Several recent studies have attempted to rate the beauty of objects (cars, buildings) by mimicking an analysis technique created for facial analysis. While these efforts are very creative and involved a lot of work, they fail to produce reliable results for two reasons. First is that human faces have a common set of features that can measured against a standard, while the objects of the study do not. Second is that the studies did not actually measure golden ratios, which in geometry and visual aesthetics are based on dividing a line segment at its unique golden ratio point. The studies instead calculated the ratios of all possible permutations of two line segments created by each combination of three marker points. The object whose mathematical average of all the ratios came closest to the number 1.618… (or its reciprocal 0.618…) was declared “the most beautiful according to science and the golden ratio.”

Since the objects identified as the “most beautiful” by the model were clearly not the most beautiful by any human judgment, the model is not evidence-based and not really science. Since it did not really use the classic definition of a golden ratio in the process of getting there, the results are not really “according to the golden ratio.” This article discusses the details of the study and discusses where the golden ratios actually appear and how they can be used in understanding nature and applied in the design arts.

Can beauty be defined by science and the golden ratio?

As a researcher, author and software developer on the Golden Ratio, I’m always very interested in stories announcing new discoveries or insights on the topic. I was thus very intrigued to find a March 2021 article in the UK’s Daily Mail announcing that UK’s Roofing Megastore had identified “The World’s Most Beautiful Buildings, According to Science and the Golden Ratio.”

I read the articles, and learned about the methodology they had used. The team that performed the study did a lot of hard work. They found candidate buildings, identified key markers and gathered many measurements. They then used a custom software program to develop a mathematical score to rank the beauty of each of the buildings. And when the work was done, they identified St. Paul’s Cathedral of London as the most beautiful building in the world:

But science requires good evidence-backed findings

In February 2020 though, I was similarly excited to see that Carwow had published an article on the “Best-looking cars of the last decade.” In this article, Carwow claimed to apply golden ratio analysis techniques used for human faces to the front-end view of automobiles. Very creative indeed! Unfortunately, my in–depth review of the Carwow study revealed that while the Carwow team had been very ingenious and detailed in their work, their methodology was fundamentally flawed. It had nothing to do at all with the way that humans perceive the golden ratio. It was based instead on an exercise in mathematical number-crunching that no human brain would use.

The results were highly suspect too. Of the 626 cars produced from 2010 through 2019, the unappealing 2016 Smart ForTwo Cabrio took the very top spot. Even the Carwow team agreed that this very utilitarian-looking car ranked very low in their personal ratings of beautiful cars. Their innate perceptions of beauty yielded very different results than their mathematical model. This was a very clear indicator that the model was flawed. There was really no basis at all in the evidence to call it “science.”

The same flawed golden ratio methodology. The same flawed conclusions.

A year later, Roofing Megastore announced in March 2021 that it had done a similar study, proclaiming this time to have identified the “World’s Most Beautiful Buildings, according to Science and the Golden Ratio.” Unfortunately, they used the very same highly flawed methodology:

As described in the article on the DailyMail:

100 of the world’s most famous architectural works were analyzed against the golden ratio using custom software, with researchers plotting the most common points found on a building, such as the top and bottom corners, the building’s centre, and equidistant points around the circumference, before comparing the distance between these markers to those laid out by the ratio.“

The unreliability of this methodology as applied to buildings should also have been immediately apparent. Their best fit to the golden ratio was only 72.28%, a failing mark in both the classroom and statistical correlation. The appeal and national pride of having London’s St. Paul’s Cathedral jump to the top of their list may have made it seem reasonable. It’s a fine looking cathedral, but there are so many others from which to choose that seem so much more stunning, majestic and … beautiful:

But here are the top winners of the Roofing Megastore search:

The 10 Most Beautiful Buildings in the World

The Taj Mahal, is known for its beautiful application of the golden ratio in many of its features. It came in at number 8 on the list, but scored only 67.45%. The Great Pyramid of Giza is known to fit a golden ratio “Kepler” triangle to within 0.025% accuracy. It came it at number 19, with a score of only 52.35%. Notre Dame Cathedral is also known for some beautiful applications of the golden ratio. It was number 46, with a score of only 34.37%. Other famous buildings that beautifully embody the golden ratio, such as the Parthenon and UN Secretariat Building, didn’t even make the list.

How could these scores and measures be so unreliable and inaccurate? Unfortunately, there were several failure points in the methodology.

She’s so beautiful. She has a face like a building!

The first failure point was in assuming that a methodology used to evaluate human facial beauty could be applied to cars and buildings. Humans may all look uniquely different, but they all have the very same basic facial structures, and positions and proportions of their features. That’s what makes us look human! Even small changes in the relative positions and proportions of facial features will result in a universal perception of their unattractiveness, and even deformity. This is illustrated by the facial images below, with each face showing their conformity to the golden ratio-based Marquardt Beauty Mask:

By contrast, there is no universal standard that defines the specific features, positions and proportions for the design elements of a car or a building. We know exactly where to expect to find the nose on a human face. A car’s emblem though can appear in the middle of the grill or the top of the hood and look equally fine. We know exactly where to find the mouth on a human face, and know there will be just one. A car though can have a single large grill or separate upper and lower grills and still look perfectly fine. So with these variations in design, there is no reason to assume that a mathematical model that measures for human facial attractiveness is going to produce reliable results in determining the attractiveness of the front end of a car.

Anthropomorphism – Making non-human things look human!

Humans though have an innate tendency to attribute human traits, emotions, or intentions to non-human entities. This is called anthropomorphism. So yes, with cars it’s very natural to envision the similarities of eyes to headlights, noses to emblems and mouths to grills. This tendency is what Disney Pixar relied upon to create their lovable Cars characters. But the animators at Disney Pixar knew enough about faces and beauty to NOT use the headlights as eyes. Instead they made the cars look human, natural and aesthetically pleasing by putting the “eyes” within the windshields. They then positioned a grill for the mouth at the golden ratio point between the eyes and the bottom of the car’s “face.” This is the very same golden ratio point that defines the position of the mouth on a human face, between the pupils of the eyes and the bottom of the chin! Do you see how the golden ratio proportions of design were applied to all the images below?

But do buildings have human features???

So what about buildings? Here the analogy to human features completely falls apart. There is nothing at all about a building that you would necessarily expect to resemble two eyes, a nose and a mouth. Windows could be eyes, but most buildings have too many to make the connection. You might think of the door as a “mouth.” Doors though can be of any number, and be positioned in the middle, off-center or at the ends of a building. Some buildings have two separate entrance and exit doors on opposite ends. So when it comes right down to it, there is nothing about the design of a building that we should expect to be analogous to the design of a human face. And therefore, there is no reason that we should think that a mathematical model that measures human beauty would be able to measure the beauty of a building. This is most likely why the “best” score achieved by a building in this study was only a failing grade of 72.88%.

And unlike faces, which we can expect to be substantially the same in basic appearance and proportions, architects take great pride in creating something new and different with important, high-profile buildings. Each building must be uniquely designed to meet different functional requirements, to fit within the context of its environment and to convey a message of its style or purpose.

So where are the golden ratios in building design?

Even if an architect consciously and religiously applies the golden ratio to every possible decision made on position and proportion, there is no reason to think that these design elements will align to the very simplistic set of markers used in the Roofing Megastore study. Let’s look, for instance, at St. Paul’s Cathedral, which earned the number one spot. It actually appears to have some golden ratios in its design. This is shown by the red, green and aqua rectangles divided at their golden ratio points:

Unfortunately, the simplistic one-size-misfits-all template used to identify building markers in the Roofing Megastore study didn’t recognize the important golden ratio markers on this cathedral. Note the golden ratios based on the outside of the front sets of columns. These were not identified in the study as markers.

Below is Wells Cathedral, also in England, showing a number of actual golden that appear to have been designed into its beautiful architecture. The set of markers though are different, illustrating that you have to approach golden ratio analysis with an understanding of each building, and not one standardized template.

The same issue undoubtedly happened with most of the other buildings in the study. Accordingly, the results of the Roofing Megastore study likely do not truly reflect the golden ratios in the building’s design. Their measure is just a mathematical average of calculations applied to the hundreds of permutations of pairs of line segments based on their standardized markers.

And what is a golden ratio anyway?

The second major failure point of both the Carwow and Roofing Megastore studies is in an incorrect assumption about the golden ratio. They assumed that any golden ratio proportion between two line segments formed by any three markers would emulate the golden ratio relationships found in nature and in the design arts. This, however, simply isn’t the case. The golden ratio, by its very definition as provided by Euclid in 300 BC, is based on this:

“A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.”

This concept is illustrated below:

This same division of a straight line at this one special golden ratio point appears in the proportions of the human face:

What if we had only achieved 72% accuracy in hitting the golden ratios in the face of the woman above? Her face would be proportioned something like this:

As you can see, a score of 72% is hardly a reliable measure of beauty, in a face, a car or a building.

The Golden Ratio in Logo Design

Look how these same basic golden ratio concepts have been applied in the design of these famous logos:

No complicated mathematics with hundreds of permutations on a dozen plus markers was required. In each case, the composition decisions on position and/or proportion of the key elements of the logo simply used the golden ratio point of the logo’s height or width.

The Golden Ratio in Geometry

This same division of a straight line into segments in golden ratio proportion also appears in the geometry of a five-pointed star. Note that multiple golden ratio proportions appear here – in the black to blue lines, the blue to red lines, and the red to green lines:

There’s certainly a beauty in the face and in the five-pointed star.

Suppose, however, that I took these golden ratio segments of the star and rearranged them into something very different:

The lines are all the same. The arrangement is still symmetrical, just like the face of a car or building would be. All the same golden ratio relationships exist between the lines. The beauty though of the five-pointed star that was there is now gone. As you can imagine, a beautiful face could also be rearranged and distorted to keep certain golden ratio relationships intact. That face would no longer be beautiful either. Why? It’s because the golden ratios that had defined beauty are no longer discernible by the human eye.

The Golden Ratio in design is all about visual aesthetics, not just mathematical calculations.

Unfortunately, the approach used in these studies replaced simple visual aesthetics with complicated mathematical mumbo-jumbo. This is what both Carwow and Roofing Megastore did in taking their measurements and then doing their calculations and ratings. They did the math, and did it very well. They missed though the very critical fundamental concept of the beauty that is associated with golden ratio. It is simply based on dividing a straight line at the one very unique point that gives it the familiar and pleasing aesthetic that we see in nature, geometry and the arts. The beauty that comes from the golden ratio is based on the visual aesthetics it produces in the human mind.

Whether it’s a face, a logo, a car a building or anything else, it’s critical to to understand the markers that give rise to the golden ratios. The Parthenon, the Taj Mahal, Notre Dame and the UN Secretariat Building all have beautiful golden ratios in their designs. The markers though that define where these golden ratios exist are different for each building. You can’t just make up one simplistic, generic set of markers. You can’t calculate the ratios of every possible permutation of two lines that can be created by each set of three jumbled markers and proclaim that the cars or buildings whose average comes the closest the number 0.618 are the most beautiful according to science and the golden ratio.

Look at the set of markers and line segments chosen for both the car and building analysis. There is only ONE straight line among ALL the line segments created by the markers. In both cases, that is the straight line from the top center to the bottom center. In both cases, for the objects rated “the most beautiful” there is a marker in between those two points that actually comes quite close to the golden ratio, as shown by gold grid below. This as the ONLY possible actual golden ratio point among ALL the line segments these studies analyzed. These studies were thus set from the start to produce results that had little to do with beauty as actually defined by the golden ratio.

Sharing the true beauty of the Golden Ratio in design

The good people at Carwow, Roofing Megastore and Propellernet (the marketing agency behind these studies) have all been a delight to work with in providing information for my analysis. They all tried very hard to make a legitimate attempt to use the golden ratio to identify the most beautiful examples of design in their respective industries. I applaud them for their creativity and effort. I truly wish I could do the same for their methodologies and results. I’ll be very happy to change my article if anyone can show my assessment to be inaccurate or incomplete.

I’ll also be very happy to consult with anyone conducting a golden ratio analysis of beauty. It should be performed in a way that is truly scientific in its methods and truly meaningful in its results. Golden ratio analysis isn’t difficult in concept. It does though require of experience and knowledge of its appearances and applications to avoid invalid methods and incorrect conclusions. Let’s use studies like these as an opportunity to educate rather than misinform people about the golden ratio. Let’s demonstrate how it is used to achieve beauty in aesthetics and design, so that people can use it to create more beauty, in their lives and in our world.

The post The World’s Most Beautiful Buildings, According to Science and the Golden Ratio appeared first on The Golden Ratio: Phi, 1.618.

