In my analysis, I’ll use the golden ratio grids created with pixel-level accuracy by my PhiMatrix software. These include golden rectangles and/or rectangles with a dividing line at the golden ratio point of their height or width. I’ll apply these to the best photos and illustrations of the Parthenon available. This requires that the images be very high resolution, and taken from directly perpendicular to the Parthenon to avoid distortions from perspective.

As a quick background, the golden ratio is defined by dividing a line at the one point at which the ratio of the larger segment (a) to the small segment (b) is equal to the ratio of the line (a+b) to the larger segment (a). Note that the PhiMatrix grid always shows the golden ratio point of any dimension:

It will help to be familiar with names of the elements of classic Greek Doric architecture as well:

Golden ratios in the entablature of the Parthenon

One of the simplest constructions of the golden ratio is below. It divides a square at its midpoint and then with an arc creates a golden rectangle with the following properties:

• The height of the rectangle is the golden ratio of its width
• The width of the square is at the golden ratio point of the rectangle’s width
• The small vertical rectangle adjacent to the square is also a golden rectangle.

We find that this basic construction of a golden rectangle divided at its golden ratio point is found in the 14 golden rectangles formed by each metope and triglyph set across the frieze of the Parthenon:

Note that even small variations from the golden ratio of 1.618… do not align well with the actual dimensions:

The architrave then was then designed to have the same height as the frieze. By doubling its height, this creates a golden rectangle with twice the dimensions of that of the frieze. This is used to create yet another golden ratio in the design:

• The distance between the columns is a golden ratio of the height of the entablature: 

A row of smaller blocks was set across the entire entablature to create the geison, which adds these golden ratio properties:

• Each block of the geison is a golden rectangle whose width is half that of the metope-triglyph golden rectangle.
• Each block of the geison has a horizontal dividing line at the golden ratio of its height.
• The height of the entablature with the geison added creates a golden rectangle that defines the width of the capital of the column.

Golden ratios of the columns

The spacing of the columns and the width of the capitals of each creates yet another golden ratio property:

• The width of two capitals on the columns defines a golden rectangle whose height is that of the columns:

Golden ratios of the overall façade

This dimensions and proportions of the entablature and columns tie into an overall structure of design that creates the following golden ratios:

• The height of the Parthenon, from the base of the second step to the top of its roof peak (projected lines), is a golden ratio to its width at the end of the entablature.
• The base the entablature is at the golden ratio point of the height of the Parthenon.
• The top of the frieze is at the golden ratio point of the distance from the base of the architrave to the Parthenon’s height.

For full disclosure and transparency:

• A bit of creative license must be taken in defining the base of the Parthenon. This is due to the fact that the base is in fact slightly curved, perhaps for drainage and also to make it look more perfect to the eye.
• The peak of the Parthenon must also be estimated based on the existing roof lines, as much of the pediment at the top was destroyed.
• The golden rectangle based on the width of two column capitals to the height of the columns is a more precise golden rectangle on the west façade than on the east façade.

Golden ratios in the floor plan of the Parthenon

In addition to the golden ratios in its façade, there are also golden ratios that appear in the floor plan of the Parthenon:

• The interior wall between the two chambers is at the golden ratio point of the distance between the center points of the exterior columns on each end.
• The centers of the interior columns and base of the statue of Athena are at the golden ratio points of the distance between the center points of the exterior columns on each side.
• The width of the entry way to the large chamber is a golden ratio of the distance between the outside of the outer walls of the interior chamber.
• The interior of the large chamber is a golden rectangle, when using the center of the entry wall as the rectangle’s perimeter.

Source of original floor plan: Argento, public domain on Wikimedia.org.

For transparency, note that a variety of Parthenon floor plans are available on the Internet, and they are not all consistent in their representations of dimensions. The above image was of very high resolution and includes a scale of the dimensions. Below are two other versions on which the golden ratios shown on the individual floor plans above are consolidated into a single view. Certain golden ratios align rather precisely across all three floor plans, including:

• The width of the entry way into the main chamber in relation to the width of the building
• The golden rectangle created by interior dimensions of the main chamber

Other dimensions vary slightly among the three floor plan versions, creating slight deviations from the golden ratios identified above:

• The exact position of the wall between the two interior chambers
• The exact positions of the four columns in the small interior chamber

Clearly only one version can be right, so I’ll gladly update this analysis for any source that can be shown to be authoritative and accurate.

Golden ratios in the interior chamber

In addition to the golden ratios in the floor plan of the interior chamber, the interior chamber has two stacked sets of columns that appear to be in golden ratio proportion to each other.

Note here too that a variety of Parthenon interior illustrations are available which are not all consistent in their representations of dimensions. The Metron Ariston version was chosen because it is a well recognized source of Parthenon dimensions and of high resolution.

