• 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:”

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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 what was not. This site studies this golden number Phi, and its mathematical cousin, the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, …) , both of which have roles in the plot of this murder mystery, and distinguishes between the myth and the math.

This Golden Ratio truly is unique in its mathematical properties and pervasive in its appearance throughout nature. The “mathematically challenged” may be more interested in the appearances of Phi in nature, its application to art, architecture and design, and its potential for insights into the more spiritual aspects of life, but let’s begin with the purest of facts about Phi, which are found in mathematics.

Most everyone learned about the number Pi in school, but relatively few curricula included Phi, perhaps for the very reason that grasping all its manifestations often takes one beyond the academic into the realm of the spiritual just by the simple fact that Phi unveils a unusually frequent constant of design that applies to so many aspects of life. Both Pi and Phi are irrational numbers with an infinite number of digits after the decimal point, as indicated by “…”, the ellipsis.

Where Pi or p (3.14…) is the ratio of the circumference of a circle to its diameter, Phi or Φ (1.618 …) is the Golden Ratio that results when a line is divided in one very special and unique way. To illustrate, suppose you were asked to take a string and cut it.  There’s any number of places that you could cut it, and each place would result in different ratios for the length of the small piece to the large piece, and of the large piece to the entire string. There is one unique point, however, at which the ratio of the large piece to the smaller piece is exactly the same as the ratio of the whole string to the larger piece, and at this point this Golden Ratio of both is 1.618 to 1, or Phi.

What makes this so much more than an interesting exercise in mathematics is that this proportion appears throughout creation and extensively in the human face and body. It’s found in the proportions of many other animals, in plants, in the solar system and even in the price and timing movements of stock markets and foreign currency exchange. Its appeal thus ranges from mathematicians to doctors to naturalists to artists to investors to mystics.

Part of the uniqueness of Phi is that it can be derived in many other ways than segmenting a line.

• Phi is the only number whose square is greater than itself by one, expressed mathematically as Φ² = Φ + 1 = 2.618.
• Phi is also the only number whose reciprocal is less than itself by one, expressed as 1/Φ = Φ – 1 = 0.618.

These two qualities of phi can be expressed algebraically as a+1=a² and a-1=1/a. This can then be rearranged and expressed as a²-a -1=0 . This is a quadratic equation, the only positive solution of which is:

Φ  = (1 + √5) /2 = 1.61803398874989484820…

Where 1.618 is represented in upper case as Phi or Φ, its near twin or reciprocal, 0.618, is often represented in lower case as phi or φ. Phi is an irrational number, a number which cannot be expressed as a ratio of two integer numbers.

The Fibonacci Sequence

The Fibonacci sequence, also a plot element in “The Da Vinci Code,” provides yet another way to derive Phi mathematically. The series is quite simple. Start with 0 and add 1 to get 1. Then repeat the process of adding each two numbers in the series to determine the next one: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on. This pattern is also found in the diagonals of Pascal’s Triangle. The relationship to the Golden Ratio or Phi is found by dividing each number by the one before it. The further you go in the series, the closer the result gets to Phi. For example:

1/1=1
2/1=2
3/2=1.5
5/3 = 1.666
13/8 = 1.625
21/13=1.615

If you go further into the series and you’ll find that 233/144 = 1.61805, a very close approximation of Phi, which to ten decimal places is 1.6180339887.

Geometry of the Golden Ratio

The Golden Ratio is also found in geometry, appearing in basic constructions of an equilateral triangle, square and pentagon placed inside a circle, as well as in more complex three-dimensional solids such as dodecahedrons, icosahedrons and “Bucky balls,” which were named for Buckminster Fuller and are the basis for the shapes of both Carbon 60 and soccer balls.