If We Built It Today – Code of the Parthenon

In April 2021, the Science Channel released an intriguing new documentary in their “If We Built It Today” series. In Episode 3 of Season 2, the “Code of the Parthenon” explores the wonders and mysteries of the Parthenon of ancient Greece, and contemplates what it would take to build it today. The show features experts in architecture, engineering, history, sculpture and more. For insights into the beauty, aesthetics and design of the Parthenon, show producer Andrew Killawee turned to Gary Meisner as his Golden Ratio expert. Gary is, of course, the founder of this site at  GoldenNumber.net, developer of PhiMatrix Golden Ratio Design and Analysis software and author of “The Golden Ratio – The Divine Beauty of Mathematics.”

Gary knows from past experience that you never know what parts of a one or two hour interview are going to end up as the few quotes or sound bites that make it to the final cut of an article or documentary. This documentary though far exceeded his expectations. The very first minute of the episode featured closeups of him, his Golden Ratio book and his PhiMatrix software in action. Marketing giants like Proctor & Gamble or Coca-Cola couldn’t have paid for better product placement and promotion!

Gary Meisner’s perspectives on the Golden Ratio

Highlights of Gary’s contribution to the documentary (appearing at about the 1, 6, 8, 12 and 22 minute markers of the Internet version) included:

• Providing explanations of the fundamental concepts and history of the Golden Ratio.
• Describing how the Golden Ratio is used as a tool for beauty and aesthetics in design.
• Explaining how our attraction to the Golden Ratio is “fundamentally human,” as it appears within the proportions of our own face and body, and even the spirals of our DNA.
• Illustrating how his PhiMatrix software can be used to provide evidence of the application of the Golden Ratio by the ancient Greeks in their design of the front face and floor plan of the Parthenon.

The entire hour long episode is fascinating on many levels, and very worth watching. For those without the time or access, a recap of the segments featuring Gary’s contributions to the documentary are shown below in the two “Selected Transcript” sections.

What’s a classic Greek story without an Antagonist?

The documentary though wasn’t all smooth sailing for Gary. The Golden Ratio is clouded with misinformation and confusion everywhere it goes. Conflict and controversy sell, and the show’s writers didn’t miss the opportunity in this classic Greek story to infuse the drama of the battle between a protagonist and antagonist. Pitted against the Gary’s quotes in support of the ancient Greek’s use of the Golden Ratio were quotes from his antagonist, Emily Varto, an Ancient Greece Historian.

Emily countered “This is an idea that the Parthenon is so beautiful and pleasing to the human eye, that it must follow one of these universal principals and beauty and proportion. Well it doesn’t have to follow a universal standard of beauty to be a thing of beauty. So I think there is a secret in the ratios, just that it’s a different ratio. If you pay attention to the measurements, which the archeologists have done within the last twenty or so years, and beyond, you’ll notice that proportion is 4 to 9, or 9 to 4. Of course, they’re interested in proportions. They just don’t seem to be that interested in the ‘so-called’ Golden Ratio.”

The scene then shifts to an illustration of the Parthenon’s length and width, with a big 4/9 to show their ratio. This is quickly followed by an animation of the Parthenon exploding into pieces to deliver the seemingly devastating blow that 9 to 4, or 2.25, is nowhere near the Golden Ratio of 1.618…!

This sounds quite condemning, but there was a fatal flaw in the antagonist’s evidence and reasoning. The ONLY proportion referenced to support her conclusion was that of the Parthenon’s length (228.14′ or 69.54 meters) to its width (101.34′ or 30.89 meters). This has a ratio of 2.2512, or about 9:4. The problem is that NO real expert on this topic has EVER claimed a Golden Ratio to exist in the ratio of the Parthenon’s length to width. To see the Parthenon’s Golden Ratios, you have to walk around the corner to the front face of the building, and look upward to see all of its beautifully designed detail. A trip inside reveals more yet. The attack was just an misinformed strawman argument, which is all too common among golden ratio skeptics, critics and debunkers.

In fairness, Emily is a very credible ancient Greece historian who added some wonderful historical content and color commentary to the documentary. Gary though has been investigating the Golden Ratio for many years, even writing his own software to do original research. And it’s a safe bet that the documentary, as good as it is, didn’t come close to capturing all that either of them had to say on this topic. Emily may well have been an unwitting victim of the surprising misinformation on the golden ratio that has been publicized by seemingly reputable sources. There truly is a general lack of good, consistent, accurate information that is available, even to those who really try to learn about the Golden Ratio.

And while Gary would completely agree with Emily that art and architecture certainly do not have HAVE to follow a universal standard of beauty to BE a thing of beauty, this entirely misses the point. The question at hand isn’t whether the Greeks HAD to use the Golden Ratio. The question is simply whether or not they DID.

The Parthenon’s Golden Ratios revealed

The documentary provided great exposure for Gary’s work and expert status, but didn’t even begin to dive into the depths of his writings and research that support his findings on the Greek’s use of the Golden Ratio in the design of the Parthenon. These are best presented in his article “The Golden Ratios of the Parthenon.” Some of the key golden ratios of the Parthenon are shown below. Note that a Golden Rectangle divided at its golden ratio point can be created with one of the most very basic geometric constructions of the golden ratio. This was shown by Euclid in “Elements” in a number of his propositions. The ancient Greeks embedded this Golden Rectangle FOURTEEN times across the face of the Parthenon, in the frieze section of the entablature resting atop the columns! An informed observer might conclude that they they thought it to be rather special and important, and didn’t want anyone to miss it!

Golden Ratios can then be found in many other design elements of the Parthenon, both on the exterior face as well as in its interior:

So then, in classic Greek drama fashion, the plot takes a turn for the protagonist as the narrator takes viewers to the story’s conclusion with quotes like these:

• “It’s clear for Gary AND the Greeks, beauty’s in the math.”
• So let’s do it. Let’s build the world’s most beautiful building according to the ancient principles of the Golden Ratio. (They cut once again to show “The Golden Ratio – The Divine Beauty of Mathematics” on Gary’s book shelf.) And that mission will guide every decision we make on our modern day Parthenon.”
• “To design it we’re going to need to bust out our calculators and do the math.”
• “Armed with the right ratios and optical refinements, we’re finally ready to settle on a design plan.”
• “We’re imagining how we’d build a modern day Parthenon. It’ll be built following the same proportions and golden ratios that defined this ancient icon.”
• “We did it. A modern shrine to democracy, freedom and beauty. We used the proportions and ratios of the ancient Greeks to reconstruct their greatest piece of architecture.
• And at the episode’s very end, “By rebuilding the Parthenon according to the Golden Ratio, we’ve revived a lost construction technique, and a pathway to beauty the world has sorely missed.”

Nashville – The Athens of the South

The fact that Gary lives in Nashville was an added bonus, as Nashville is the home to the world’s only full size replica of the Parthenon. This provided the Science Channel production team with great resources and footage, as well as interviews with other local experts, including Wesley Paine – Nashville Parthenon Museum Director, Betsy Robinson – Ancient Architecture Historian and Alan LeQuire – Sculptor, whose 24 foot sculpture of the Greek goddess Athena graces the interior chamber of Nashville’s Parthenon.

Where to watch the show

The best place to watch “The Code of the Parthenon” is on The Science Channel itself, through your cable TV provider or other subscription resource. This 50 minute version is the most complete.

If you do not have access to The Science Channel from a cable TV subscription, you can also watch a slightly shorter 42 minute version on The Science Channel’s web site here:

https://www.sciencechannel.com/tv-shows/if-we-built-it-today/full-episodes/code-of-the-parthenon

You can also watch the Science Channel live without cable with one of the following options: fuboTV, Philo, Sling TV, or Hulu with Live TV.

“Code of the Parthenon” will be gradually distributed throughout the world in 2021 through 2023. International versions of the Science Channel currently exist in Southeast Asia, Europe, France, United Kingdom, Italy, India, Sweden, Turkey, Canada, Latin America and Australia. Unlike the U.S. network, the international channels are branded under the Discovery Science name.

For those without the access or time to watch, a recap of the segments featuring Gary’s contributions to the documentary are shown below in the two “Selected Transcript” sections.

Thoughts on the Parthenon’s lasting impact

Justin Shubow, President of the National Civic Arts Society, was another expert interviewed for the show. He expressed a very important point as to why the Parthenon and other buildings of this period are so important to architecture. Buildings throughout the world have been inspired by the architecture of ancient Greece. In Washington DC alone, these include the Supreme Court Building, the Lincoln Memorial, The Jefferson Memorial, The National Gallery of Art. He noted that these buildings all look like they’ve been there forever. They’re not a matter of fashion. These classic styles have been used whenever the architecture needs to present this sense of timeless, classic strength.

Sculptor Alan LeQuire was quoted with this insight at the show’s conclusion: “There is something there that’s ageless, and that is their sense of beauty. I think that most people who see the building get that immediately. We’re not using beauty as a guide, but if we did I think it would have a beneficial effect on everyone. Beauty, it makes your live much better on lots of levels.” The narrator concludes the episode with, “By rebuilding the Parthenon according to the Golden Ratio, we’ve revived a lost construction technique, and a pathway to beauty the world has sorely missed.”

With a spoiler alert, the show proposes to build this new version of the Parthenon in Washington, DC, with a focus of honoring the democracy of the United States of America and those who made it great. To match the original Parthenon’s 15 year construction period, they estimated that its cost would be $161,000,000. With just a small budget overage, I think the team could easily get the final cost to come in at $161,803,398.87, in near perfect homage to the golden ratio of 1.6180339887… !

I’d like to thank Andrew Killawee for the opportunity to be his Golden Ratio expert for this documentary. I’ve long studied and admired the Parthenon, both through the work of others as well as through my own original research. It was great to be able to share just a small part of what I’ve learned in this great documentary. I hope my participation along with all those involved in its production, will inspire others to learn more about it, and the role of the Golden Ratio in beauty, design and life itself.

For more information on my work on the Parthenon, see my articles here:

• “Phi and the Golden Section in Architecture,” published on this site in 2003, as recorded on Archive.org.
• “The Parthenon and Phi, the Golden Ratio,” published on this site in 2013.
• “Dr. George Markowsky’s “Misconceptions about the Golden Ratio” Reviewed,” published on this site in 2016.
• “The Golden Ratio – The Divine Beauty of Mathematics,” published by The Quarto Group in 2018. See pages 104-108.
• “Donald Duck visits the Parthenon in ‘Mathmagic Land,” published on this site in 2020.
• “The Parthenon and the Golden Ratio: Myth or Misinformation?,” published on this site in 2020.
• The Golden Ratios of the Parthenon, published on this site in 2020.

“Code of the Parthenon” – Selected Transcript of 42 Minute Internet Version

• 0:23 PhiMatrix software overlaying the face of the Parthenon
• 0:25 Feature shot of The Golden Ratio – The Divine Beauty of Mathematics book, along with the golden mean gauges on Gary’s bookshelf
• 0:32-0:36 – Gary at computer analyzing Parthenon floor plan
• 1:08-1:15 – Gary in office saying, “The Golden Ratio appears many different places in the Parthenon. If understood, and if applied, it can be like a secret weapon.”
• 6:10-6:21 – Gary in office saying, “The Parthenon is said to be the most mathematically perfect building or temple ever constructed. The proportions that appear in it use many, many different numbers …”
• 8:07-8:11 – Gary at computer analyzing the Rose Window of Notre Dame
• 8:12 – Feature shot of collectible shelf on Gary’s bookshelf, feature Vitruvian Man and other items.
• 8:13 – Feature shot of The Golden Ratio – The Divine Beauty of Mathematics book and golden mean gauges on Gary’s bookshelf
• 12:05 – Narrator saying, “For most of us, beauty really is in the eye of the beholder, but for guys like Gary it’s all in math.” Gary Meisner is a writer and software developer obsessed with the Golden Ratio, or Phi, an ancient Greek formula discovered by Euclid, the forefather of geometry,” with feature shots of Gary working at his computer, Gary’s Golden Ratio book and bookshelf collectibles.
• 12:30-13:08 – Gary saying, “Euclid defined the Golden Ratio as follows: A line is said to be cut in extreme and mean ratio when the length of the entire line to the larger segment is the same as the larger segment to the small segment. That’s a lot of words, but it’s really not that complicated. It’s just a matter of dividing a line at the one unique point where the ratios are all consistent. There are certain elements of design that use position and proportion, and when the golden ratio is used in those cases you end up with kind of a foolproof way of achieving beauty in aesthetics.”
• 13:16-13:17 – Feature shot of The Golden Ratio – The Divine Beauty of Mathematics book and golden mean gauges on Gary’s bookshelf
•  22:02-22:20 – Narrator saying, “And while some contemporary architects believe the Golden Ratio has become less important over time, other experts like Gary Meisner think it’s the most useful framework for defining beauty. He’s even developed software based entirely on the concept.”
• 22:21-23:00 – Gary, at computer, saying, “What the software does is (provide) a transparent overlay that appears on the screen, and provides a set of grids that are in golden ratio proportion.” Showing an image of the Parthenon, “Here, I’ve applied a golden rectangle to the face of the Parthenon, going from where the roof line would be projected to be to the bottom of the second set of steps here. So this is a Golden Rectangle that frames the Parthenon, but in addition to that the Golden Ratio of the height defines the top of the columns.” Narrator saying, “It’s clear for Gary and the Greeks, beauty’s in the math.”
• 41:10-41:22 – Gary on screen at computer with Narrator saying, “By rebuilding the Parthenon according to the Golden Ratio, we’ve revived a lost construction technique, and a pathway to beauty the world has sorely missed.”