Support for application of the Golden Ratio by the Greeks

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

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

The first known recorded references to the golden ratio did not appear until about 350 BC, when Euclid’s included numerous geometric constructions in his treatise Elements. According to Britannica though:

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.

Other research

For other background on this topic, please see the evidence for golden ratios in the design of the Parthenon and other historical information and perspectives in my previous 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 Parthenon and the Golden Ratio: Myth or Misinformation?,” published on this site in 2020.

 An open invitation for research and collaboration

Research on the Parthenon is difficult because there are so many sources of information that are not consistent, and not well documented as to the authority or accuracy of the sources. Perhaps some of the best documentation is only available in Greek or in academic circles, which limits its availability. I welcome the opportunity to share and discuss findings with anyone on this topic, and can be reached through the Contact page of this website.

Credits:

Image of Parthenon East face used for analysis: Greece, Athen, Acropolis, Parthenon, from east, Date 9 October 2017, Author Berthold Werner, 6016×4000 pixels. This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Image of Parthenon West face used for analysis: Western facade of the Parthenon during its restoration, Acropolis of Athens, Greece. Date, 20 July 2008, 2000×1579 pixels. Taken by Eusebius (Guillaume Piolle). Attribution: © Guillaume Piolle / CC BY 3.0

Interior view of Parthenon: Metron Ariston, a book written in 2002 and published on 11 February 2003 in Athens,Greece (275 pages – in Greek). ISBN 960-8286-06-9.

Interior view of Parthenon: GreatBuildings.com

The post The Golden Ratios of the Parthenon appeared first on The Golden Ratio: Phi, 1.618.

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.

The post Carwow, best-looking beautiful cars and the golden ratio. appeared first on The Golden Ratio: Phi, 1.618.

Stamp collecting is one of the world’s most popular educational hobbies, and is enjoyed by people of all ages and nationalities. The most popular stamp ever is US 1993 Elvis stamp, with over 124 million collected. So, the Golden Ratio and Fibonacci sequence may not compete with Elvis in popularity anytime soon, but they too have been recognized with commemorative stamps issued by governments around the world. In addition to adding some “phi” to “philately,” they offer some interesting expressions of golden ratio concepts, as well as its application in design.

Liechtenstein 2013 Commemorative Fibonacci Sequence and Phi Stamp set:

The Principality of Liechtenstein, a landlocked micro-state bordered by Switzerland and Austria, issued a set of three stamps in 2013 that illustrate the Fibonacci sequence and its relationship to the golden ratio.

Note that interesting presentation of concepts:

• The first stamp lists the numbers of the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, etc. Alternating bold letters were used to show the numbers.
• The second stamp shows the ratios of each successive pair: 2:1, 3:2, 5:3, 8:5, 13:8, 21:13, etc. Alternating bold letters were used to show the pairs.
• The third stamp shows the golden ratio (1.6180339887… ). The ratios of the Fibonacci pairs converge on as the series increases. Curiously, the decimal representation of Phi (Φ) on this stamp has two errors. 3653 should be 36563, and 0911 should be 0917. These differences would not impact anyone’s practical use of the number, but here they are:
  • 1.61803398874989484820458683436563811772030917 is the golden ratio, as rounded.
  • 1.6180339887498948482045868343653811772030911 is shown on the stamp.
• Note that the final numbers in purple, green and red are denominations and not part of the Fibonacci sequence or digits of Phi.

Note: Click on the images on this page to see the full resolution images.

Use of the golden ratio in the design of the Liechtenstein stamps

Interestingly, the designer very creatively applied the golden ratio in the design of the stamps as well. This is shown in the grid overlays created with PhiMatrix software.

• Stamps 1, 2 and 3 (top row): The center stem of the leaf in all three stamps is positioned at the golden ratio of the width of the stamp. The point where the stem meets the leaf is also at a golden ratio of the height to the stamp.
• Stamp 4 (bottom row left): The purple leaf design within the circle aligns to golden ratios from the center of the leaf to the outside of the circle.
• Stamp 5 (bottom row middle): The points of the purple leaf aligns to a template based on the golden ratio. The concentric circles, divided into 36 sections, have diameters are based on the square root of the golden ratio. This means that every other circle is a golden ratio of the diameter of the one before it.
• Stamp 6 (bottom row right): The dimensions of the stamp itself are a golden rectangle. The blue rectangle inside perforations has a length to width ratio of 1.618.

All in all, a rather very creative and thoughtful expression of the golden ratio and its application in design.

Macao China 2007 Commemorative Golden Proportion Stamp set

Macao China in 2007 issued a two sheet series called “Ciência E Tecnologia- A Proporção Dourada,” or “Science and Technology – The Golden Proportion.” Macau, spelled Macao in English, is officially the Macao Special Administrative Region of the People’s Republic of China. It’s an autonomous territory in East Asia, bordered by the city of Zhuhai in China and the Pearl River Estuary.

