The Visual Spectrum of Light

Most people know the familiar colors of the rainbow and the visual spectrum of light as red, orange, yellow, green, blue, indigo and violet. This is often referred to by the acronym “ROYGBIV”:

Have you ever noticed though that the individual ROYGBIV colors we readily distinguish are not evenly dispersed across the spectrum? Some colors are tightly grouped within a small range of wavelengths and frequencies. Other colors spread across a wider band. Keep that thought.

Chakras and the Human Body

The concept of Chakras originated in the early traditions of Buddhism and Hinduism. The Sanskrit word “chakra” means “disk” or “wheel,” and refers to energy centers of the body, each corresponding to specific nerve centers and major organs. Early Sanskrit texts refer to them in meditative visualizations and also as physical entities within the body. In Hindu thought, there are seven primary chakras. These run from the base of the spine to the crown of the head. They are are often depicted in an image like the one below:

Note that the seven colors associated with the seven chakras correspond directly to the seven ROYGBIV colors of the rainbow and visible spectrum. The chakras appear evenly spaced on this diagram, but their actual positions on the human body vary in spacing. Keep that thought too.

The Golden Ratio Connection to Light, the Body and the Chakras

The amazing connection in these concepts came to Kalana Dido when he began to “connect the dots” in the pattern of spacing that was common to both the visible spectrum and the chakras of the human body.

The visible spectrum extends from the lower red frequencies of about 380 to 450 nanometers to the higher ultra-violet frequencies of about 625-700 nanometers. Kalana searched for a pattern to explain the uneven spacing in the spectrum and actual positions of the chakras on the human body. He noticed that the breaks that best defined the colors fell at alternating gaps of about 26 and 42 nanometers. Curiously, he noted that these numbers, which are double the numbers 13 and 21 of the Fibonacci sequence, have a ratio of about 1.615. This is a variance of only 0.16% from the golden ratio. He revised his model ever so slightly and found that it matched even better using golden ratio proportioned gaps of 26.18 and 42.36 nanometers.

He created this illustration of his concept. Note that the lines between the golden ratio-based circles indicate the primary color breaks, not what is within the circles themselves:

Note how the alternating gaps of the ROYGBIV colors of the spectrum align very closely with the positions of the chakras on the body:

• Red – Root (Muladhara) chakra, at the base of the spine. (Trust, belonging)
• Orange – Sacral(Swadhishthana) chakra, located just below the navel. (Sexuality, creativity)
• Yellow – Solar plexus (Manipura) chakra, located at the solar plexus. (Wisdom, power)
• Green – Heart (Anahata) chakra, located at the heart (Love, healing, relationships)
• Blue – Throat (Vishuddha) chakra, located at the throat (Communication, expression, truth)
• Indigo – Third Eye (Ajna) chakra, located between the eyes (Awareness, intention, purpose)
• Violet – Crown (Sahastrara) chakra, located at the top of the head (Spirituality, connection to the divine)

Note too how the seven ROYGBIV colors of the visible spectrum are spaced according to same alternating gaps of 26.18 and 42.36 nanometers. These numbers have relative ratios of 1 and 1.618. Oddly enough, these numbers also beautifully mirror the square and cube of the golden ratio, 2.618 and 4.236.

It has long been clear that the colors of the spectrum correspond to the colors of the chakras. It’s amazing though to connect the varied spacing gaps of the ROYGBIV colors in the spectrum to the same spacing in the chakras. To find that the spacing is related to the golden ratio is beyond amazing. What led the ancients of Hinduism to pick these particular positions to define the chakras? By contrast, various Buddhist traditions use 4, 5, 7 or 10 chakras. The 5 Buddhist chakras are sequenced from the base of spine to crown with colors of green, yellow, blue, red and white, which do not even correspond to the visible spectrum of light. It begs this question: Did the ancients of Sanskrit and Hindu tradition had a much deeper insight into the nature of energy, light and the human body than Western tradition may give them credit for?