“Code of the Parthenon” – Selected Transcript of 50 Minute Cable TV Version

• 0:37 – Feature shot of collectible shelf of Gary’s bookcase, featuring Parthenon, PHI 1618 license plate, shell and more.
• 0:38 – Feature shot of collectible shelf of Gary’s bookcase, featuring Vitruvian man and Eiffel Tower.
• 0:39-0:40 – PhiMatrix software overlaying the face of the Parthenon
• 0:41 – Profile of Gary working at computer
• 0:43 – Feature shot of The Golden Ratio – The Divine Beauty of Mathematics book, along with the golden mean gauges on Gary’s bookshelf
• 0:46 – Gary in office saying, “The Golden Ratio appears many different places in the Parthenon. I truly think this was done by intent by the designers.
• 1:03-1:10 – Narrator saying, with Gary analyzing Leonardo’s “John the Baptist” on the computer, “Could an ancient mathematical formula help us design and build the most beautiful building in the world?”
• 1:31-1:37 – Gary saying, “So the golden ratio, if understood, and if applied can kind of be like a secret weapon.”
• 6:34-6:45 – Narrator saying, with profile of Gary and then PhiMatrix overlay on Parthenon and his Golden Ratio book, and bookshelf collectibles on screen, “But still today, brilliant minds wonder if we can use numbers and geometry to quantify beauty.”
• 6:47-6:55 – Gary saying, “The proportions that appear in it use many, many different numbers but among these you find the golden ratio repeated again and again.”
• 6:56-7:13 – Narrator saying, “Gary Meisner is a writer and software developer obsessed with the Golden Ratio, aka Phi, an ancient Greek formula discovered by Euclid, the forefather of geometry.”
• 7:14-7:57 Gary saying, “Euclid defined the Golden Ratio as follows: A line is said to be cut in extreme and mean ratio when the length of the entire line to the larger segment is the same as the larger segment to the small segment. That’s a lot of words, but it’s really not that complicated. It’s just a matter of dividing a line at the one unique point where the ratios are all consistent. The Golden Ratio has been in used in architecture, really across many centuries. We think it was used in the Great Pyramid of Giza. The Taj Mahal, built in the 600’s, has beautiful examples of the Golden Ratio. Notre Dame has the same patterns, in both the exterior of the building as well as in the rose window that shows Golden Ratios.”
• 8:59-9:11 – Narrator, with Gary on screen, saying, “So can we use the ancient Greek principles of mathematics and geometry to construct the most beautiful building in over 2,000 years?”
• 12:51- Narrator saying, “For most of us, beauty really is in the eye of the beholder, but for guys like Gary it’s all in math.”
• 13:00-13:14 – Gary saying, “There are certain elements of design that use position and proportion, and when the golden ratio is used in those cases you end up with kind of a foolproof way of achieving beauty in aesthetics.”
• 13:15-13:25 – Narrator saying, “So let’s do it. Let’s build the world’s most beautiful building according to the ancient principles of the Golden Ratio.” Cut to show “The Golden Ratio – The Divine Beauty of Mathematics” on Gary’s book shelf.
• 13:26-13:32 –  Narrator saying, “And that mission will guide every decision we make on our modern day Parthenon.”
• 13:53-13:56 – Narrator saying, “To design it we’re going to need to bust out our calculators and do the math.”
• 13:57-13:44 – Gary in office saying, “The Parthenon is said to be the most mathematically perfect building or temple ever constructed.”
• 23:44-23:55 – Narrator saying, “But Gary Meisner still believes the Golden Ratio is central to the Parthenon’s design. He’s even developed a software based entirely on the Golden Ratio.”
• 23:56-24:58 – Gary, at computer, saying, “What the software does is (provide) a transparent overlay that appears on the screen, and provides a set of grids that are in golden ratio proportion. I’ve used my PhiMatrix software to get high resolution images of the Parthenon and to do that analysis for myself. And I’ve confirmed the presence of a number of golden ratios.” (Showing an image of the Parthenon) “Here, I’ve applied a golden rectangle to the face of the Parthenon, going from where the roof line would be projected to be to the bottom of the second set of steps here. So this is a Golden Rectangle that frames the Parthenon, but in addition to that the Golden Ratio of the height defines the top of the columns. The evidence becomes even more interesting and compelling if we look at the floor plan of the Parthenon. If we take a rectangle through all the centers of the columns, the golden ratio of that rectangle defines the dividing wall of the two inner chambers.”
• 24:59-25:09 – Narrator saying, “Gary believes there’s something fundamentally human about our attraction to this ancient Greek formula.” (Cut to the Golden Ratio Book on screen, then image of a human face.)
• 25:10-25:39 – Showing an x-ray image of a human hand, Gary saying. “Here we see that the length of each segment of your finger is also in roughly a Golden Ratio proportion. Showing an image of a human forearm, Gary saying, “This illustration shows that the length of your hand in relation to your forearm is also generally in Golden Ratio proportion. Showing an image of a DNA spiral helix, Gary saying, “Even the two spirals of our DNA are offset by positions based on the Golden Ratio.”
• 25:40-25:45 – Narrator saying, “It’s clear for Gary and the Greeks, beauty’s in the math.”
• 47:59-48:05 – Gary on screen at computer studying Leonardo’s “John the Baptist,” with Narrator saying, “By rebuilding the Parthenon according to the Golden Ratio, we’ve revived a lost construction technique, and a pathway to beauty the world has sorely missed.”

The post The Science Channel Parthenon documentary features Gary Meisner as Golden Ratio Expert appeared first on The Golden Ratio: Phi, 1.618.

Here are some questions from Elena from Austria, who was working on her pre-scientific work as one part of her final exam to finish high school, along with my responses to her. She covered a variety of interesting topics regarding appearances and applications of the golden ratio, including phyllotaxis, identifying meaningful appearances, human attractiveness, the pyramids, art and future expectations and implications.

More importantly, she asked questions on how one can know whether a claimed appearance of the golden ratio has validity, and what significance we should find when the golden ratio is found so frequently in nature.

Occurrences of the Golden Ratio

Phyllotaxis

Pineapples: It is a well-known fact in mathematics that you can find the golden ratio and the Fibonacci-sequence on pineapples. What significance to you assign to that? (Whether mathematically or spiritually speaking)

Meisner: To understand the significance, we first need to understand the processes involved. Spirals in nature in non-biological forms (e.g., galaxies, hurricanes, waves, etc.) are generally the result of equalization of forces (e.g., gravitational, nuclear, etc.). Spirals in nature in biological forms generally produce efficiency or optimization in growth processes or use of resources, and that pattern has to be encoded in the DNA of the organism.

In the case of the alternating Fibonacci numbers that appear in plants, this pattern is the most efficient way for the plant to add additional rows, seeds or stems as it grows. The rather amazing fact about this is that these Fibonacci patterns appear only because each successive growth appears at the golden angle to the previous one. The golden angle is 137.5 degrees, which is determined by taking the 360 degrees of a circle and dividing it by 1.618… to get two sections of 222.5 degrees and 137.5 degrees.
This is illustrated in the interactive model here:

https://demonstrations.wolfram.com/PhyllotaxisSpirals/

Look how different the results are when the angle is varied only slightly:

Source: http://ci47.blogspot.com/2018/04/most-rare-mathematical-proof-of-golden.html

See also an animation of the above effect at http://cmuems.com/2015c/deliverables/deliverables-09/

So now as to significance, we can see that plants are optimized in their patterns for growth processes and access to resources such as sunlight. We know that these patterns are encoded in their DNA. If this were achieved solely by evolution, we should expect to see that very early plants did not have this optimization. If this was achieved by design, we have to ask what was the source of the original information that was encoded in the DNA. We each need to study the evidence and come to our own conclusions about its significance.

From a spiritual perspective, the appearance in nature of mathematical optimization in biological forms gives perhaps an evidence of a greater design, and a designer. More important though is our innate ability to perceive beauty. We need to ask why we are able to perceive and experience beauty at such a deep level within our experience of life, whether that be in a face, a work of art, the spirals of a plant, the sound of music or the colors of a sunset. Is the experience of appreciating so many forms of beauty a necessary result of evolution, or was it given to us as a means to appreciate creation, and the work of our Creator? To me, that is a question we each should explore with an open heart and mind.

Meaningfulness of measures that find the golden ratio

Leaves and Branches: A lot of scientists, mathematicians and biologists have measured different leaves and branches to see if they could find the Golden Ratio. What do you think about the quote “”That there remains a certain margin in the choice of the distances to be measured and that one will seek to select the measurement in such a way that it gives the desired result.” (Source: (Beutelspacher & Petri, 1996, S. 133) quoted after. (Timerding, 1919))?

Meisner: I agree with that quote. There are a number of easily documentable appearances of the golden ratio in nature, but sometimes people set out looking for the golden ratio and find it via means that would be difficult to call statistically valid, undeniable or even convincing. This generally occurs in situations where:

• the measurement points were somewhat arbitrary, and weren’t based on the simplest or most obvious features of the object, such as it height and width,
• the results weren’t as close to the actual golden ratio as they should be to have made the claim, (e.g. 1.6 and not 1.618),
• the sample size was too small to be statistically validity or provide confidence in the conclusions,
• achieving the results required a complicated approach, or
• the observations were isolated and not part of a generally repeated pattern.

It is for all these reasons that in my work, and on page 57 of my book (The Golden Ratio – The Divine Beauty of Mathematics by Gary B. Meisner) that I apply four principles to all measures that claim to identify golden ratio proportions:

• Relevance: Appearances should be based on the subject’s most prominent or relevant features.
• Ubiquity: Appearances should appear in more than one place to demonstrate design, knowledge and/or intent rather than coincidence.
• Accuracy: Appearances should be within about 1% of the golden ratio, measured with as much accuracy as possible, and by using the highest-resolution images available.
• Simplicity: Appearances should be based on the simplest possible approaches, those that most likely would have been applied by the process, artist or designer.

When these principles are applied it drastically minimizes or eliminates the impact of a “margin in the choice of the distances to be measured,” and the “opportunity to select measurements in such a way as to give the desired result.”

Human Attractiveness

Human Beings: Which studies proof or show that the Golden Ratio on the human body seems attractive? (May you refer to some if possible)

Meisner: The relationship of the golden ratio to human facial attractiveness has been shown in studies by Dr. Stephen Marquardt, Dr. Eddy Levin, Dr. Kendra Schmid and others, including my own work. See these references:

Marquardt:

https://www.beautyanalysis.com/
https://www.discovermagazine.com/the-sciences/the-math-behind-beauty
https://www.goldennumber.net/beauty/

Levin:

http://goldenmeangauge.co.uk/about-golden-mean-gauge/
http://goldenmeangauge.co.uk/wp-content/uploads/2015/07/May2011_Arcticle.pdf

Schmid:

https://digitalcommons.unl.edu/cgi/viewcontent.cgi?&article=1098&context=csearticles

Meisner:

https://www.goldennumber.net/meisner-beauty-guide-golden-ratio-facial-analysis/

A number have studies have also been done that tried to discredit or deny the golden ratio relationship to attractiveness, but every study I’ve seen like this have been flawed in its analysis and conclusions. See my examples and analysis:

https://www.goldennumber.net/facial-beauty-new-golden-ratio/

https://www.goldennumber.net/meisner-beauty-guide-golden-ratio-facial-analysis/

https://www.goldennumber.net/golden-ratio-design-beauty-face-evidence-facts/

https://www.goldennumber.net/celebrity-faces-golden-ratio-beauty/

A Golden Universe

Golden Universe: Is it possible that we are living in a Golden Universe where the Golden Ratio is connecting everything?  

Meisner: In my twenties plus years of investigation, I have found many appearances and applications in which the golden ratio does appear, but I have also found many places in which the golden ratio does not appear. Based on that, I would describe its appearance as surprisingly common and pervasive, but not universal in connecting everything.

It does though continue to found in new areas, including at the level of quantum mechanics and in cosmology, such as in black holes. Given this, it may be that we will indeed discover in the future that the golden ratio connects even more than we now know. See these links as examples. The first link in particular makes these statements:

“The golden ratio may be the fundamental constant of nature.”