One sheet has just a single stamp, surrounded by illustrations. It has the digits of phi as its background. The other sheet has 8 stamps. It has a nautilus shell as its background.

The full sheet has four wonderful illustrations of basic golden ratio concepts.

• Definição / Definition (upper left): This shows the a line divided at its golden ratio. This is based on the ratios of the three points A, B and C. AB is created by the dividing point C, where AB/AC=AC/CB. This equation can be solved using the quadratic formula. The only positive solution is the golden ratio, also known as phi, and is calculated as (1+√5)/2.
• Divisão do todo AB / Division of entire AB line (lower left): The geometric construction to divide a line at the golden ratio, where DB=AB/2.
• Traçado de maior AC / Tracing of line AC (upper right): The geometric construction to extend a line to its golden ratio, where BG=CB/2.
• Traçado de menor CB / Tracing of line AC (lower right): The geometric construction to extend a line to its golden ratio from its midpoint, where CF=AC and OC=AC/2

The sheet with four stamps shows:

• Yellow: The rabbit population growth model identified by Leonardo Fibonacci. This led to his discovery of the sequence that bears his name.
• Red: A pattern of alternating Fibonacci number spirals. These appear in the seed pods of flowers, pine cones and other plants. This particular example has 21 counter-clockwise spirals in green and 34 clockwise spirals in gold.
• Blue: A pattern illustrating five-fold symmetry based on the solution identified by Roger Penrose. This is found in his phi-based “kites” and “darts.” These are called Penrose tiles.
• Green: A Fibonacci spiral formed from squares of the successive Fibonacci numbers 1, 1, 2, 3, 5, 8 and 13. A spiral arc connects the corners of each square. A shell pattern with chambers based on a second spiral (the thinner line) illustrates the relationship to spirals in sea shells.

Use of the golden ratio in the design of the Macao stamps

Here too the designer used the golden ratio in the design of the stamp itself:

• Left: The width of the stamp is based on the golden ratio of the width digits of Phi shown on the sheet.
• Middle: The width of the golden spiral is based on the width of the digits of Phi shown in the white area.
• Right: The width of the Penrose tiles is based on the width of the digits of Phi shown in the white area. In this case, it is at the golden ratio of the primary golden ratio.

Monaco 2000 Commemorative International Year of Mathematics Stamp

Monaco issued a 6.50 Franc / 0.99 Euro postage stamp in 2000 titled “Année internationale des mathématiques” (International Year of Mathematics) among their stamps issued for that year.

The stamp features Leonardo da Vinci’s famous illustration Vitruvian Man, which is based on the mathematical proportions of human design by Vitruvius.

While it covers aspects of mathematics beyond the golden ratio, it includes:

• A golden ratio construction, with the note that dimension AK = 1.618 and φ. This would be in relation to the diameter HK of the circle.
• A pentagon, and five-pointed star, the dimensions of which are all related to the golden ratio.
• A dodecahedron, whose dimensions and the xyz positions of its vertices are all related to the golden ratio.

It appears that the stamp designer, Claude Andréotto, may have used the golden ratio in several aspects of his composition of the stamps:

Italy 1994 Commemorative Luca Pacioli Stamp

Italy issued a 750-lira postage stamp in 1994 to commemorate the 500th anniversary of the publication of Luca Pacioli’s Summa de Arithmetica. The inscription on the stamp “Della Summa de arithmetica, geometria, proportioni et proportionalita” means “The Summary of arithmetic, geometry, proportions and proportionality.” This book on mathematics was written by Luca Pacioli and first published in 1494. It contains a comprehensive summary of Renaissance mathematics, including practical arithmetic, basic algebra, basic geometry and accounting. Pacioli continued his work in this area with a study of the golden ratio in his publication of “De Divina Proportione”, or “The Divine Proportion” in 1509. While this stamp specifically relates to Pacioli rather than the Divine Proportion itself, it references his earlier work on proportions and proportionality.

San Marino 1992 Commemorative Divine Proportion Stamp

The Republic of San Marino, an enclaved microstate surrounded by Italy, issued a sheet of stamps in 1992 commemorating the life of Piero della Francesca. Much of his work was later absorbed into the writing of others, notably Luca Pacioli. Piero’s work on solid geometry appears in Pacioli’s “De Divina Proportione”, which has illustrations by Leonardo da Vinci. This work is recognized in the middle stamp of the set, which is captioned “de Divina Proportione”:

Switzerland 1987 Commemorative Golden Rectangle Design Stamp

Switzerland issued a stamp in 1987 to commemorate the 150th anniversary of the Swiss Society of Engineers and Architects.

