moon 1,735.97 0.272 Earth + Moon 8,114.07 1.272 B Hypotenuse 10,320.77 1.618 (Φ) C Hypotenuse / (Earth Radius + Moon Radius) 1.618 (Φ) A²+B²=C² This geometric construction is the same as that which appears to have been used in the construction of the Great Pyramid of Egypt. Source: Hidden Nature by Alick … More on the Cosmos

#### Unique Properties in Geometry

## Phi and Geometry

Phi (Φ) was described by Johannes Kepler as one of the "two great treasures of geometry." (The other is the Theorem of Pythagoras.) Phi appears in many basic geometric constructions. 3 lines: Take 3 equal lines. Lay the 2nd line against the midpoint of the 1st. Lay the 3rd line against the midpoint of the 2nd. The ratio of AG to AB is Phi, the Golden Ratio. (Contributed by Jo Niemeyer) 3 sides: Triangle Insert an equilateral triangle inside a circle, add a line at the midpoint of the two sides and extend that line to the circle. The ratio of AG to AB is Phi. 4 sides: Square Insert a square inside a semi-circle. The ratio of AG to AB is Phi. 5 sides: Pentagon Insert a pentagon inside a circle. Connect three of the five points to cut one line into three sections. The ratio of AG to AB is Phi. When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry shown below creating … More on Geometry

#### Unique Properties in Mathematics

## Mathematics of Phi, 1.618, the Golden Number

Phi, Φ, 1.618…, has two properties that make it unique among all numbers. If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or Φ² = Φ + 1. If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or 1 / Φ = Φ - 1. These relationships are derived from the dividing a line at its golden section point, the point at which the ratio of the line (A) to the larger section (B) is the same as the ratio of the larger section (B) to the smaller section (C). This relationship is expressed mathematically as: A = B + C, and A / B = B / C. Solving for A, which on both sides give us this: B + C = B²/C Let's say that C is 1 so we can determine the relative dimensions of the line segments. Now we simply have this: B + 1 = B² This can be rearranged as: B² - B - 1 = 0 Note the various ways that this equation can be rearranged to express the relationship of the line segments, and also Phi's unique … More on Mathematics