“In the past the appearance of this ratio has usually been ignored by scientists because they had no way to explain it, so interest in it was considered the stuff of amateur scientists. But now, for the first time, a rigorous quantum gravity is being developed which predicts the golden ratio’s existence is literally everywhere.”

https://www.goldennumber.net/quantum-gravity-reality-golden-ratio/

https://www.goldennumber.net/quantum-matter/

https://www.goldennumber.net/quantum-time/

Uses of the Golden Ratio

The Pyramids of Egypt

Pyramids: Do you think that the ancient Egypt knew about the Golden Ratio and used it or do these significances appear just randomly? If a proof could be found what do you think would change in mathematics nowadays?

Meisner: Based on my investigations, I suspect that the ancient Egyptians had an awareness and understanding of the golden ratio, but that it was not advanced in the way we understand it now. I say this because the golden ratio is very present and observable in basic geometries and in nature, even in the simple most significant proportions of the human face. The ancients understood the relationship of a circle’s circumference to its diameter, but estimated it with integers such as 22/7. In the case of the Great Pyramid of Egypt, the base, height and hypotenuse form a golden ratio based triangle, now known as a Kepler triangle, within a precision of 0.025%:

A common method at this time though to measure and construct angles for pyramids was the seked, a measure of height to width of base. It appears that the Egyptians constructed the Great Pyramid using a seked of 7 to 5.5, expressed in integers as 14 to 11. This produces a ratio of 1.272727…, which is within 0.8% of the actual golden ratio measure of 1.27202…

In my article at https://www.goldennumber.net/golden-ratio-misconceptions-by-george-markowsky-reviewed/, I show that there are only 5 right triangles based on integers from 1 to 50 that produce a variance from golden ratio dimensions of less than 1%. Of these, the triangle with a 14 to 11 ratio is the most accurate of all. So even if the Egyptians didn’t understand the golden ratio from a perspective of algebra and irrational numbers, we have to look at the evidence and ask how and why they chose a measure so very close to the golden ratio for their greatest of pyramids if it wasn’t with intent, rather than just by coincidence or accident.

The Golden Ratio in Art

Art: You pictured a few paintings from different artists and included the Golden Ratio. How do you tell if those artists included the Golden Ratio on purpose or if it was just a coincidence? Can you name artists where it is proven that they knew about the Golden Ratio and actively included it in their paintings?

Meisner: I use the same principles I listed above when analyzing the application of the golden ratio to artwork:

• Relevance: Appearances should be based on the subject’s most prominent or relevant features.
• Ubiquity: Appearances should appear in more than one place to demonstrate knowledge and intent rather than coincidence.
• Accuracy: Appearances should be within about 1% of the golden ratio, measured with as much accuracy as possible, and by using the highest-resolution images available.
• Simplicity: Appearances should be based on the simplest possible approaches, those that most likely would have been applied by the artist or designer.

You’ll notice on my image analysis that I most often use indisputable markers, such as the height and width of a canvas, or very clear and significant features of the composition. When the golden ratio is seen with pixel-level accuracy on high resolution images in the position or proportion of multiple key markers of an artist’s compositions, it becomes quite undeniable from the evidence that this could have happened by chance. Once this is established, the application by the artist of other golden ratios within a work becomes easier to establish.

Artists which I’ve found to use the golden ratio in their compositions include Leonardo da Vinci, Michelangelo, Botticelli, Raphael, Seurat, Dali and others. See the Design/Art section of my website for a variety of articles that show the evidence for this:

https://www.goldennumber.net/category/design/

See also my video on Leonardo da Vinci’s Salvator Mundi:

https://youtu.be/nXDmAtTJ6JY

A key point in that video, and in most of my investigations, is that if you find compelling evidence of golden ratios using my four principles above, you may find some that weren’t intended by the artist, but can have a high degree of confidence that at least some or most of the appearances are intentional. Unless an artist explicitly documented their use of the golden ratio it can be difficult to “prove”, but as with everything from science to court cases we make reasonable assumptions based upon the available evidence and can come to solid conclusions.

Impacts of the Golden Ratio in the future

Future: What significance do you think will the Golden Ratio have in the future? In which areas will it have an important influence?

Meisner: Throughout history, the golden ratio has continued to show up in a variety of new and unexpected places. While mankind has known of it for thousands of years, just in the past 50 years it has shown up in new areas, including DNA, Penrose tiles, acoustics, quasi-crystals, quantum mechanics, black hole mathematics and more. I suspect we will continue to discover new appearances of the golden ratio, and also find new ways to apply it, in the sciences, engineering and the arts. Like most things in life, it will be limited only by our curiosity, creativity and initiative.

The post Golden Ratio Interview – December 2020 appeared first on The Golden Ratio: Phi, 1.618.

The application of the golden ratio by the designers of the Parthenon is a common point of confusion and dissension. Multiple seemingly credible sources state that the golden ratio was not used in the design of the Parthenon. Most of these, however, trace back to a single very inaccurate and incomplete article as their source. Once this information was published by a few recognized authorities, more people yet then quoted them as a “reputable” source, and the misinformation grew.

No disrespect is intended for those who have unknowingly created or passed along inaccurate or incomplete information. The objective here is to investigate the sources, examine the evidence, present the facts, and allow readers to have all information needed to come to their own evidence-based conclusions.

What the expert sources say

As of this writing in August 2020, Wikipedia’s article on the Parthenon says this:

Some studies of the Acropolis, including of the Parthenon and its façade, have conjectured that many of its proportions approximate the golden ratio. However, such theories have been discredited by more recent studies, which have shown that the proportions of the Parthenon do not match the golden proportion. ^[66][67]“

Wikipedia’s article on the Golden Ratio says this:

Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. ^[114]“

Checking the dissenting sources

Let’s look more closely at the references presented in the Wikipedia articles to understand the basis for their claims that the golden ratio’s appearance is unsupported and now discredited.

Dr. George Markowsky on the Parthenon

Reference 66 is to the 1992 article “Misconceptions about the Golden Ratio” by Dr. George Markowsky. I reviewed this article in 2016, and found that Dr. Markowksy had performed only a single relevant calculation of the Parthenon’s width to height. He referenced seven sources who claimed that the Parthenon did embody the golden ratio. He presented Figure 6 alone below to illustrate the claimed fit. Among his references are similar illustrations by David Bergamini in “Mathematics” and Theoni Pappas in “The Joy of Mathematics.”

From the various sources, Dr. Markowsky calculated the ratio of the width of 101′ 3.75″ from Tractenberg and Hyman to the “height of the apex above the stylobate” of 59 feet from Rossiter. This yields a ratio of 1.717, which is not the golden ratio of 1.618…+. He references a quote by Christine Flon in “The World Atlas of Architecture“:

On the basis of a small number of ancient texts, an effort has been made to find (in buildings sufficiently well preserved) a coherent system of proportions based on the golden number, pi or on the universal ratios of the Pythagoreans. Almost always, when all possible measurements have been taken, some system of geometric figures or some modular common denominator has come to light. However, the validity of this research remains uncertain: it is easy to overestimate the importance of an architectural speculation, It is not unlikely that some architects, in imitation of sculptors such as Polycleitos, should have wished to base their works on a strict system of ratios, but it would be wrong to generalize. In the conservative environment of ancient Greece, architectural activity was an empirical practice in which experience and intuition, that is to say “mastery”, played a large part.”

I found the “World Atlas of Architecture” to be a wonderful resource on the history and style of architecture, but there is not one building in its 408 pages which references the mathematics of a building’s design. This is true even on its coverage of the UN Secretariat Building, whose architect, Le Corbusier publicly advanced his golden-ratio based system of design, called” Le Modulor”. So as good as this book is, it’s approach to the mathematics of design was to punt rather than to tackle the topic.

With Flon’s opinion on the “uncertain validity” and “speculation” in support of his finding though, Markowsky summarily dismissed the golden ratio of the Parthenon without any investigation into any of its other elements of design, and moved to the next topic in his article, which I review separately here.

In doing so, Markowsky notably omits the words of Flon that follow:

One thing is certain though: in the great civil and religious buildings there was a module, often the diameter of the column at its narrowest point— a basic dimension from which all the others were calculated. This made the designing and the dressing of the blocks easier, and permitted both the application and variation of proven formulae. In most cases the architect would not venture beyond simple arithmetic proportions which would ensure the uniformity of the building.”

Also of interest is that while Flon says that we should not generalize, she also notes that “it is not unlikely that some architects, in imitation of sculptors such as Polycleitos, should have wished to base their works on a strict system of ratios.“

Polykleitos wrote a treatise around 450 BC called “The Canon,” which comes from the Greek word kanon, meaning measure, rule, or law. It is a theoretical work that discusses ideal mathematical proportions for the parts of the human body, and aesthetic theories of the mathematical bases of artistic perfection. Though the text is lost to history, he is quoted as saying,

Perfection … comes about little by little (para mikron) through many numbers”.

According to the Getty Museum, references to “The Canon” in other ancient books:

imply that its main principle was expressed by the Greek words symmetria and rhythmos. By this Polykleitos meant that the work should be composed of clearly definable parts, all related to one another through a system of ideal mathematical proportions and balance.”

Symmetria means ‘common measure’. The Greeks interpreted this word, as the harmony of the different parts of an object, the good proportions between its constituent parts. This is exactly what the closer examination of the Parthenon (below) reveals. Rhythmos refers to the proportions and patterns as an exploration of the nature of motion.

Polykleitas worked alongside other Athenian sculptors, including Phidias, who designed the status of Athena and others for the Parthenon. Work on the Parthenon began in 447 BC, so the impact of Polykleitos’ treatise on artistic perfection could very well have had a very timely and direct impact on the design on Parthenon, and not be subject to the “generalization” Flon made in reference to all Greek architecture.

Dr. Mario Livio on the Parthenon

Reference 67 that follows on Wikipedia is to pages 74-75 of Dr. Mario Livio’s 2003 book, “The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number.” Dr. Livio references books by Adolph Zeising (“The Golden Section”) and Matila Ghyka (“The Golden Number“) that claim this golden rectangle, and yet another golden ratio, but he then turns to Dr. Markowsky’s article. Dr. Livio includes yet another single, simple illustration of the Parthenon bordered by a golden rectangle. He refers to the same width to height measurements and ratio as Markowsky. He too quotes Christine Flon, and thus concludes “this is far less certain than many books would like us to believe and is not particularly well supported by the actual dimensions of the Parthenon.” Again, none of the many other design elements of the Parthenon are illustrated or addressed.

Reference number 112 in the Wikipedia article on the Golden Ratio appears in the caption of the the photo of the Parthenon, but we find that the reference simply says “Livio 2003, pp. 74–75,” duplicating the reference 67 that has already been given.

Dr. Keith Devlin on the Parthenon

Reference number 114 in the Wikipedia article on the Parthenon is from page 54 of Dr. Keith Devlin’s book, “The Math Instinct.” In the copy I purchased, the reference to the Parthenon appears on page 108. There the text simply states anecdotally, “Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and the golden ratio seems to be without foundation.” Dr. Devlin presents no discussion at all of any evidence. No illustrations of the Parthenon are presented. No measurements or anything else are offered to support this statement.

I did a Google search to find that Dr. Devlin revealed his source in two of his articles on the Mathematics Association of America site:

• “Good stories, pity they’re not true,” June 2004, where referring to the Parthenon, architects in general, Leonardo da Vinci and painters in general, he states, “there’s not a shred of evidence to support any of these claims, and good reason to believe they are completely false, as University of Maine mathematician George Markowsky pointed out in his article “Misconceptions About the Golden Ratio”, published in the College Mathematics Journal in January 1992.”

• “The Myth That Will Not Go Away,” May 2007, where he states, “Then I read an excellent article by the University of Maine mathematician George Markowsky, titled “Misconceptions about the golden ratio”, published in the College Mathematics Journal in January 1992. In his article, Markowsky subjected many of the common claims about the golden ratio to a fairly rigorous review, and found that quite a few of them come up decidedly short. Further evidence against many of the common claims you see made about the golden ratio were provided by writer Mario Livio in his 2002 book “The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number.”

So it turns out that Devlin’s “two” sources were Markowsky and Livio, and Livio’s source was Markowsky. Unfortunately, as of this writing many other sources are now quoting this “evidence” and accept the conclusions of Markowsky, Devlin and Livio as credible and established.

Devlin is a noted author on mathematics and NPR’s “The Math Guy.” Livio is also an accomplished author on mathematics and science, and widely recognized astrophysicist. These gentlemen are highly intelligent, and accomplished and recognized in their fields. Their words have broad impact, even if in this particular situation their positions were lacking in evidence and analysis.

In the end, none of these very recognized sources who stated that the golden ratios are not supported by the measurements did anything to go beyond a very simplistic and superficial calculation of the Parthenon’s width to height, and even this was done inaccurately. This lack of evidence and analysis presented to support their statements is indeed unfortunate.