Testing the model for validity and accuracy

I credit Kalana with much ingenuity and insight in finding these fascinating relationships, and thank him for bringing to my attention. I’m always excited to learn about new appearances and applications of the Golden Ratio. I’m also just as passionate about using my auditing background and analytic skills to assure that what is published on GoldenNumber.net is as accurate as possible. Kalana’s model fits the chakra positions on David’s classic male body above. It also fits the visible spectrum illustration quite well, but we put the model to the test.

Analysis of alternative spacings

Kalana tested his model with base wavelengths of 432 to 438 nanometers, and spacing gaps ranging from 25 to 27 nanometers. In doing so he found that the optimal spacing gap for alignment to the ROYGBIV colors occurred with a base of 438 nanometers and alternating gaps based on 26.2 and 42.3 nanometers. Below is a sample of his analysis:

Analysis of correlation to stated color wavelengths

To test the theoretically derived wavelengths from Kalana’s model, I compared the wavelengths for each color in his model to the stated wavelengths for each color of the visible spectrum as listed on Wikipedia:

I then created the graph below to show how the theoretical wavelengths in Kalana’s model compared to the Wikipeida color ranges:

The chart reveals that all of the theoretical wavelengths derived in Kalana’s model (in blue) fall cleanly between the actual low (in red) and high (in green) wavelengths for each respective color, with one exception. His model has blue at 506.54 nanometers while Wikipedia shows the actual color range for this color, which its calls cyan, has a range of 485 to 500 nanometers. This is a slight variation indeed of only 1.3% above the maximum, and is likely not that noticeable to the eye.

Analysis of spectrum images to actual spectral photos

In my analysis of Kalana’s work, it became very clear, and very frustrating, to find that there were many variations in images showing the visible color spectrum. Further research led me to learn that the spectrum of colors generated by RGB color monitors do not correspond exactly to the spectrums of sunlight that appear in nature. RGB monitors create 16,777,216 colors, but this is all based on all the mathematical combinations of 256 variations of red, blue and green. This realization just about led us to abandon the work on this model altogether. I suggested though that we use the nature spectrum of sunlight from rainbows rather than the various RGB renditions and artist renditions of spectrums. Kalana developed a new set of models based on photos of rainbows, and it still corroborated the results.

The image below applies Kalana’s model to the statue of David again. It uses three actual photos of rainbows, as shown in the inset at the lower right, as the source for the color spectrum:

Analysis of impact of individual body variations

Another potential challenge to the model was that the body proportions of individual people are of course not identical. To put this to the test, Kalana found a set of models from a swim suit site, and created yet another set of images. Despite the small range of individual differences, the results were still consistent and compelling. The images below compare a sample of individual bodies to actual photos of rainbows. Click on any image for the full size version:

Background

As a background to this story, Kalana Dido is interested in colors and their effects on the human body. He wanted to see if there was a correspondence between the visible light spectrum and the chakras. He aligned the chakras and the spectrum by shifting and resizing the images with the ratios in proportion. It was then that he noticed that the symmetries took shape.

He then measured the gaps and noticed that it was very close to the golden ratio. By shifting only 1 or 2 pixels with the chakras aligned to the spectrum, the golden ratio appeared everywhere in only two proportions. At this first stage, he noticed a spacing of 26nm and a second at 42nm, with its ratio of 1.615…, as being very close to the golden ratio of 1.618…

Kalana then wanted to solve the question of the exact value of the base by taking the reference spectrum and calibrating the wavelength of the yellow. He did this by taking 10 bases every 0.2 nm from 25 to 27 nm. After having establishing a corresponding set of yellows, he oriented his work to the 26.2nm base which seemed to best correspond visually to a real yellow. Soon thereafter, he made the connection between 26.2nm and 2.618 (phi squared) and 42.36nm and 4.236 (phi cubed).

Kalana isn’t taking these findings too seriously without additional scientific confirmation. He has a level of certainty though that the chakras are distributed according to a pattern of yxxyyx from base to crown, where y=Φx and Φ represents the golden ratio.

Conclusions

Kalana and I both regard this as a fascinating insight. It allows us to understand the association of several concepts. It appears to be yet another appearance of the golden ratio in nature, with its correlations to the human body, the seven key colors of the visible spectrum of light and the seven chakras of Hindu and Sanskrit traditions. It is not presented as a proven scientific concept or body of scientific evidence.

