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 (See Index)

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 very unique properties in **Mathematics** and **Geometry**,

Appreciate its many appearances in **Life**, **Theology** and the **Universe**,

Apply it in **Art**, **Design**, **Beauty** or **Financial Markets**, and

Keep current with **recent news on the golden ratio**.

Share your thoughts on the Discussion Page, or on any of the posts. Try the golden ratio design app.

#### 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

## “The Golden Ratio” book – Author interview with Gary B. Meisner on New Books in Architecture

Gary B. Meisner, author of the website GoldenNumber.net and developer of PhiMatrix software, did an interview on the New Books Network (NBN) for his 2018 book, “The Golden Ratio – The Divine Beauty of Mathematics.”To listen to the podcast, click on the image above or the links below: Apple iTunes Podcast New Books Network Podcast The interview was conducted by Tricia Keffer, as host for NBN's Architecture Podcast. The podcast aired on November 23, 2019. Coincidentally, this is celebrated by some as Fibonacci Day as the date 11/23 corresponds to the first digits in the Fibonacci sequence of 1, 1, 2, 3. (Some references start with zero). NBN is in the top 2% of podcast downloads. Questions that Tricia posed in the interview included: What is the Golden Ratio, aka Phi, and why is it important to understand and study? Can you talk about some examples of divine ratio in Architecture? Can you talk about how paintings are designed? Seurat and Monet, Da Vinci? Where else can we find divine … More on this article

## “The Golden Ratio” book – Author interview with Gary B. Meisner on The Authors Show

Gary B. Meisner, author of the website GoldenNumber.net and developer of PhiMatrix software, recently did his first author interview for his 2018 book, "The Golden Ratio - The Divine Beauty of Mathematics." The interview was conducted by Danielle Hampson, Executive Producer of TheAuthorsShow.com on November 5, 2019, and appeared on November 13. It can be heard for free at this link or the image below: A fascinating topic and interview on the golden ratio: Danielle's fascination with the golden ratio led her to conduct the interview herself, rather than turning to one of her other interviewers. She covered a variety of interesting questions in an fast-paced and engaging 17 minute interview, including: Why should people other than maybe professionals like mathematicians, architects etc. even be interested in the Golden Ratio? What is the Golden Ratio, and why is it important? What do we find inside The Golden Ratio book? What motivated you to write this book, and what are your … More on this article

## Gary Meisner’s latest Tweets on the Golden Ratio

Here are the latest Tweets on the golden ratio, Fibonacci sequence and related topics by Gary Meisner. Gary's qualifications include: Founder and author of the website GoldenNumber.net, Developer of PhiMatrix Golden Ratio Design and Analysis software and Author of the book "The Golden Ratio - The Divine Beauty of Mathematics:" Tweets by GaryBMeisner … More on this article

## Dr. George Markowsky’s “Misconceptions about the Golden Ratio” Reviewed

Debunking the Debunkers of Golden Ratio Myths. Overview: This article reviews Dr. George Markowsky's article on eight misconceptions about the golden ratio. He starts his analysis by establishing a very generous ±2% range for considering evidences of the golden ratio as acceptable. In four cases though, evidences for the original claims are missed or ignored. In two cases, the measurements are taken incorrectly. In two cases, the measurements taken are too limited to be conclusive. In one case, no measurements are taken at all. The arguments are accompanied by weak or incomplete evidence, which is used to dismiss the measurements that vary from the golden ratio by less than 1%. A flawed assumption is made that there are "infinitely numbers near Phi" that could have been used instead. This is of course true in theoretical mathematics, but false with respect to physical measurements in the real world, which are limited by significant figures of accuracy. In addition, the number of … More on this article

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

## Carwow, best-looking beautiful cars and the golden ratio.

In November 2019, Samantha James of Carwow published a very interesting article titled "Best-looking cars of the last decade." What made it more interesting yet is that their review applied golden ratio analysis techniques used for human faces to the front end view of automobiles. Headlights became eyes. Emblems became noses. Grills became mouths. Makes good sense, right? I've been showing the golden ratio in human faces since the late 1990s. London plastic surgeon Dr. Julian Silva is recognized for giving a golden ratio score of famous celebrity faces, as shown below left. Carwow used a very similar approach, as shown on the right: Carwow is UK company that helps consumers to make choosing and buying their perfect car easy and enjoyable. Stefanie Finch of their PR agency, Propellernet.co.uk, contacted me on behalf of Carwow to ask if I'd be interested in covering their story on this site. I of course said "YES!" She and the Carwow team were kind enough to provide me with the … More on Art and Design