Johannes Kepler (1571-1630), discoverer of the true elliptical nature of the orbits of the planets in the solar system described it as such: “Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”  (See other Quotations)

Nature and Life

There are many other fascinating mathematical relationships and oddities in both Phi and the Fibonacci series that can be explored in more depth, but for now let’s now take a step away from the purely mathematical and venture into nature, where Phi and the Fibonacci series manifest themselves pervasively, but not universally. Fibonacci numbers frequently appear in the numbers of petals in a flower and in the spirals of plants.  The positions and proportions of the key dimensions of many animals are based on Phi. Examples include the body sections of ants and other insects, the wing dimensions and location of eye-like spots on moths, the spirals of sea shells and the position of the dorsal fins on porpoises. Even the spirals of human DNA embody phi proportions.

Perceptions of Beauty

More intriguing yet is the extensive appearance of Phi throughout the human form, in the face, body, fingers, teeth and even our DNA, and the impact that this has on our perceptions of human beauty. Some would argue that beauty is in the eye of the beholder, but there is evidence to support that what we perceive as beauty in women and men is based on how closely the proportions of facial and body dimensions come to Phi. It seems that Phi is hard-wired into our consciousness as a guide to beauty. For this reason, Phi is applied in both facial plastic surgery and cosmetic dentistry as a guide to achieving the most natural and beautiful results in facial features and appearance.

Art, Architecture and Design

With all the unique mathematical properties of Phi and its appearance throughout creation, it’s little wonder that mankind would not only take notice of this number and the Golden Ratio it creates, but also use it to capture the beauty and harmony of nature in our own creations in art, architecture and other areas of design.  In some cases, mankind’s application of Phi is undeniable. In other cases, it is still the subject of debate. The Great Pyramid of Egypt appears to embody the Golden Ratio in the ratios of its base, height and hypotenuse, but its missing outer shell and the absence of the mention of Phi in ancient Egyptian writings make it difficult to prove conclusively that this was by design. The Greeks knew of Phi at the time of the building of the Parthenon, and while its overall dimensions only approximate the golden ratio, there appears to be evidence of it use in the design of the beam atop the columns. The confusion on these points leads to those who contest this as well. It is recognized that Leonardo Da Vinci used Phi, known in the 1500’s as “The Divine Proportion,” in a number of his paintings. Other artists, including Raphael, Sandro Botticelli and Georges Seurat did as well. While this is undeniable, some people creatively overlay golden spirals to images where others do not believe they were intended by the artist. The dimensions of the treasured Stradivarius violins built around 1700 show Phi relationships. It plays a role in music and acoustics.  More modern applications of the Golden Ratio in architecture can be seen in Notre Dame in Paris, the United Nations Headquarters Secretariat building in New York and the CN Tower in Toronto. It’s commonly used in the design of products and logos and by many major corporations.  It has even been used in high fashion clothing design, such as in the “Phi Collection” announced in 2004 and covered by Vogue, Elle and Vanity Fair. It’s also the basis for The Fashion Code, a style guide to women’s dress. Various studies have tested to see if a golden rectangle is the most pleasing rectangle to the human eye. Results of the studies are mixed, but generally point to rectangles with shapes close to the golden rectangle as being most pleasing.

The Solar System and Universe

Curiously enough, we even find golden ratio relationships in the solar system and universe. The diameters of the Earth and Moon form a triangle whose dimensions are based on the mathematical characteristics of phi. The distances of the planets from the sun correlate surprisingly closely to exponential powers of Phi. The beautiful rings of Saturn are very close in dimension to the golden ratio of the planet’s diameter. NASA released findings in 2003 that the shape of the Universe is a dodecahedron based on Phi.

New Discoveries involving the Golden Ratio

The Golden Ratio continues to open new doors in our understanding of life and the universe. It appeared in Roger Penrose’s discovery in the 1970’s of “Penrose Tiles,” which allowed surfaces to be tiled in five-fold symmetry, a task previously thought impossible. It appeared again in the 1980’s in the three-dimensional molecular arrangement of quasi-crystals, a newly discovered form of matter. As we enter the 21st century, Phi seems to be having a rebirth in integrating knowledge across a wide variety of fields of study, including time and quantum physics.