Flaws in the Markowsky analysis

Before we examine what the evidence actually shows for golden ratios in the design of the Parthenon, we should understand this: If the golden ratio DOES exist, why did Dr. Markowsky not find it? The answer is surprisingly simple.

Note that Markowsky’s height measurement of 59 feet is based on “height of the apex above the stylobate.” This means that he measured from the base of the columns.

Note too that in ALL of the Parthenon illustrations presented, including Markowsky’s, that the golden rectangle is NOT based on the base of the columns “above the stylobyte.” Rather, the base of the golden rectangle INCLUDES the steps of the stylobyte that lead up to the Parthenon.”

Let’s determine what his results would have been if his measurements had been consistent with the supporting illustrations.

According to the book Metron Ariston (2002), the height of the first two steps descending from the stylobyte are 1.811 feet and 1.680 feet, respectively. If we add that to the height included in the Parthenon’s height, we get 62.491 feet instead of 59 feet.

The ratio of the width of 101′ 3.75″ to revised height of 62.491′ is 1.621, a variance of only 0.20% from the golden ratio of 1.618 (to four digits)! This is well within the tolerance that Dr. Markowsky established for the basis of his investigation and conclusions.

It makes perfect sense too that any measure of the Parthenon’s height to width should include its stylobye/base, for this is critical to what the observers sees as “The Parthenon.” If asked to draw the Parthenon, anyone with good observational skills would draw the structure with its base foundation, not just columns on the ground.

More importantly, if Dr. Markowsky had expanded his investigation beyond a very simplistic measure of height vs. width, he likely would have found a much richer body of evidence for appearances of the golden ratio in the Parthenon, just as did Donald Duck in the 1959 film, “Donald in Mathmagic Land.” Both Devlin and Livio would have been quoting different evidence, measurements and conclusions as well, as would be the many who have since quoted them.

Curiously Dr. Markowsky, in his March 2005 review of Dr. Livio’s Golden Ratio book, acknowledges being exposed to the golden ratio in “a comic book entitled Donald in Mathmagic Land, which later became an animated cartoon.” In his defense, the comic book unfortunately didn’t show the same level of detail in the golden ratios of the Parthenon that are presented in Disney’s animated film by the same name. One can’t help but wonder if his evidence and conclusions might have been different if he’d seen the film.

What the evidence shows

I’ve presented evidence for golden ratios in the design of the Parthenon in several of my writings over the years:

• My article “Phi and the Golden Section in Architecture,” published in 2003, as recorded on Archive.org.
• My article “The Parthenon and Phi, the Golden Ratio,” published on this site in 2013.
• My article “Dr. George Markowsky’s “Misconceptions about the Golden Ratio” Reviewed,” published on this site in 2016.
• My book “The Golden Ratio – The Divine Beauty of Mathematics,” published by The Quarto Group in 2018. See pages 104-108.
• My article “Donald Duck visits the Parthenon in ‘Mathmagic Land,” published on this site in 2020.
• My article “The Golden Ratios of the Parthenon,” published on this site in 2020.

Note that “The Golden section in architecture – The Parthenon and Greek Architecture,” published by Dr. Ron Knott (PhD, Mathematics) in 1999 also showed the some of the same basic golden ratios.

You can find more detailed high resolution image analyses on the pages listed above, but many of the most apparent golden ratios identified in the proportions of the Parthenon are illustrated in these images. The golden rectangles and golden ratio divisions on these images were created with pixel-level accuracy using my PhiMatrix software.

Parthenon Golden Ratio Summary

So between Disney’s “Donald in Mathmagic Land” and my articles listed above, we find evidence that appears to substantiate the application of golden ratios in the following design elements of the Parthenon:

Façade:

• Façade – Overall height vs width, if you accept starting from the second step. Using that approach, there is then another golden ratio from the bottom of the second step to the top of the columns to the top of the roof.
• Triglyph – Rectangles formed by the Triglyphs.
• Metope/Triglyph – Rectangles formed by each pair of Metopes and Triglyphs.
• Entablature/Architrave – Height of the Entablature to the height of the Architrave.
• Frieze/Roofline – Height from the Column top to the top of the Frieze to the height of the roof top.
• Capitals/Entablature – Width of each pair of column Capitals in relation to the height of the Entablature.

Floor Plan:

• Perimeter Outside Columns/Chamber wall – Center points of the exterior columns in relation to the position of the wall between the two interior chambers.
• Perimeter Columns/Inner columns – Center points of the exterior columns in relation to the centers of the four columns in the smaller interior room.
• Building width/Entry width – Width of the inner enclosed structure of the Parthenon in relation to the width of the entry way into the large interior chamber.

Even if you don’t accept every single appearance presented above as a golden ratio, it would be inaccurate to say that “no evidence exists” or that this is a story “without foundation.” It would also not be reasonable to assume that these appearances were all due to coincidence, without any awareness or intent by the designers.

A geometric perspective

Frankly, we should be more surprised if the golden ratio did NOT appear among the Parthenon’s numerous precise mathematical proportions. It is truly one of the easiest ratios to create with simple geometric constructions, and likely the most ubiquitous ratio in mathematics, geometry and nature. The constructions below illustrate illustrate this fact:

Interestingly, one of the simplest geometric constructions of the golden ratio appears to be applied and illustrated in design of the frieze. It appears fourteen times actually, so it’s hard to miss:

The Greeks were clearly very intentional and precise in the numbers and proportions they incorporated into the Parthenon’s design. If the golden ratio were not used, a more compelling argument to support this premise would require developing a theory for what the Greeks DID in fact use, rather than to simply say that no evidence exists. The analysis in the photo below shows that other simple integer ratios like 3:2 and 5:3 do not match the proportions used in the frieze, but the golden ratio does:

An application perspective

Once a golden ratio has been determined with a simple geometric construction, replicating it for application in the arts is even simpler yet. All that is required is two sticks or rods of equal length that are attached at their golden ratio point. This then creates a measuring divider that can identify the golden ratio of any dimension in a second or two. Dividers have been in common use in construction since the time of the Greeks and Romans.

A historical perspective

Beyond the physical evidence itself, history tells us that followers of Pythagoras were using the golden ratio-filled pentagram as their secret symbol around 500 BC.

In about 450 BC, Polykleitos wrote “The Canon,” his treatise on aesthetic theories of the mathematical bases of artistic perfection.

It wasn’t until 300 BC that Euclid had written the first known recorded references to the golden ratio in numerous geometric constructions in his treatise Elements, but according to Britannica:

Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (flourished c. 440 BCE), not to be confused with the physician Hippocrates of Cos (c. 460–375 BCE). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle (384–322 BCE).”

Hippocrates thus compiled the first known work on the elements of geometry nearly a century before Euclid, right around the time when Polykleitos wrote on the mathematical basis of artistic perfection and when the Parthenon was being constructed between 447 and 432 BC.

So it’s reasonable to assume that knowledge of the golden ratio could have existed when the Parthenon was designed and built.

A forensic analysis perspective

For those who like TV crime shows, you may know that there are three elements of evidence often required to establish that a crime has been committed. While applying the golden ratio to the Parthenon would certainly not be a crime, we can use a forensic analysis approach to assess whether there is a case for its use. The three elements of evidence are:

• Motive – Having a reason to commit the act.
• Means – Having the tools or methods to commit act, and
• Opportunity – Having the occasion present itself to allow the act to take place.

For someone to become a suspect in a criminal investigation, all three must be established.

So in the case of the Parthenon, we know the Greeks valued incorporating mathematics and geometry into their art and architecture (motive). We know that the pentagram with its golden ratio proportions was recognized and revered for its special properties at least one hundred years before the construction of the Parthenon, and that this evolved into a very sophisticated set of constructions and propositions in Euclid’s Elements by one hundred years after (means). We know that the Parthenon, as the pinnacle of Greek architecture, would have presented a wonderful opportunity to apply the golden ratio in its design (opportunity).

Myth or Misinformation?

So hopefully you now have enough evidence and information to decide for yourself:

Does it seem most likely that the Greeks applied the golden ratio in the design of the Parthenon, or is this just a myth that won’t go away?

I look forward to your comments below.

Update on new research

Note: See an update with my latest research on the Parthenon here:

• My article “The Golden Ratios of the Parthenon,” published on this site in October 2020.

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Donald discovers the Golden Ratio

At about 9 minutes into the film, Donald Duck visits the Parthenon of ancient Greece. Here, he is shown in a 15 second segment that its design embodies a number of golden rectangles. In the next 5 minutes, he learns about other appearances and applications of the golden ratio. Click below to watch on YouTube:

As popular as the film was in mathematics classes, it seems that some students who went on to earn their PhDs in Mathematics may have missed it, or perhaps they just disagree. On Wikipedia, for instance, they’re quoted saying things like “the entire story about the Greeks and golden ratio seems to be without foundation” and “such theories have been discredited by more recent studies, which have shown that the proportions of the Parthenon do not match the golden proportion.“

Could Walt Disney and Donald Duck have been so seriously mistaken? Did they lead millions of innocent school children astray? Let’s take a look at the evidence. Then you can decide for yourself whether Donald should be awarded an honorary PhD (a Ducktorate of Philosophy?) in Mathematics or be quickly discredited as a “quack.”

To do this, we’ll look at these golden rectangle overlays using high resolution photos of the Parthenon and the pixel-perfect measurements of PhiMatrix software to assess their accuracy.

A quick lesson in Doric architecture

Before we start, it may help to review the nomenclature of the elements of classic Doric architecture:

The Eastern Facade

Disney first shows that the height and width of the facade of the Parthenon form a golden rectangle. I presented the same finding in article my 2013 article on the Parthenon. I also found that the base of the entablature is at the golden ratio point of the Parthenon’s height.

It’s not clear though why the Parthenon’s architect would have used a golden rectangle that starts at the base of the second step. It might make a more compelling case to begin the golden rectangle at the base of the columns, or at the very first step. In any event, the Disney video shows a golden rectangle using same approach I used in my analysis. Let’s call this one intriguingly close, and see how the rest of the evidence stacks up.

Column to Capitals

The Disney video shows that another golden rectangle based on each pair of columns. The height of the columns in relation to the width of the capitals that crown each pair of columns form this golden ratio. I analyzed two high resolution photos of the Parthenon. On the Western facade, a golden rectangle aligns to the columns to the pixel. Curiously, the rectangle based on the center columns on the Eastern facade has a proportion closer to 1.645 than 1.618. This variation of about 1.7% could be the result of camera angles and the impact of perspective. Disney’s claim appears to be a reasonably good fit to the golden ratio, but let’s keep looking.

Entablature to Column

The entablature, the beam-like structure, lies horizontally above columns and rests on the capitals. Disney shows that the height of the entablature in relation to the width of the capital forms a golden rectangle. The golden rectangles shown in blue on two capitals in the high resolution image below reveal that these dimensions on the Parthenon do in fact embody the golden ratio very accurately. Note that there is a dividing line on the entablature between the frieze on top and the architrave on the bottom. This line is very closely aligned to the golden ratio of the height of the entablature. You’re golden on this one, Donald!

Triglyphs

Disney shows that the fifteen triglyphs within the frieze are also golden rectangles. In the article I wrote on this in 2013, I showed that the triglyphs are in fact golden rectangles in portrait orientation. Beyond that, the triglyph and metopes together create yet another golden ratio in landscape orientation. Donald is golden on this one too.

This is particularly interesting because this illustrates one of the simplest constructions of the golden ratio. Look at this the geometric construction below to see the similarity:

Euclid used the same concept for his first golden ratio construction in Elements. It’s in Book II, Proposition 11 (with the red arc added by me):

Parthenon Floor Plan

My research for my 2018 book on the golden ratio revealed that there are also golden ratios to be found in its floor plan.

Parthenon Golden Ratio Summary

So between the Disney’s “Donald in Mathmagic Land,” my 2013 article and my 2018 book, we find the following golden ratios in the Parthenon:

Facade:

• Facade – Overall height vs width, if you accept starting from the second step. Using that approach, there is then another golden ratio from the bottom of the second step to the top of the columns to the top of the roof.
• Triglyph – Rectangles formed by the Triglyphs.
• Meotope/Triglyph – Rectangles formed by each pair of Meotopes and Triglyphs.
• Entablature/Architrave – Height of the Entablature to the height of the Architrave.
• Frieze/Roofline – Height from the Column top to the top of the Frieze to the height of the roof top.
• Capitals/Entablature – Width of each pair of column Capitals in relation to the height of the Entablature.

Floor Plan:

• Perimeter Outside Columns/Chamber wall – Center points of the exterior columns in relation to the position of the wall between the two interior chambers.
• Perimeter Columns/Inner columns – Center points of the exterior columns in relation to the centers of the four columns in the smaller interior room.
• Building width/Entry width – Width of the inner enclosed structure of the Parthenon in relation to the width of the entry way into the large interior chamber.