We hope that this presentation of Kalana’s original research and findings may lead to work by others. Key concepts to take away include:

1. The research reveals an apparent alignment of colors and positions in the chakras and the visible spectrum of light based on actual photos of rainbows and corresponding visible spectrum images.
2. The research demonstrates apparent golden ratio relationships in that alignment, with a base gap of approximately 26 nanometers, and with 42 nanometers as its golden multiple (26 * 1.618 = 42.07).
3. Testing was conducted to find the most accurate value for the gap in the wavelengths by comparison of results in 0.2 nm steps from 25nm to 27nm.
4. It was determined that the 26.2nm base gap produces the most consistent results.
5. The model then uses Φ²x10nm (2.618 nm) and Φ³ x10 (42.36 nm) as the golden ratio-based theoretical measures of the gaps. 
6. The model uses a scale of 10^-10 meters for x and y, where in the illustrations x = Φ² and y = Φ³.
7. A base of 438nm was chosen as the reference point based on best fit. The exact wavelengths of the colors cannot be defined without knowing the base. It could vary from 438nm slightly, but this was not found to impact the results significantly.
8. We acknowledge that the connection of the gaps in nanometers of 26.18 and 42.36, and the association of those numbers to the square and cube of the golden ratio of 2.618 and 4.236 is interesting and convenient, but unexpected and not necessary to the model. Nanometers are of course just one unit of measure, and the actual digits would be different if measured in inches or some other unit of measure. France is credited with the invention of the meter, which in 1790 was defined as one ten-millionth of the distance between the North Pole and the Equator through Paris. There is thus no obvious or implied reason for this connection in these mirrored digits other than coincidence … or a grander cosmic plan that is beyond our comprehension.

An invitation to continue the research

We invite anyone to test these results, further refine them and to share any new findings with Kalana and I.

You may also want to try this: In his research, Kalana made two sections of string in golden ratio proportion and used them to measure his own body. He found his own chakras fit the model as well. We’ll just invite you to take your own measurements and share your results in the comments below.

The post Light, the Human Body, Chakras and the Golden Ratio 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.

The post The Parthenon and the Golden Ratio: Myth or Misinformation? appeared first on The Golden Ratio: Phi, 1.618.

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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Carwow is UK company that helps consumers to make choosing and buying their perfect car easy and enjoyable. Stefanie Finch of their PR agency, Propellernet.co.uk, contacted me on behalf of Carwow to ask if I’d be interested in covering their story on this site. I of course said “YES!” She and the Carwow team were kind enough to provide me with the inside story on what they did, and answered my many questions to understand and validate their study.

Now here’s the spoiler. Of 626 cars analyzed, Carwow identified the best-looking car of the last decade to be the 2016 Smart ForTwo Cabrio:

To be completely honest, of all the cars I’ve ever seen in my life, this car would probably strike me as being in the bottom 10% when it comes to good-looking cars. So how did it score so high? Does the golden ratio work when it comes to cars? Time to investigate!

How could the 2016 Smart ForTwo Cabrio be the best looking car of the past decade?

As it turns out, the methodology used by Carwow was very impressive. It was thoughtful and creative, and their research and analysis was very detailed and exhaustive. I’ve reviewed facial studies by major universities on the presence of golden ratios in the human face whose methods didn’t even begin to compare to those used Carwow’s Mat Watson and team. Here’s what they did:

Identify 14 key, common markers that appear in the frontal design of a car:

• Windshield – Top Left / Top Center / Top Right
• Wing Mirror Midpoint – Left / Right
• Headlight Midpoint – Left / Right
• Grill – Left / Center / Right
• Logo/Emblem
• Body Bottom – Left / Center / Right

They then:

• Computed the ratios between every possible permutation of those points. Note that taking 3 points with 14 to pick from results in 364 combinations and 2,184 permutations!
• Identified the most significant 50 ratios found in these 2,184 sets of points, as ranked by their conformity to the Golden Ratio.
• Averaged these same 50 ratios for each car to get the car’s score for conformity to the Golden Ratio
• Converted that to a score between 0 and 100.