Spiritual Aspects

The description of this golden proportion as the Divine proportion is perhaps fitting because it is seen by many as a door to a deeper understanding of beauty and spirituality in life, unveiling a hidden harmony or connectedness in so much of what we see. That’s an incredible role for a single number to play, but then again this one number has played an incredible role in human history and in the foundations of life itself. The line between its mathematical and mystical aspects is thus not easily drawn.

Phi does not appear explicitly in the Bible or other ancient scriptures, yet we find that the dimensions given by God to Noah for the Ark and to Moses for the Ark of the Covenant both reflect a 5 to 3 proportion, Fibonacci numbers with a ratio of 1.666, and a reasonably close approximation to Phi.  The Kaaba, the most sacred site of Islam in Mecca, is located very close to the golden ratio of the distance between the Earth’s north and south poles.  Curiously enough, even the symbol for Phi, a circle with a line drawn through it, can be thought to represent a zero, or void, divided by one, or Unity, to create beauty, analogous to God creating the universe from nothing.

A Journey of Discovery

In matters of reason, seeing is believing but in matters of faith, it is believing that first opens the door to seeing. Just as we need two eyes to add depth to our perception in vision, both faith and reason serve us in adding depth to our understanding of life and the universe in which we live. Some will tell you that appearances of the Golden Ratio are proof of God’s design. Most religions though reveal that we come to know God through faith, not proofs. Some will tell you that there isn’t even any evidence at all for the Golden Ratio, that we’re just perceiving patterns that don’t really exist. Probe further though to ask if their views are based in evidence, or just a reflection of their own atheistic or naturalistic beliefs about life.

The best way to know for yourself where Phi appears and where it is imagined is to explore with an open mind, learn and reach your own conclusions on the facts and implications. You can learn more about the Golden Ratio here at www.GoldenNumber.net and download a free trial of PhiMatrix golden ratio design software to easily discover and apply Phi proportions for yourself. Enjoy the “phi”nomemon, whether to enhance your own understanding and appreciation of beauty and harmony in life or to apply it to your own artistic creations, like Leonardo Da Vinci and other masters before you.  I welcome you to contribute your own insights and findings with our online community.

Gary Meisner
The Phi Guy
Author of www.goldennumber.net
Developer of PhiMatrix software at www.phimatrix.com

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Golden ratio proportions of the teeth/lips and nose in relation to
the distance from the pupils to the bottom of the chin.

1. Does the human face reflect golden ratios in its proportions?

Not everyone will agree, but there is much evidence to demonstrate that the answer is “Yes.” There’s some truth to the adages “beauty is in the eye of the beholder” and “beauty is only skin deep,” yet there is no denying that there’s almost universal agreement when asked to distinguish between a very attractive face and a very unattractive face. Let’s acknowledge that every human face has many unique measurements and proportions. As a result no single measure can be said to apply to every aspect of every face. The real question is whether the golden ratio represents an average or ideal in human facial dimensions. The even more interesting question is whether golden ratios in facial features impact our perceptions of attractiveness and beauty. Debunkers will point out the variations, and perhaps suggest that those who find golden ratios in a beautiful face did so because they were looking for them. Consider the following studies though, which demonstrate that the connection to beauty is based on much more than looking for data to fit the desired result:

• Dr. Stephen Marquardt, a maxillofacial surgeon and recognized expert on beauty, studied hundreds of faces in his studies on attractiveness to develop his patented “Marquardt Beauty Mask.” The mask identifies archetypes for the faces perceived as most beautiful. The mask is based on dodecagons, which are based on golden ratios. Below is an example of a face that has been morphed to fit the beauty mask. Make your own assessment and see the YouTube video of the step-by-step transformation or click here for the article.:

• In 2009, a university study identified ideal facial proportions, as selected by study participants who chose the most attractive face from a series of photos. The researchers were not looking for the golden ratio. To the contrary, they concluded that it didn’t exist and announced their own two “new golden ratios” of attractiveness. My subsequent analysis, however, shows that their ideal facial features reveal a dozen golden ratios, in both horizontal and vertical dimensions of key facial markers. See image and details further below in this article.
• In 2012, a UK cosmetic company ran a competition to find Britain’s most perfect face. The winner was selected from 8,000 contestants. Her face shows almost two dozen golden ratios in both the horizontal and vertical dimensions of key facial markers, as shown in an accompanying YouTube video to the article.