A historical timeline perspective

Beyond the physical evidence itself, history tells us that followers of Pythagoras were using the golden ratio-filled pentagram as their secret symbol around 500 BC.

By 300 BC, Euclid had written the first known recorded reference to the golden ratio, defining it and presenting it in numerous geometric constructions in his treatise Elements.

The Parthenon was constructed between these two dates, from 447 to 432 BC.

Quack … or not so quack here?

So what do YOU think? Is this a story that is “without foundation?” Is Donald a “quack?” Or, is there reasonable evidence to conclude that the architects of the Parthenon used the golden ratio in its design?

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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 ratios in nature? Why is it important?
• What about the design of musical instruments? I’ve noticed that even my pen has a divine proportion to it.
• What is your favorite golden mean and Phi research topics?
• In your book, you included a section for the skeptics. What do you say to them about the meaning of your research?

The interview was fast-paced and engaging, and its 40 minute duration allowed for a deep dive into some of the topics, especially as it related to where the golden ratio appears in architecture and how it can be applied by architects, and anyone else in the design arts.

Classic applications of the golden ratio include some of the most notable buildings in the world, including the Parthenon, the Taj Mahal and the UN Secretariat Building. It was even used in the futuristic design of Star Trek’s USS Enterprise.

One of the key takeaways, for architects or for anyone in the design arts, is that any design or composition they create involves many decisions on position and proportion. The golden ratio, with its unique mathematical properties and inherent harmony of proportions, provides a tool like no other to incorporate visual harmonies into all aspects of their designs.

• Leonardo da Vinci used this technique in his composition of The Last Supper and Salvator Mundi.
• Michelangelo used it in his composition and design of the many paintings on the Sistene Chapel.
• Google used it when it announced a complete revision in 2015 of its logos, icons and website layouts.
• Although not covered in the interview, cartoonist Steven Silver (of Disney, Sony and Nickelodeon) even uses it in his creation of cartoon characters. He refers to the golden ratio as his “secret weapon of design” The golden ratio prevents him “from making things so even.” He’s refers to this as “avoiding the ladder” in describing the tendency to create identical spacing in design and composition, which “makes everything feel so ordinary.” He notes that the golden ratio, by contrast, is flawless because it gives the “small, medium and large” that good design is built upon.

As Meisner pointed out in the interview, using the golden ratio may seem somewhat limiting at first, but it’s really not. Consider that just 12 notes of the musical scale have been used to create limitless songs and styles of music. Yes, there are many other frequencies that could be used, but what can be created from those 12 notes is limited only by your creativity, and all the music ever composed with the Western musical scale is in harmony with that design, and thus inherently familiar to the ear. The same is true of the golden ratio. It can be applied to every decision being made on position and proportion, and the creative results are truly unlimited. This makes it far better than the “Rule of Thirds,” which is a simple way to approximate the golden ratio, but that lacks many of its benefits.

Meisner also discussed the historical applications of the golden ratio in architecture, its appearances in nature, and a number of topics that are most subject to confusion and debate.

Tricia shared that her professor, Dr. Chang, learned about the golden ratio and its applications from his art training at the Beau Arts School in Paris. His hero was Leonardo da Vinci. While he now has a golden mean caliper, he had learn it first by doing geometric constructions with a string. Tricia intends to use the golden ratio in her career in landscape architecture, which is certainly another great application for it use.

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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:

1. 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?
2. What do we find inside The Golden Ratio book?
3. What motivated you to write this book, and what are your credentials to do this?
4. Many books have been written on this topic. What makes yours different from the rest?
5. The golden ratio has many different appearances and applications. Which ones generate the most interest or are most surprising?
6. Would you please explain your cover?
7. What has surprised you most in studying the golden ratio over the past twenty years?
8. This type of topic can be seen as a little heavy to read. How would you describe your writing style?
9. Will you share with us an excerpt from The Golden Ratio?
10. What is the biggest benefit the reader will take away from reading your book?
11. You mentioned your target audience earlier, but what other reader demographics do you think would also benefit from reading your book?
12. Other than of course selling your book, what do you hope to accomplish with it?
13. Where can readers find out more about you and your work, and where can they purchase your book?

The book is available in the US on Amazon, and in other countries as well, with translations available in Spanish, German and Dutch.

Selected excerpts of answers to the questions above.

• Introduction – This provides some basic definitions, and sets the stage for a journey the reader is about to embark on.
• Geometry – Chapter One covers the golden ratio’s unique qualities in geometry, covering what Euclid first described back around 300 BC to more recent discoveries in the past century.
• Mathematics – Chapter Two covers the golden ratio’s unique properties in mathematics.
• The Classic Arts – Chapter Three then dives into its application in the arts. It open with the quote, “Without mathematics there is no art” and then explores high resolution color images that show the application of the golden ratio in the paintings of classical artists, including, Leonardo da Vinci, Michelangelo, Raphael and Botticelli.
• Architecture and Modern Design Arts – Chapter Four opens with a quote from French painter Georges Seurat, “Some say they see poetry in my paintings. I see only science.” It covers applications in architecture, from ancient architecture to modern, and then shows the reader how the golden ratio can be applied to photography, product design, logo design, fashion design and more.
• Life and Nature – Chapter Five then reveals where we find appearances of the golden ratio in life, from our DNA to our faces, and how this impacts our innate perceptions of what seems aesthetically pleasing and natural to us as humans.
• Chapter Six explores other appearances in nature and closes with some thoughts to inspire the reader to ponder implications of the design we see, and how they might apply that in their own lives and creative pursuits.

4. Many books have been written on this topic. What makes yours different from the rest?

• The visually engaging coffee-table book style with over 250 beautiful color illustrations in a 224 page , 9.5 x 11.7” 2.9 pound book.
• The breadth of topics covered.
• The simple, non-technical approach to the topic.
• The diverse background and perspective I bring to this topic, from having connected with thousands of people over the past twenty years. I’ve done everything from helping students with their science projects to consulting with artists and designed products, commercial logos, commissioned sculptures and more. It’s been an incredible journey that took on a life of its own and I’ve learned a lot along the way.
• The original research that I’ve done, some of which has never been revealed before this book, and which uses software that I wrote to do the analysis.

7. What has surprised you most in studying the golden ratio over the past twenty years?

• How many different topics it has led me to learn about.
• How many people in different disciplines I’ve connected with along the way.
• How few people know much at all about the golden ratio
• How much misinformation and confusion there is on the golden ratio, and
• Probably most surprising of all is how much bitter debate and polarizing conflict there is on this topic. It seems to be a topic that inspires people in a wide range of disciplines to write about, but also one that most people don’t go far enough outside their own areas of expertise to fully understand. So we find articles and videos by some people claiming the golden ratio to be much more than it really is, and articles and videos by others trying to refute and debunk everything about it. I’ve always sought to be the voice of reason, evidence and fact in the middle.

Listen to the full interview on TheAuthorsShow to hear more.

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• 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

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Which face is the most attractive? Why?

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 reasonable, viable options based on integers, simple fractions, square roots or geometric constructions are far more limited. Here there are less than two dozen likely choices, and the geometric construction of the golden ratio is the simplest of all of them. Many of the conclusions presented are incorrect and incomplete, and new misconceptions are created.

Readers pressed for time: Just read the titles and overviews in bold and view the images shown as supporting evidence.

One of the more commonly referenced articles to “debunk” the golden ratio is “Misconceptions about the Golden Ratio” by Dr. George Markowsky. Dr. Markowsky has a B.S. in Mathematics from Columbia, both a M.A. and Ph.D. in Mathematics from Harvard University and worked in IBM’s research center for ten years. These are very solid credentials indeed. The article’s title, however, is rather unfortunately descriptive of much of its content and conclusions, as you will see in the analysis below.

To be fair, Dr. Markowsky provides accurate information on some common misconceptions about the golden ratio. The article was written before the Internet was widely used, so perhaps some of the information wasn’t as readily available as it is now. To his credit, he actually gathers and analyzes some measurements as evidence, which is far more than can be said of many skeptics and debunkers.

He notes, with the precision of a theoretical mathematician, that “measurements of real objects can only be approximations.” He accordingly proposes a guideline, saying:

I will consider a claim for the golden ratio to be at least reasonable if the computed ratio is within about 2% of Φ.”

He then generously rounds that tolerance up to an “acceptance range” of 1.58 to 1.66. I’m admittedly a golden ratio enthusiast, and that’s a much greater range than I allow for any of my own articles. I usually like to see accuracy to within a pixel or two on high resolution images, or variance of less than 1% from Phi at most. We should be off to a good start.

Dr. Markowsky then adds one other qualification:

Since the acceptance range includes infinitely many numbers near Phi it is necessary to justify that the claim that Phi is the preferred number. Some other ratio coincidentally near Phi might be the important one.”

This sounds reasonable enough on the surface, but this is where the article begins to lose validity. It introduces a flawed assumption that Phi is just like “infinitely many other numbers,” and misses the fact that real world measurements are limited to a small number of significant digits. It also changes the nature of the investigation from one of objective evidence to subjective opinion, deciding what evidence should be included and what constitutes a justification. Many of the “justifications” do not demonstrate a solid investigation and understanding of the claims that are being refuted. As a result, many of his assessments are incomplete and inaccurate.

Let’s review Dr. Markowsky’s analysis of these purported “misconceptions.”

#1. The Name “Golden Ratio” was used in Antiquity

Overview: It is correctly pointed out that the term “golden ratio” didn’t not come about until its use in the 1800’s.

The term to describe the number or ratio 1.618…, what we now usually call the golden ratio, has gone under a variety of different names. Euclid described it as the “extreme and mean ratio.” Renaissance artists of the 1400 and 1500’s knew it as the Divine proportion. It has also been called the golden mean, golden section and golden proportion, and has been represented by the Greek letters phi and tau.

#2. The Great Pyramid was designed to conform to Phi

Overview: The Pyramid, completed in 2450 BC, measures to within 0.09% of the golden ratio. This relationship is dismissed though, largely because of ambiguity in the interpretation of a Greek text written about it 2,000 years later in 425 BC. In addition to the golden ratio that appears in the dimensions of the Great Pyramid, golden ratios can also be found in the relative sizes and positions of all three pyramids.

Dr. Markowsky researched the dimensions of the Great Pyramid of Giza and determined that the best available measurements are this:

• The base is 755.79 feet, so the half base of 377.895. (a)
• The height is 481.4 feet. (b)

The Pythagorean theorem (a² + b² = c²) reveals the hypotenuse (c) of this triangle to be 612.005 feet. The ratio of the hypotenuse (c) to the height (b) is thus 1.6195.

Eureka! The variance from the golden ratio of 1.618 is only 0.09%. This is a difference of just 7″ in the height of the pyramid from the exact golden ratio.

So can we chalk this one up to a good example of the golden ratio? No, not so fast. Dr. Markowsky references a quote from Herodotus’ History (written about 425 BC) which describes the dimensions of the Great Pyramid. This is followed by a page and a half of narrative discussing the merits of various interpretations of the Greek text. One interpretation reveals a very simple geometry that produces an exact golden ratio relationship – when the area of the face equals the area of the square formed by the height. He concludes, however, that ambiguities in the wording make this interpretation uncertain and summarily dismisses it ALL – despite the FACT the measurements themselves have an astonishingly high degree of accuracy and were well within the acceptance range he just proposed. No other theory is offered to explain its very close correlation to the golden ratio. No other ratio or geometry is offered as a better explanation of its proportions. No rationale is offered as to why a single Greek text written 2,000 years after the pyramid’s construction should establish the validity of the claim.

Dr. Markowsky’s opening premise was that the appearance of Phi has to be justified as the “preferred number” because some other ratio might be the important one, but none is presented. The typical approach in science is to use the best theory available that fits the evidence, not to reject the only reasonable explanation you have without proposing another one. A more scientific approach would be to develop and compare competing theories and then select the one that best explains the known data. The Occam’s Razor principle would require that the simplest of competing theories be preferred to the more complex. As discussed below in this article, golden ratios are easy to create with a variety of simple geometric constructions. One select integer ratio (14/11) is within 0.06% of the height of this Phi-based triangle, and is related to a known method of pyramid construction. This may have been an integer approximation of the golden ratio the Egyptians found in nature, just as the ancients used 22/7 as an approximation of Pi. These approaches provide a much simpler and more rational explanation than the notion that one of the other “infinitely many numbers” available was used.

See my full articles on the Great Pyramid of Giza and on the evidence of the golden ratio in the design of the Giza site. The latter article provides evidence that the use of the golden ratio at Giza is not based on just one isolated instance. The apexes of the pyramids form golden rectangles to within 0.5%, and the bases of Khafre and Khufu are in golden ratio proportion to within 0.3%.