Carwow said that the ratio between 0 and 100 was built to give a scale of conformity. They used an average of the ratios and then calculated how close that was to the golden ratio as an absolute number.

Carwow also used a min-max scaling methodology between the ratio furthest from the golden ratio and the ratio closest to the golden ratio. This provided a theoretical limit of how close to golden ratio conformity the cars could get. This was then applied to the average of all the 50 ratios for all 626 cars. In practicality, these two measures were almost the same thing, but they wanted to use their actual measurements to keep things feeling ‘fairer’. The score for first place SmartForTwo Cabrio was a 98.83% conformity to the golden ratio.

A great methodology, so why the questionable results?

I must compliment Mat and the Carwow team on doing such an exhaustive evaluation of car “facial marker” possibilities, and for identifying the ratios in their body of evidence that most closely aligned with the golden ratio in then doing their assessment. Many university studies of the human face by major universities and industry professionals might have produced much better results if such rigor had been applied.

Still, I was puzzled as to how a car that I found to be so visually unappealing could have been the best-looking car of the past decade.

I did some research to find the best looking cars as judged by other major automotive magazines, including Car and Driver, AutoGuide and US News.

Among them, I found three cars that were recognized by both Car and Driver and AutoGuide for their styling and beauty. I found that these great looking cars all rated below #200 in the Carwow rankings:

• Lexus LC – Ranked #214 by Carwow with a score of 96.50%
• Alfa Romeo Giulia – Ranked #240 by Carwow with a score of 96.20%
• Infiniti Q60 – Ranked #224 by Carwow with a score of 96.38%

I also asked Carwow’s project lead, Mat Watson, for his personal favorites, which were:

• Peugeot 508 – Ranked #123 by Carwow
• Aston Martin Vantage – Ranked #554 by Carwow
• Range Rover Velar – Ranked #270 by Carwow

I asked, “Ignoring the analysis for a second, and just going on feelings, how would you rate the beauty of each of your top 3 cars against the Carwow top three, simply using a 1-10 scale, where 10 is the highest?”

Iain Reid, Carwow’s site editor, provided his rankings:

• Peugeot 508 – 8/10 by Ian, but only #123 by Carwow
• Aston Martin Vantage – 9/10 by Ian, but only #554 by Carwow
• Range Rover Velar – 9/10 by Ian, but only #270 by Carwow
• Smart Fortwo Cabrio – 2/10 by Ian, but #1 by Carwow
• Volkswagen Up – 4/10 by Ian, but #2 by Carwow
• Vauxhall Mokka – 2/10 by Ian, but #3 by Carwow

There continued to be a clear disconnect here. I asked why their personal tastes didn’t align with the Golden Ratio results, and the Carwow team shared this perspective with me:

“For many people, it’s impossible to ignore the whole chassis of the car, or dismiss the beauty of the individual features within the headlights, the hood ornaments or bumper details. When determining the beauty of something, humans take in a much wider view, noticing all the smaller design details that get lost in a pure ratio/distance analysis. Furthermore, other non-design factors evoke an emotional reaction into how someone responds to a car’s aesthetic. For example the noise they imagine the engine makes, how they imagine the leather seats might feel or smell like, or the thrill of adrenaline they might feel at opening up into 6th gear on an open coastal road. It’s these extra imagined layers of emotion that influence people’s perception of a car’s beauty – whilst your brain might not read the face of a Smart Car as being ugly or ill-designed, the model doesn’t evoke those powerful stirs of emotion in the way Aston Martin or Range Rover does.”

I agree that there are more factors in a car’s beauty that its front view. It seemed though that we should be able to make a reasonably consistent assessment of a car’s visual appeal and “beauty” without smelling the leather or hearing its engine roar.

The Golden Ratio – Beautiful or Ugly?

So it was very clear that the cars that Carwow identified as being closest to the golden ratio in their key markers were in fact rather ugly cars, when judged by the Carwow team’s natural instincts as car buffs and automotive professionals. Conversely, the cars that this team thought to be the most beautiful ranked surprising low on the Carwow list.

So is this proof that the golden ratio has no real correlation to beauty?