• My own research shows a very simple and clear approach to facial analysis that reveals over one dozen common facial golden ratios.

2. Does the design of the Parthenon reflect the golden ratio?

Yes, but only in part, and not in the way you often hear. The front facade of the Parthenon is often shown with a golden rectangle framing it to illustrate its golden ratio proportions. Unfortunately, a precise golden rectangle would need to start at the second of four steps approaching the Parthenon, rather than the base of its columns. The golden rectangle would then end at the peak of the Parthenon, a point which has to be estimated because the structure is in ruins. Even as a golden ratio enthusiast, this is rather arbitrary and unconvincing as evidence. It’s close to a golden rectangle, but not close enough to conclude that it was used in its design. That, however, is not the end of the story.

• When this golden rectangle is placed as just described, the bottom of the supporting beam at the top of the columns is at the golden ratio of its height.
• The top of the supporting beam structure across the columns has dimensions that appear to be very close to the golden ratio of its base to the apex of the roof.
• More interesting yet, the fourteen rectangles across the horizontal beam each appear to be divided exactly at their golden ratio point, as closely as can be determined from photos.

These golden ratios may or may not have been included intentionally among all the other numbers and ratios represented in the Parthenon. The first written record of the golden ratio was by Euclid, almost 100 years after its construction though, so the architects of the Parthenon did not learn of it from him. Plato was born not long after it’s construction and adopted the pentagon star as the symbol for his group of followers, so that suggests there could have been awareness of special geometries using the golden ratio in that era. The Parthenon has a complex design that embodies many numbers and mathematical relationships. It’s not unreasonable at all that the golden ratio would have been among those represented.

3. Is the spiral of the Nautilus shell based on the golden ratio?

It can, but not in the way you often hear. The traditional “golden spiral” is constructed from a series of adjacent golden rectangles. This creates a spiral that increases in dimension by the golden ratio with every 90 degree turn of the spiral. The spiral of the Nautilus shell does not match this golden spiral, which is a source of the confusion on this question. There is, however, more than one way to create a spiral with golden ratio proportions. As one example, you can create a spiral that expands by the golden ratio with every 180 degree turn of the spiral. This spiral is a closer match for the spirals of many Nautilus shells, especially in the first several rotations of the spiral. An even closer match yet is found when when measuring to spirals on opposite sides of the center point.

4. Did Renaissance artists (Botticelli, Da Vinci, Michelango) use the golden ratio in composition of their paintings?

Yes. This is often contested by debunkers as being completely unsupported. One frequently referenced article by a Ph.D. in mathematics points out that there is no mention of the golden ratio in two biographies written about da Vinci. Wouldn’t it be better to actually measure the paintings than to read a biography?  Another debunker, also with a Ph.D. in mathematics, claims there is “no evidence” that da Vinci used the golden ratio, but in his presentation shows only da Vinci’s simple sketches rather than his masterpiece paintings on religious subjects. Unfortunately, most commentary on da Vinci’s use of the golden ratio in his paintings is based on The Mona Lisa, which has very few linear composition elements from which to definitively measure a golden ratio. Golden ratios are really quite easy to see if you have the right tools.

• Da Vinci’s “Last Supper”, “The Annunciation” and Salvator Mundi are much better examples and show clear composition lines at golden ratio points.
• Botticelli’s “Birth of Venus” was painted on a canvas which is itself a golden rectangle, and key elements of the painting are at golden ratio points of the painting’s height and width. Botticelli also created several renditions of “The Annunciation” which also have clear golden ratios in their composition.
• Michelangelo’s “Creation of Adam” has God finger touching Adam’s finger at the precise golden ratio point of the width of the area in which they are framed. That is just one example of how Michelangelo used this Divine proportion in the Sistine Chapel.
• Raphael’s “The School of Athens” features a golden rectangle placed front and center in the foreground of the painting, and the main scene above reveals many golden ratios in its composition.