## Golden Ratio, Phi and Fibonacci Commemorative Postage Stamps

Governments around the world commemorate the Golden Ratio, Phi and Fibonacci sequence. Stamp collecting is one of the world's most popular educational hobbies, and is enjoyed by people of all ages and nationalities. The most popular stamp ever is US 1993 Elvis stamp, with over 124 million collected. So, the Golden Ratio and Fibonacci sequence may not compete with Elvis in popularity anytime soon, but they too have been recognized with commemorative stamps issued by governments around the world. In addition to adding some "phi" to "philately," they offer some interesting expressions of golden ratio concepts, as well as its application in design. Liechtenstein 2013 Commemorative Fibonacci Sequence and Phi Stamp set: The Principality of Liechtenstein, a landlocked micro-state bordered by Switzerland and Austria, issued a set of three stamps in 2013 that illustrate the Fibonacci sequence and its relationship to the golden ratio. Note that interesting presentation of concepts: … More on Art and Design

## The Golden Ratio in Character Design, Cartoons and Caricatures

The golden ratio is Stephen Silver's secret weapon of character design. Stephen Silver is a character designer and art teacher who has designed characters for Disney Television Animation, Sony Feature Animation and Nickelodeon Animation. In this role he has designed the style of the shows such as "Kim Possible", Kevin Smith's "Clerks" the animated series, "Danny Phantom" and many others. He is also the author and artist of seven books on the art of character design, caricature, sketching and life drawing. In addition to his freelance work and lectures, he owns an art school called Silver Drawing Academy in Simi Valley, CA. He has two online character design courses online at www.schoolism.com. Those are some great credentials. And what does he say is his "secret weapon" for character design? Yes, it's the golden ratio. The golden ratio is flawless because it gives the "small, medium and large" that design is built upon. He describes the golden ratio is just perfect, flawless, … More on Art and Design

## Golden ratios in Great Pyramid of Giza site topography

Overview: Analysis of the site of the Great Pyramid of Giza reveals that the positions and relative sizes of the pyramids may be based on the golden ratio. Evidence of the Golden Ratio in the Great Pyramid complex. There are many pyramid theories and questions as to who built the pyramids in ancient Egypt. It's commonly known though in Egyptology that the proportions of the Great Pyramid of Egypt are within inches of a golden ratio-based pyramid. This is discussed in detail in my article Phi, Pi and the Great Pyramid. Some say, however, that this single piece of evidence is just a simple coincidence. The primary rationale given to deny this claim is that there is no written historical evidence that the ancient Egyptians had any knowledge of the golden ratio. That's a reasonable objection, but what if the evidence showed that the application of the golden ratio at Giza was not limited to the Great Pyramid? That evidence is presented in this article. The golden ratio connection … More on Art and Design

## Michelangelo and the Art of the Golden Ratio in Design and Composition

New study unveils Michelangelo's extensive use of the golden ratio in the Sistine Chapel. In 2013, I reported that Michelangelo used the golden ratio in his painting "The Creation of Adam" in the Sistine Chapel. This Divine Proportion appears at the point at which Adam's finger is touched by the finger of God, as God breathes life into Adam. In July 2015, Clinical Anatomy published an article by a team from Brazil titled "More than a neuroanatomical representation in The Creation of Adam by Michelangelo Buonarroti, a representation of the Golden Ratio." (PDF here.) The article claimed, incorrectly, that there was no previous association of Michelangelo's works with the golden ratio. It identified the very same golden ratio touch point as the one reported here two years earlier. It also claimed, however, a new finding. It claimed that the fingers touched at the golden ratio point of the length of all the paintings on the ceiling of the Sistine Chapel. My analysis of this claim … More on Art and Design

## 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

See the YouTube video "Leonardo da Vinci's "Salvator Mundi" and the Divine Proportion." 2017 Update - Salvator Mundi, "the last Leonardo" sells for a record-breaking $450,312,500, decimating the prior world record for the most expensive work of art sold at auction. See Christie's article and learn below some of what makes this piece so incredible and valuable. 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 … 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

#### Stock Market and FOREX Trading Analysis

## Phi and Football Strategy’s Golden Ratio Rule

Using the Golden Ratio to win at football. The golden ratio is found in many places, but here's one you might not have expected: It can be found in football gaming strategy when making the "point after touchdown" decision as to whether to kick for 1 extra point or go for a 2 point conversion. This applies to the situation when a team is trailing by 14 points late in a football game, and then scores a touchdown to be trailing the other team by 8 points. So what should you do: Kick for 1 point after touchdown (PAT) or go for a 2 point conversion? Most football teams kick the PAT in this situation, but in fact it is (almost) always better to go for 2. The reason is related to … More on Markets/Gaming