Fortunately, having studied this topic for over twenty years, I was able to identify the problem. While Carwow’s methodology was very ingenious and rigorous, one of their most fundamental assumptions had a fatal flaw.

The Carwow team assumed that any golden ratio relationship between two line segments formed by three of their 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 the very definition 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:

Auto manufacturers have used this technique in the design of their logos:

Aston Martin applied this same golden ratio concept to the design of their 2014 Rapide-S and DB9 models:

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, the logos, the Aston Martins and 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 front end of a car would be. All the same golden ratio relationships exist between the lines, but the beauty 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, but the face would no longer be beautiful either. These golden ratios are no longer discernible by the human eye, especially when placed among the 364 possible combinations of lines.

Golden Ratio design is all about visual aesthetics, not just mathematics.

Unfortunately, this type of visual distortion is pretty much what Carwow did in taking their measurements and then doing their ratings. They did the math, and did it very well, but they missed the the very critical fundamental concept that the beauty associated with golden ratio is 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 not based on just a jumble of lines with mathematical ratios in their relative lengths that happen to come close to 1.618.

To illustrate the point, here’s another set of lines from the original five-pointed star whose golden ratio relationships have been distorted. The blue lines are 95% of their original size, the red are 90% and the green are 85%.

Using Carwow’s scoring methodology, this set of lines above would have a much lower correlation to the golden ratio and should thus be not nearly as attractive as the jumbled version of the star above. That’s clearly though not the case. This star is still more appealing because the lines better represent the concept of dividing a line at its golden ratio point. That is likely why cars that were in fact designed with better aesthetics produced lower conformity scores than the cars that were really quite unattractive.

Reverse engineering the golden ratio data and proving the concept.

I reverse engineered the methodology of Carwow’s results and confirmed with them that I’d done it correctly. From this, I found that if we look at the 50 markers that produced the highest correlation to golden ratios for the Smart ForTwo Cabrio, we find 23 relationships that are with 0.02 of the golden ratio:

The two images below illustrate the actual configuration of each of the above 23 “golden ratio” markers above, with a different color for each set of two lines that they identified as being a “golden ratio:”

For the 23 Smart ForTwo Cabrio markers whose golden ratios varied by less than 0.02 from the golden ratio, only ONE was in a straight line. This was the Grill Center – Emblem Center – Windshield Center measure, with its two permutations appearing twice in the list above.

Golden ratios, yes, but not perceptible by the human eye.

This means that all of the rest of the data points that drove the Smart ForTwo Cabrio to the #1 position in Carwow’s list were actually based on measures of markers that were visually confusing by being at odd angles to one another and not forming a straight line. As such, the lines formed by these markers would have had no real visual impact on the human perception of golden ratios in the car’s design. There was thus no reason to rank the cars as beautiful based on the measurements used by the Carwow team. That, I truly believe, is why there was no correlation between what everyone perceived as being beautiful and what their “golden ratio measurement” rankings said.

Great efforts don’t always produce great results, but we still win by learning.

Despite my findings on the lack of validity in the methodology and the results of the Carwow beauty rankings, I want to compliment the entire Carwow team on their ingenuity on coming up with this approach, and their very hard work in the exhaustive measurements and calculations that they performed.

Perhaps by applying the concepts I’ve described above, they could rerun the numbers and find out which cars really come closest to golden ratio proportions based on the visually perceptible division of a straight line into its golden ratio proportions, as we see in the beauty of a human face.

Applying the Golden Ratio in design.

What if we were to to apply traditional golden ratio design techniques to analyzing the beauty of car design? Mat might find that his personal favorite, the Peugot 508, has more golden ratios than the Carwow team realized that explain some of its beauty, as illustrated below:

I would like to thank the Carwow team for so generously sharing their research, time and perspectives with me.

I’d be happy to consult with the Carwow team in approaching this initiative with a more traditional and classic use of the golden ratio. I am happy to make the same offer to any parties who are interested in studying appearances of the golden ratio and its relationship to human perceptions of beauty and aesthetics. Golden ratio analysis isn’t difficult in concept, but it does require a depth of experience and knowledge of its appearances and applications to prevent the analysis from leading to the wrong conclusions.

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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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