As noted by “Civilisation” author Kenneth Clark, “This union of art and mathematics is far from our own way of thinking, but it was fundamental to the Renaissance.”

5. Are the spirals seen in nature based on the golden ratio?

Some are, but most are not. The spirals most commonly seen in nature are equi-angular (aka logarithmic) spirals. This simply means that the spiral expands at a constant rate. This occurs because it creates an even flow of energy or distribution of tension. This has nothing at all to do with the golden ratio. Accordingly, all the illustrations of spiral arms of galaxies, curves of ocean waves, spiraling hurricanes, etc. that are incorrectly identified as a “Golden Ratio” or “Golden Spiral” should be relabeled or removed to avoid further confusion in perpetuating this golden ratio myth. There is another type of spiral that is related to the golden ratio, however, that occurs very commonly in nature. The spirals that appear in pine cones, pineapples, seed pods and similar plant structures are usually based on two successive numbers of the Fibonacci sequence. For example, if there are eight spirals in a clockwise direction, you will find thirteen spirals going in the counter-clockwise direction.  Neither of these spirals is necessarily a “golden ratio” spiral on its own, but the ratio of these successive Fibonacci numbers approximates the golden ratio.

6. Did French painter Georges Seurat use the golden ratio in the composition of his paintings?

Yes. One art critic claimed that Georges Seurat “attacked every canvas with the golden section.” At least one debunker says this is a simply “myth that needs to be dispelled.” The truth is somewhere in the middle.

• My examination of over 100 Seurat paintings reveals that over 30 of them show very clear golden ratio composition lines. This analysis was based solely on the height and width of the canvas, where there is no room for interpretation or error.
• Seurat chose to paint about one-quarter of his work on golden rectangle panels. Of his 103 paintings listed at Wikimedia, at least two dozen of them had a height to width ratio to within 0.1″ of the golden ratio.
• Other paintings show golden ratios in the elements within the composition, including both their positions and dimensions.

So the correct answer is that Seurat frequently used the golden ratio in his artistic compositions, but probably did not attack every canvas with it.

7. Is the Great Pyramid of Giza based on a golden ratio?

Possibly, but unknown. This is also subject to debate and interpretation, and there is no certain answer. There is no mention of the golden ratio in the existing historical record of the Egyptians. The outer limestone skin of pyramid is now missing, which makes its exact original dimensions subject to some estimation. The outer skin is still intact at the top though, and thus provides good guidance as to the original form. What we do know is this:

• The estimated original dimensions of the Great Pyramid per Wikipedia reveal a pyramid with an average base measure of 230.4 meters and height of 146.5 meters.
• This height is different from a phi-based triangle by only 0.0367 meters, a variance of only 0.025%.
• There’s another school of thought that the Pyramid’s proportions are based on the arc of a circle whose circumference is equal to the perimeter of the base of the pyramid.
• If that’s true, then the estimated height is different by only 0.177 meters), a variance of 0.1%.

Other geometries are also possible, one based on surface areas and another on gradients using Egyptian units of measure. There is also evidence that the relative sizes and positions of the pyramids at the Giza site embody golden ratio proportions. The truth is that there is much that we do not know about these pyramids or their construction, so the dimensions could be based on any of these methods and geometries. The Great Pyramid was clearly constructed with some specific geometry in mind, and with exacting precision, so we can reasonably assume that its architects did not create a pyramid with these very special and unique geometries by blind chance. A very interesting question still remains: Even if the pyramid wasn’t consciously designed with a knowledge of phi, why did they pick a geometry for the Great Pyramid that so closely embodies the golden ratio, the one number that has a ubiquitous presence in geometry, and that is associated with nature and beauty?