## 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 Markets/Gaming

#### What determines our perceptions of beauty?

## Meisner Beauty Guide for Golden Ratio Facial Analysis

Facial golden ratios are simple, clear and abundant. Since my first investigation of human facial proportions in 1997, it seemed clear that the most important features that define a human face are the simplest and most obvious: An oval head and the positions and dimensions of the eyes and the mouth. As evidence, consider the "smiley face." It expresses a variety of facial expressions with nothing more than a circular head, two eyes and a mouth line: We immediately recognize this icon as a face, but it clearly does not represent the proportions of a human face. Let's instead express these same facial markers with normal human proportions, and with the simplest possible representation. This requires a vertical oval, two lines for the eyes and a line for the mouth. The resulting face will look something more like this: This simple facial configuration is easily mapped to very natural human proportions with the use of four simple golden ratio … More on Beauty

## How to take photos and measurements for facial beauty analysis and identification of golden ratio proportions

Best practice techniques for facial and beauty analysis. Numerous studies have been done to investigate the existence and/or relationship of golden ratio proportions in the human face. Surprising few get it right, so the results of many of these studies are inaccurate and misleading. There are four important keys to selecting photographs that are suitable for facial measurements. Next, it is critical to understand and measure the facial markers that commonly reflect golden ratio proportions. In this article I'll cover these key points, the common mistakes made and the best practices to yield results that are both accurate and meaningful. 1. Orientation of the head, and eliminating the impacts of perspective. As simple as it sounds, one of the most important aspects of any image used for facial analysis is that the subject be looking directly into the camera. The front of the face must be parallel to the lens of the camera. There must be no tilt of the head either up, down, left or … More on 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 in 2015, 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 a … 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 Marquardt Beauty Mask

A template for human beauty is found in phi, the pentagon and dodecagon Dr. Stephen R. Marquardt is the world's most recognized expert on facial analysis and beauty. He has appeared in many documentaries and been referenced in countless articles, including the Beauty episode of BBC's 2001 documentary "The Face." Dr. Marquardt, also CEO of Marquardt Aesthetic Imaging, has studied human beauty for decades in his practice of oral and maxillofacial surgery, and now continues his studies full time. His work includes cross-cultural surveys on beauty, in which he found that all groups had the same perceptions of facial beauty. He also analyzed the human face from ancient times to the modern day. Through his research, he discovered that beauty is not only related to phi, but can be defined for both genders and for all races, cultures and eras with the beauty mask which he developed and patented. This mask uses the pentagon and decagon as its foundation. Both of these geometric shapes embody … 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 included … More on Beauty

## Facial Beauty and the "New" Golden Ratio (or is it just 1.618 in disguise?)

University study declares a "new" golden ratio for facial beauty but validates Phi, the Golden Ratio, as the basis for perceptions of beauty. A university study (PDF) by Pamela M. Pallett, Stephen Link and Kang Lee at the University of Toronto and University of California, San Diego announced that a "new" golden ratio had been found in perceptions of beauty in the human face. The study also said there was "little support for the Golden Ratio" that "dates back to antiquity, when the ancient Greeks believed beauty was represented by a Golden Ratio of 1:1.618." Science Daily, in reporting on this study, quoted the one of the researchers as saying "People have tried and failed to find these ratios since antiquity" and "there was never any proof that the golden ratio was special. As it turns out, it isn't." But all that glitters isn't gold ... With this article, I think you'll agree that the search is over, the proof exists and the real Golden Ratio is still quite special. … 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 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 … 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. Several university math professors say no, but they 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 … 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 … 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 … 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

## Fibonacci 60 Repeating Pattern

The last digit of the numbers in the Fibonacci Sequence form a pattern that repeats after every 60th number: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1 This pattern can be seen in the following list of the first 72 Fibonacci … More on 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

## Powers of Phi

Phi has a unique additive relationship. The powers of phi have unusual properties in that they are related not only exponentially, but are additive as well. We know that: Phi 2 = Phi + 1 Which is the same as: Phi 2 = Phi 1 + Phi 0 And this leads to the fact that for any n: Phi n+2 = Phi n+1 + Phi n Thus each two successive powers of phi add to the next one! n Phin 0 1.000000 1 1.618034 2 2.618034 3 4.236068 4 6.854102 5 11.090170 6 17.944272 Powers of Phi and its reciprocal: Another little curiosity involves taking phi to a power and then adding or subtracting its reciprocal: For any even integer n: Phi n + 1 / Phi n = a whole number For any odd integer n: Phi n - 1 / Phi n = a whole number Examples are shown in the tables below: for n = even integers n Phin 1/Phi n Phi n + … More on Mathematics