#3. The Greeks used Phi in the Parthenon

Overview: The appearance of the golden ratio in the Parthenon is dismissed because of a single measure that is taken incorrectly from the base of the columns rather than from the base of the steps, which is the traditional approach. Golden ratio relationships in other elements of its construction are ignored.

Dr. Markowsky references nine sources that claim the dimensions of the Parthenon to be based on the golden ratio. He then provides the measurements and presents these dimensions:

• The length is 228′ 1/8″, or 228.0104′ (a)
• The width is 101′ 3.75″, or 101.3125′ (b)
• The height to the base of the triangular apex from the base of the columns (stylobate) is 45′ 1″, or 45.0833′ (c)
• The height to the top of the triangular apex from the base of the columns (stylobate) is 59′ (d)

He shows an image (Figure 6) of the Parthenon’s “supposed fit.” For whatever reason, the image has been stretched horizontally to deform it to a ratio of 1.77, and does not represent the Parthenon’s real proportions. This is revealed by an overlay of his image to a photograph of the Parthenon, below center. Note in the image on the right how closely the claimed golden ratio fits the actual structure, with a golden rectangle enclosing the Parthenon, and the two golden ratios (base-column top-apex and column top-beam top-apex) in its height. Click on images for full resolution.

He then makes these calculations:

• The ratio of the length (a) to width (b) is 2.25
• The ratio of the width (b) to height (c) is 2.25
• The ratio of the height (b) to width (d) is 1.72

These ratios are all outside his acceptance range. The only one that was even relevant though was the third one (b) to (d), the fit of the face of the Parthenon at its apex within a golden rectangle. He concludes that the claim for use of the golden ratio here is erroneous and proven false. How could all nine of those sources have missed it by so much?

There’s a critical fact that he misses or doesn’t mention. The claims for the appearance of the golden ratio in the Parthenon are not based on just this one measurement. Furthermore, the one measure he does take is from the base of the columns on the floor, while the other claims measure from the steps leading up to the floor. Even his Figure 6 illustration uses the base of the steps, so why is this not used? There is admittedly some room here for interpretation in the proper starting place, but the corrected analysis has a much closer correlation to the golden ratio, as shown in the images above and below. Starting from the second step, we find alignment with the golden ratio, and it’s not just in its overall dimensions fitting inside a golden rectangle. It’s also from the bottom to the top of the columns to the top of the beam, and again from the top of the columns to the top of the beam to the apex of the roof.

These other elements of the building’s design actually provide better evidence yet of its golden ratio relationships, as shown below. None of these examples were addressed in the article, or considered in its conclusion that the golden ratio cannot be found in the Parthenon’s design.

See my full article on the golden ratios in the design of the Parthenon here.

#4. Many painters, including Leonardo da Vinci, used Phi

Overview: Insignificant unfinished works by Leonardo da Vinci are submitted as evidence, along with the lack of any reference to his use of the golden ratio in two biographies written many years later about him. Conclusions are drawn without ever taking a single measurement of the most famous works of any of the artists. 

In this instance, Dr. Markowsky quotes claims from various sources and takes no measurements at all. From all of Leonardo da Vinci’s brilliant paintings and illustrations, we’re presented with just these examples as our evidence:

With these two feable strawmen images now selected for his argument, their shortcomings are presented and the conclusion is drawn that Leonardo did not use the golden ratio in any of his other creations. He also points out that neither of the two biographies on Leonardo da Vinci he reviewed give any mention of Leonardo using the golden ratio.

If some simple measurements of Leonardo’s most famous paintings had been taken, particularly those of religious topics, it would have revealed more than enough evidence to show that Leonardo used this “Divine Proportion” quite extensively and exquisitely in their composition. See the examples below and my full articles on Leonardo’s use of Divine Proportion and his Salvator Mundi. The video below shows dozens of golden ratios incorporated into the exquisite painting “Salvator Mundi.” Furthermore, Leonardo created the illustrations for Luca Pacioli’s book “The Divine Proportion.” The first third of the book is devoted to the importance of the golden ratio in art, so Leonardo would certainly have known of it.

It was said that artist Georges Seurat “attacked every painting with the golden ratio.” Dr. Markowsky summarily dismisses Seurat’s use of the golden ratio without taking a single measure of Seurat’s hundred plus paintings and illustrations. He simply references one study that found “no justification” for the golden ratio lines drawn by someone on Seurat’s paintings, and another study whose author was unable to find any golden ratios at all. These findings are surprising, because it’s not that difficult to find clear examples. Seurat’s use of the golden ratio are shown below.

In addition, Seurat created two dozen or more paintings on small wood panels that were cut to golden rectangle proportions, as illustrated below:

See my article on Seurat’s use of the golden ratio, which shows many more examples.

The use of the golden ratio by Botticelli, Raphael, Michelangelo, Dali or others is never explored. He simply concludes this topic with one final sentence referencing someone else’s study of “the alleged use of the golden ratio by other painters, including Le Corbusier.”

Ironically, Le Corbusier is best known as an architect. Among his most prominent projects is the UN Secretariat Building, which brings us to #5.

#5. The UN Building embodies the Ratio Phi

Overview: The ratio is simplistically and incorrectly calculated using the wrong dimensions of the building. The correct measures are within 0.9% of the golden ratio. No awareness is shown that the building’s lead architect was a pioneer and advocate of the golden ratio in design.

Dr. Markowsky called the UN and received these measurements for the UN Secretariat Building:

• Height of 505′ (with heights at other elevations of 544′ and 550′)
• Width of 287′
• Depth of 72′

He then very simplistically divided 505 by 287 to get a ratio of 1.76. He tried again with 544′ and 550′ and calculated ratios of 1.90 and 1.92. He saw that these ratios were outside his acceptance range, aborted any efforts to understand the claims and concluded “the Secretariat building does not appear to be designed on the basis of the golden ratio.” He also consulted “several architecture books” and could not find the golden ratio in their indexes. End of story. Another myth debunked.

Here’s what was missed:

• The design is based on three stacked golden rectangles – Anyone who’s ever seen the very popular 1959 Disney classic educational short “Donald in Mathmagic Land” is likely to know this: The design of the UN Building is based on three golden rectangles stacked on their sides, not its simple height and width. This Academy Award-nominated movie has been shown in math classes for decades. Dr. Markowsky even acknowledges his familiarity with this work in a book review he later wrote. The three stacked golden rectangles result in the following relationships:
  • Divide the width of 287 by 1.618 to get 177.38 feet as the rectangle’s side.
  • Multiply 177.38 by 3 to get the height of the three stacked rectangles at 532′.
  • Take the elevations of 505′ and 550′ to get an average of 527.5′.
  • The golden rectangle-based height of 532 is only 0.9% greater than the average height of 527.5′, which is well within Dr. Markowksy’s “acceptance range.”
• The building’s architect was a golden ratio advocate – Le Corbusier, one of the building’s lead architects was passionate about the golden ratio in design.
  • He developed a golden-ratio based design system called “Le Modulor” in 1943.
  • He presented it in the United States in 1946.
  • He led the architectural design project for the UN Secretariat building, which was constructed in 1947 through 1952.
• The evidence is easily measured – Image analysis of the UN Secretariat Building shows that it embodies many golden ratios in its design.

See my full article on the UN Secretariat Building’s design.

#6. A Golden Rectangle is the Most Aesthetically Pleasing Rectangle

Overview: A survey from 1860 of people’s “favorite rectangle” showed a preference for the golden ratio, and the ratios closest to it. The survey is recognized as having questionable relevance and lacks context. Dr. Markowsky expands the selection to 48 choices, making the selection more difficult and less relevant yet. Data gathered from surveys he conducted with this 48 choice version is used this to conclude that a preference for the golden ratio is false. A more relevant survey would be to test this with images that have context, such as a human face.

Dr. Markowsky references five articles/books which discuss studies that show that people prefer a golden rectangle to rectangles with other dimensions. He notes that most of these are based on the original research in the 1860’s by Gustav Fechner that had study participants choose from 10 rectangles. Rectangles with ratios of 0.57, 0.62 and 0.67 (i.e. approximating the golden ratio of 0.618) were chosen 76% of the time. Dr. Markowsky expanded this test to include 48 rectangles, which obviously makes it hard to even distinguish a golden rectangle from all the others. Is there any science to suggest that this plethora of choices makes the test more accurate or meaningful? Would the test be better yet with 100 rectangles from which to choose? He concluded that people prefer rectangles in a certain range, but found no evidence to show people specifically preferred a golden rectangle.

This is an interesting study, but I have to question its validity. Others do as well, including Gustav Fechner who developed it. It shows a collection of rectangles with no context. Asking people to select a favorite rectangle from a group of random rectangles is a bit like playing dozens of musical notes and asking them to select their favorite. Musical notes only have context in relation to music created. Golden ratios only have context in relation to aesthetics of the creations in which they appear.

People have favorites of many things, but have you ever heard anyone talk about their favorite rectangle? For what? For a dinner plate? A memo pad? A desktop? An eraser? Wouldn’t anyone’s “favorite” rectangle depend on its context and purpose?

The human head often has near golden ratio proportions in its height to width. To test if the golden ratio is intrinsically related to aesthetics and beauty, a much more relevant and meaningful question would have been this: “Which of the following facial images do you find most attractive?”

Which one do you think most people would select as the most attractive rectangle, now that there is some context to the question?

The rectangles of the heads (from hair top to chin, and where the inner upper ears meet the head) have the following ratios:

2.59, 2.30, 2.06, 1.80, 1.62, 1.40, 1.26, 1.11

The fifth head from the left has golden rectangle proportions.

The appearance of the golden ratio in nature and the arts is generally based on a dividing point of a line or dimension into two parts, not by forming a rectangle. As another example, which ratio below results in the most attractive position of the mouth in relation to the eyes and chin?

This brings us to #7.

#7. The Human Body Exhibits Phi

Overview: Dr. Markowsky’s own family members average to within 0.7% of the golden ratio on the measure of their navel to their height. The relationship to the golden ratio is then dismissed because of general uncertainties as to exactly where to take the measurements, and the fact that a book he reviews uses too many significant decimal places (1.61803) in their examples of the golden ratio in two old statues and several animals. Other appearances of the golden ratio in the human form and facial features are ignored.

Dr. Markowsky knew that the navel is often said to be positioned at a golden ratio of one’s height. He took measurements of his own family members and found all their measures to be within his “acceptance range.” The average was 1.63, which is only a 0.7% variance from the golden ratio. Eureka! This claim must be true, right? Not so fast.

He raises an objection about the “precise location of the navel” rather than simply using its midpoint. A ±2% variance on a 5’6″ person gives us 2.64″ range in which to find a valid golden ratio measuring point with Dr. Markowsky’s “acceptance range.” It would be a rare person with a navel anywhere near that large. He then proceeds to discuss unseen sketches of two Greek statues with missing body parts as his evidence. His main objection is that some body parts do not offer just a single measurement point, and that the proportions in the book he reviewed cannot be accurate to the five significant decimal places of Phi, as was referenced by the authors. What happened to the “acceptance range?” Next he makes reference to some unseen images of golden ratios in animals and there takes objection again to the author’s reference to 1.61803 because of it’s excessive significant decimal places of accuracy. He ignores his own body measurement experiment that was within the “acceptance range” and dismisses the concept of the golden ratio in the human body entirely.

Dr. Markowsky never mentions, let alone measures, the perhaps most interesting appearances of the golden ratio in living creatures. These are found in the human face, and impact our perceptions of beauty.

See my series of articles on the golden ratio and beauty.

#8. Virgil’s Aeneid Exhibits Phi

Overview: Dr. Markowsky references a claim by one person that Virgil’s Aeneid is based on Phi. This straw man claim is rarely used as a good example of the golden ratio, so it doesn’t merit trying to prove or disprove it.

This claim is based on the assertion by one person, Professor George Eckel Duckworth, that Virgil used Phi as a key element in this writing. Maybe, or maybe not, but the example is so arcane that it is rarely offered as a great example of the golden ratio. It hardly belongs in any overall assessment of the validity of the golden ratio’s appearances or applications. In my fifteen plus years of running GoldenNumber.net, the only people I’ve ever heard make reference to this claimed application are the debunkers who are looking for a quick and easy win. Let’s let them have this one.

Recap of the “Misconceptions” about the Golden Ratio.