8. Is there a “new golden ratio” that can determine facial attractiveness with two measurements?

No. The 2009 study claimed to have debunked the relationship of the golden ratio to beauty and claimed to have discovered two “new golden ratio” measurements that defined attractiveness. The faces identified as the most attractive, however, had over a dozen golden ratios in the dimensions of their key facial features. The researchers apparently didn’t realize these golden ratios existed and devised their own alternative measurement points. It should be obvious that human facial beauty cannot be reduced to a single vertical and single horizontal measurement, and it’s quite easy to prove this with a few simple examples of unattractive faces that still happen to fit those two measurements.

9. Is Mecca at the golden ratio point of the Earth?

No, at least not exactly. There are two golden ratio points in latitude between the Earth’s poles, one in each hemisphere. Mecca is approximately 12 miles from the golden ratio latitude in the northern hemisphere. The golden ratio of the longitude is not easy to determine as there is no natural reference point like a rotational pole. The claim for Mecca was based on a Mercator projection of the Earth, which spans from dividing point between the US and Russia. Using this as the basis, Mecca is 938 kilometers from the golden ratio point of the longitude. Using other starting points as a basis there are many other golden ratio longitude points on the Earth.

10. Is the design of the UN’s Headquarters building based on the golden ratio?

Yes. This was supposedly “debunked” in an an article by Dr. George Markowsky, a Harvard Ph.D. in Mathematics, who calculated the simple ratio of the building’s height to width. He noted that the ratio was 1.76 and not 1.618, came to his conclusion, and missed many historical and mathematical facts. One of the lead architects, Le Corbusier, created a design system based on the golden ratio just years before his involvement with the UN project. A well-known, award-winning Disney education film illustrates that the golden ratio relationship is based on three horizontal golden rectangles stacked vertically, not its height vs. width. Golden ratios appear in the visual dividing points on the face of the building, and these are easily can be measured and calculated. Other very clear golden ratios are present in the dimensions of  the curtain wall, windows and front entrance.

Is the golden ratio unique and special, or a myth to be debunked?

It quite definitely a special number with unique properties and it is found in nature, in some quite unexpected places. Even those whose goal is to debunk golden ratio myth will agree that the golden ratio has properties that make it very unique in mathematics and geometry. This one number has intrigued and inspired countless people throughout mankind’s history, and earned a special place in their hearts and minds. It is probably for that very reason that some people have taken it beyond fact to fantasy and mysticism. That, in turn, has led others to summarily dismiss it with just as much zeal. Both extremes in position are flawed and keep us from knowing the truth and applying it.

Is the golden ratio a universal constant of design in nature? No. Must all features of a face be in perfect golden ratio proportion to be beautiful? No. Can it help us to better understand and appreciate beauty in nature, and apply it in our own lives? Yes.

Is the golden ratio required to create great art? No. Can it help to better appreciate some great art and to create with better composition and design? Yes. It’s used for that purpose in art, photo cropping and composition, logo design, product design and other areas of design.

Like many things in life, it provides us with insight into the nature of things, the opportunity to better appreciate them, and the ability to use them to advantage. With all the misinformation and misconceptions that exist, it is best to keep an open mind, and not accept claims made without investigating evidence on both sides, experimenting on your own and coming to your own conclusions.

Articles referenced:

The Myth That Will Not Go Away by Keith Devlin

Misconceptions about the Golden Ratio by George Markowsky

Debunked Legends About the Golden Ratio by Julia and Jesse Galef

Meandering Through Mathematics by Phil Keenan, Ph.D.

GurneyJourney series: Mythbusting the Golden Mean by James Gurney

Other GoldenNumber.net articles on understanding the evidence, and debunking the debunkers:

Dr. George Markowsky on “Misconceptions about the Golden Ratio”

Dr. Keith Devlin on “The one formula that’s supposed to ‘prove beauty’ is fundamentally wrong.”

Fastco Design on  “The Golden Ratio: Design’s Biggest Myth – The Golden Ratio is Total Nonsense in Design.”

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