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
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
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 reasonable ±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
Discovery of the Golden Ratio in Michelangelo’s “The Creation of Adam.”
2015 study announces the discovery of the Golden Ratio in the Sistene Chapel. Clinical Anatomy published an article on July 17, 2015 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 excitement of the authors in making this discovery for themselves is understandable, and their creativity is commendable. The discovery though was not entirely new, and it does not appear to be entirely accurate, nor complete. The Brazilian study though motivated me to take a deeper look at Michelangelo's composition of his paintings in the Sistine Chapel. This led to much richer evidence yet of Michelangelo's use of the golden ratio, also known as the Divine Proportion. These new findings are presented in a separate article, Michelangelo and the Art of the Golden Ratio in Design. Analysis of the findings of the 2015 Brazilian study. The authors of the Brazilian study … More on this article
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
Applications for Better Art, Design and Composition
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
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
Golden Ratio in Art Composition and Design
"Without mathematics there is no art," said Luca Pacioli, a contemporary of Da Vinci. Just as the Golden Section is found in the design and beauty of nature, it can also be used to achieve beauty, balance and harmony in art and design. It's a tool, not a rule, for composition, but learning how to use it can be a great Art 101 lesson on laying out a painting on a canvas. For those with a deeper understanding yet, the golden ratio can be used in more elegant ways to create aesthetics and visual harmony in any branch of the design arts. As you'll find in the examples below, it has been used by some of the greatest artists the world has known. Oddly enough, you may also find critics who say that the golden ratio cannot be found in art at all. Such statements often come from Ph.D.s in mathematics who hold a very theoretical viewpoint that nothing in the real world can be a golden ratio. Why? Simply because it has an infinite number of digits. (See a review/rebuttal on art and … 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?
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 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
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
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