Dr. Markowsky presents what looks to be an objective, scholarly attempt to investigate and correct various misconceptions about the golden ratio. There are only eight examples given, and in my analysis we find:

• Four cases in which evidences for the original claims are missed or ignored (Parthenon, UN Secretariat Building, famous artists, human body)
• Two cases in which the measurements were taken incorrectly. (Parthenon, UN Secretariat Building)
• Two cases in which the measurements taken were far too limited to be conclusive. (Parthenon, human body)
• One case in which no measurements were taken at all. (Famous artists)
• Five cases in which actual results are far better than his acceptance range, but were dismissed due to subjective reasoning, inadequate evidence or misunderstanding of the basis of the original claim. (Great Pyramid, Parthenon, famous artists, UN Secretariat Building, human body)
• Two cases in which the claim examined was not viewed by most informed people as a meaningful or significant claim anyway. (Favorite rectangles, Virgil’s Aeneid)

You can decide whether this meets your own “acceptance range” for an objective, credible academic analysis of appearances and applications of the golden ratio.

Flawed assumptions can lead to flawed conclusions

In addition to the above errors in method, there is an critical flaw in Dr. Markowsky’s opening statement:

Since the acceptance range includes infinitely many numbers near Phi it is necessary to justify that the claim that Phi is the preferred number. Some other ratio coincidentally near Phi might be the important one.”

With this assumption, he turns his investigation into one of looking for a needle in an infinitely large haystack. The odds of finding anything that fits the golden ratio exactly, with its infinite number of decimal places, become infinitely small. This is not the case in the real world, where we routinely use meaningful, discernible, finite physical measurements. That’s why your ruler only has a limited number of dividing lines. There are physical and engineering limitations on our ability to build something with much more than four or five decimal places of accuracy, and there is often any no need for any more precision. When you’re measuring the Great Pyramid, getting a golden ratio to within mere inches of its 481.4′ height (5776 inches) should be close enough to come to a reasonable conclusion that it might well have been a factor in its design. It only takes four significant decimal places to do that, not an infinite number. There are only 33 numbers with four significant decimal places that have a variance from Phi of less than 1%, not “infinitely many numbers.” These range in increments of 0.001 from 1.602 to 1.634, so those are really the only relevant ratios to use. As, I’ll illustrate below, there are only a handful of integer relationships and triangles that produce a result this close to phi. If phi is not the likely or “preferred” number, then wouldn’t a scientific approach require that one propose a better or competing theory rather than simply dismissing the best available explanation?

Many basic geometric constructions create golden ratio proportions

It’s also unreasonable to assume that any artist or architect would set out to base a proportion on number that was very, very close to phi, but decidedly not phi. Would the ancient Egyptians, Greeks or even a genius like Leonardo da Vinci set out to create a work of art based on some number close to Phi but not Phi, such as 1.618123456789? The exercise would be meaningless because no one could achieve that level of precision. Any proportion these artists would have used would likely be the result of some integer ratio, say 8/5, or the result of a simple geometric construction. Golden ratios are rather easy to find and create in geometry, even if you’re not trying. Look at all these relatively simple constructions which all produce golden ratios (AG/AB):

Note how this section of the Parthenon below mirrors the two most basic golden ratio constructions above. These are formed by simply swinging an arc on a rectangle with one side twice as long as its other side. There’s simply no need here for complicated ratios or theoretical mathematics with infinite possibilities.

Parthenon: Alternative ratios of design

In practice, it’s frankly much easier and more feasible to incorporate a golden ratio in a design and be off by a very small margin in the execution, than it is to find some other ratio close to Phi and hit it with exacting precision. Can anyone show a single simple geometric construction that produces a proportion close to Phi, but not Phi? I’ve not seen one so what else would these designers and architects have been using?

See also my articles on Phi and Geometry.

Very few opportunities exist to construct ratios that are near the Golden Ratio

There may be infinitely many numbers near Phi, but precious few that can be created by simple integer ratios. Consider this:

• Only 10 ratios near Phi – If you take all possible ratios of the integer numbers from 1 to 50, there are 1275 that are greater than or equal to one. Only 10 of these result in a unique ratio with a variance from Phi of less than 1%.

Note: 13/8 can also be expressed as 26/16 and 39/24. 21/13 can also be expressed as 42/26. Fibonacci sequence numbers are in bold.

• Only 5 right triangles near Phi – If you take all the possible right triangles based on integer numbers of 1 to 50 for two sides or the hypotenuse, there are also 2550 unique combinations. Only 5 of the resulting triangles are like the Great Pyramid. with a variance of less than 1% from Phi.

Note: 8.660 – 11 – 14 can also be expressed as its multiples of 17.321 – 22 – 28 and 25.981 – 33 – 42.
The 11 x 14 triangle can also be expressed as its multiples of 22 x 28 and 33 x 42.

Curiously, one theory for the dimensions of the Great Pyramid proposes it is based on a triangle with a hypotenuse and side of 7 and 5.5. This is the same ratio as the 14 and 11 triangle above. If this is the case, we then have to ask why they chose these numbers, other than as an amazingly accurate rational representation of Phi.

The Golden Ratio is infinitely more probable to appear than “infinitely many other numbers”

Overview: Despite the “infinitely many numbers” near Phi, there are only less than 20 ratios that can be created with integers less than 50. Phi is the most likely choice when compared to all the other close alternatives.

So in reality, that “infinitely many numbers,” or ratios, that artists and architects have to choose from in creating something that looks like Phi, but isn’t Phi, is probably less than 20. Take a good look at the lists above and see if any of the numbers involved strike you as likely candidates, numbers that have some special significance on their own that would have made them more worthy of selection than Phi itself.

Another problem with the “infinitely many numbers” premise is that Phi is not just another number among an infinite set of others, or even among the 20 sets of numbers that get you close to it. To the contrary, Phi is one of the most unique numbers in mathematics, geometry and life, with properties that no other number has. It can be derived mathematically through a variety of limits. It appears in dozens of geometric constructions. It’s found in spiral growth patterns in nature. It’s found in the world of quantum physics, crystals and DNA molecules. It’s found in the relative dimensions of the moon and Earth and beyond. Even without knowledge of all that, it’s often found in the proportions of a beautiful face, or perhaps in the proportion of one’s own navel to their height. It’s a very unique and very ubiquitous ratio, and it’s innately familiar to us. It’s properties create efficiency in design, as well as visual harmony and beauty like no other number. It’s been recognized by mankind since the time of the ancients for its relationship to nature and its value in aesthetics.

So when you think about it that way, when we see something that is less than a percent away from Phi, the likelihood that an artist or architect used Phi instead of these close substitutes is really quite high. Even if they did chose another, we’d have to ask what made them select a proportion so very close to Phi rather than an entirely different proportion altogether (e.g., 1.414 as the square root of two, 1.5,  1.732 as the square root of three, 2, etc.).

A Sherlock Holmes view of the evidence

Phi appears extensively in mathematics and geometry. The universe is based on mathematics and geometry. Look to the evidence and reason in determining if Phi is at present. As with a good detective novel, we find the means, the motive and the opportunity.

Dr. Markowsky says that it’s “necessary to justify that the claim that Phi is the preferred number,” but in reality there aren’t that many options that make any more sense than Phi. No other options are are even offered in his analysis. We can’t get a signed affidavit from the ancient Egyptians and Greeks, Leonardo da Vinci, Georges Seurat, Mother Nature or God Himself to verify that Phi was in fact used. Nor should we expect the answers to be found in books written 2,000 years after the Great Pyramid was built, or biographies written hundreds of years after Leonardo lived, in order to see and understand intent in design with reasonable certainty. We must make the most reasonable assumptions possible given the evidence available. Le Corbusier presented his golden ratio-based design system, Le Modulor, just years before he lead the design of the UN Secretariat Building, but that didn’t qualify as justification either.

Phi appears extensively throughout mathematics and geometry. The physical universe is based on mathematics. What basis in reason is there to then suspect that an infinite number of other very close ratios with no particular significance in mathematics and geometry would then express themselves in the simple, fundamental patterns in the universe?

Has mankind also applied this ratio in its own creations? We need to look at the evidence itself and apply reason. As with any detective novel, we just need to ask if there was the means, motive, and opportunity. The means can be as simple as a marker and a piece of string. The motive can be simply to appreciate and recreate the beauty and harmony of all we see in the world around us, consciously or not. The opportunities to create with Phi are then “infinitely many.”

I invite your commentary and will correct anything in the article that is not accurate or fairly presented. Post your thoughts below.

Note on images used for illustration and evidence

All illustrations with overlaying grids in this article were created with PhiMatrix, which is accurate to within a pixel. Each line shown is in golden ratio proportion to the ones of either side of it. PhiMatrix is available for with a two week free trial through this link. Do your own tests, gather your own evidence and come to your own conclusions.

Quotations

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

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

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

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

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

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

References

Dr. George Markowsky – Misconceptions about the Golden Ratio – MAA article (Also at Markowsky site)

Excel file (zipped) with Ratios of integers and triangles near Phi 1.618 the Golden Ratio by Gary B. Meisner at www.goldennumber.net

The Great Pyramid of Giza

The Parthenon

The UN Secretariat Building

Le Corbusier

Le Modulor

Leonardo da Vinci

Georges Seurat

Google Shortcode for this Article: http://goo.gl/EGS1kI

The post Dr. George Markowsky’s “Misconceptions about the Golden Ratio” Reviewed appeared first on The Golden Ratio: Phi, 1.618.

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 stated,

while the beauty of Michelangelo’s works is indisputable, there is no information in the literature regarding any association between his works and the Golden Ratio. Thus, adopting a pioneering method, we decided to analyze one of Michelangelo’s most famous works (The Creation of Adam) painted on the ceiling of the Sistine Chapel to see if there is any relationship with the Golden Ratio.“

The study reported that:

• the fingers of God and Adam touch at the golden ratio of the width of the painting.
• the fingers touch at a point that is approximately at the golden ratio of the length of all the paintings on the ceiling of the Sistine Chapel.

Part of their “discovery” was presented here in 2013. Part of it was inaccurate.

The appearance of golden ratios in “The Creation of Adam” was published on this site in August 2013, as shown at http://www.goldennumber.net/art-composition-design/. In addition to the horizontal golden ratio they reported at the touch point between God and Adam based on their widths, my 2013 article also revealed a vertical golden ratio based on the height. This finding was contributed by reader Mikko Bojarsky. More evidence on this finding is presented below.

Their analysis of the golden ratio in the length of the Sistene Chapel’s ceiling took the analysis one step further than my 2013 article, but it turns out that it’s not very accurate. I used PhiMatrix to verify their analysis. PhiMatrix quickly and easily finds and applies golden ratios, or any custom ratio, and is accurate to the pixel. This revealed their finding of the golden ratio in length of the ceiling to be somewhat inaccurate.

Any good analysis on dimensions within a photograph requires that the image be as free as possible from optical distortion. The image selected by the Brazilian researchers for their analysis has a high degree of distortion, as shown by the curvature of the straight lines on the ceiling. Compare their image on top to the non-distorted image below it :

The Brazilian study shows that the proportions they found in the ceiling are closer to 1.6, as indicated by their annotation. The true golden ratio point is shown by the gold dividing line, slightly to its left.  Click on the image for full resolution.

Let’s do a more precise analysis yet using the non-distorted image.  This reveals that the golden ratio point of the entire width of the paintings is well to the left of the point reported in the Brazilian study, as shown below. The gold line shows the golden ratio from the outside edge of the paintings. The green line shows the golden ratio from the area of the ceiling that frames all the paintings. Click on image for high resolution.

The place at which the fingers actually touch is at a ratio of 1.68, based on the outside edge of the paintings. The touch point is 1.707, based on the area of the ceiling that frames all the paintings.

So, did Michelangelo intend this to represent a golden ratio? Possibly, but probably not with as much likelihood or precision as the Brazilian study suggests.

Background on the discovery announced in my 2013 article

Following is some detail on the background of the discovery of this information presented in 2013. The date on my original article on this topic shows as May 4, 2014. That is only because the entire GoldenNumber.net site was redesigned at that time using a new platform. The Archive.org Wayback Machine shows the original content on an archived version that page on August 8, 2013 at:

https://web.archive.org/web/20130813191911/http://www.goldennumber.net/art-composition-design/

Additionally, a Google search for “creation of adam golden ratio” in August 2015 revealed that Google already had my article and image as their top listing:

Click image for full resolution or see where it is at the current moment at https://www.google.com/?gws_rd=ssl#q=creation+of+adam+golden+ratio

This discovery was originally submitted to me on November 25, 2011 by Mikko Bojarsky, a GoldenNumber.net contributor. He used my free golden ratio template to make his finding, and sent this:

“Thought you might be interested in this application of your golden ratio ruler to Michelangelo’s Creation of Adam.  Many thanks for inventing and distributing this helpful device.” Mikko Bojarsky

The golden ratio tends to be “discovered” again and again, by many individuals across history. It is entirely possible that these discoveries have been made before mine. This does not make them any less exciting or worth repeating. As with the discovery by the team in Brazil, this is often the way that our understandings grow deeper yet. I thank them for the initiative and creativity in their work, and for inspiring me to take the study further yet.

References:

http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1677-55382012000300004

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