It includes references to:

• Helvetia, the female national personification of Switzerland, and also its name in Latin.
• Stephan Bundi, a Swiss poster designer and graphic designer.
• Jules-Ami Courvoisier, a Swiss designer who often used black-and-white effects or just two colors to create an expressive, figurative design. In addition to numerous posters, he also created stamps for the Pro Juventute and illustrated books.
• The Schweizerische Ingenieur und Architektenverein (SIA), or Swiss Society of Engineers and Architects.
• And last, but not least, the connection of the golden ratio to design. This was done with a golden rectangle spiral, with an arc connecting each 90 degree arc of its corners.

Dominica 1986 Commemorative Leonardo Fibonacci Stamp

The Commonwealth of Dominica, is a sovereign island country and part of the Windward islands in the Lesser Antilles archipelago of the Caribbean Sea. It issued a stamp in 1986 to commemorate Leonardo Fibonacci, with the Greek letter Phi representing the golden ratio.

Japan 1986 Commemorative Golden Section Stamp

Japan issued the “Golden Ratio Centenary of the Architecture” stamp in 1986 to commemorate the 100 year anniversary of the founding of Architectural Institute of Japan. It too recognized the relationship of the golden ratio to design. It did so with its illustration of the classic construction of a golden ratio using an arc swung from the midpoint of a square.

In addition to the golden ratio construction, the designer of this stamp very creatively incorporated the golden ratio into the design of a number of its elements. This is shown by the red grid lines, each of which is in golden ratio proportion to the ones on either side of it:

Nichtsburg & Zilchstadt 2004 Fibonacci Commemorative Stamp

In 2004, the non-existent nation of Nichtsburg & Zilchstadt issued a stamp bearing the image of Leonardo Fibonacci. This falls in the realm of fantasy issues, also referred to as “Cinderellas.” This stamp was included in a set of four stamps on mathematics that were created by Inzander Berenku, of Occussi-Ambeno’s Imperial Government Printing Office (KDPN). “Nicht” and “Zilch” mean “nothing,” so it was quite appropriate that the stamps commemorated mathematicians who advanced the concept of zero, or nothing. The 30 Midenika stamp honors Leonardo Pisano Fibonacci. Fibonacci brought nothing, or zero more precisely, to Western Europe from the Arabian world. Their decimal place-value numeral system quickly replaced the Roman numeral system. The “Midenika” is the plural of Miden, the modern Greek word for “zero”, and is a fictitious monetary unit. That’s a whole lot of nothing, isn’t it!

What about the United States Postal Service and other major countries?

You may have noticed that the world’s largest countries are absent from the list of those commemorating the golden ratio in their postage stamps. In fact, some of the countries that issued commemorative stamps are among the very smallest in the world, and one is even non-existent. Perhaps that’s because these small countries do not have their own rock stars, movie stars and super heroes to honor with a stamp. Yes, perhaps that’s the reason. On the other hand, maybe the pop-icon cultures of major countries just keep them from fully appreciating the finer things in life, like the golden ratio, that are truly timeless in their beauty and impact!

25 most popular stamps in the US

With a few exceptions, there’s a striking contrast in the subjects of stamps that are the most popular in American culture. Here’s the list of the 25 most collected stamps issued by the United States Post Office:

And here are the top 10 from the list:

Submitting ideas for new stamps

So how can we get more mathematically-oriented subjects on stamps? The Postal Service receives suggestions for new stamps from about 50,000 people every year. Only 20 to 25 of those make the cut. This selection is made by the Postmaster General’s Citizens’ Stamp Advisory Committee, which includes designers, historians and academics. They review the submissions and make their recommendations to the Postmaster General for final approval.

So there’s a lot of competition, but if you’d like to recommend that the Golden Ratio and Fibonacci series be recognized with a commemorative stamp in the good old USA, the Postal Service encourages those with ideas for new stamps to submit their suggestions in writing to:

Citizens’ Stamp Advisory Committee
Stamp Development
U.S. Postal Service
1735 North Lynn St Rm 5013
Arlington VA 22209-6432

Every ten years there is an international celebration of stamp collecting at the World Stamp Show, which was hosted in 2016 in New York City. The show is attended by stamp collectors, dealers, exhibitors, postal administrations and other societies from around the world. There they sell, buy and discuss all aspects of this hobby, which is known as philately. Maybe if enough of you reading this article write to the Postmaster General, we’ll see some more golden ratio commemorative stamps on display at the next show.

Create or purchase your own custom USPS postage stamps!

For those of you who just can’t wait that long or take the competition, there is still another option. You can create and order your very own custom US postage stamps at Zazzle.com. They’ve already created some beautiful Golden Ratio Stamps, and Fibonacci Stamps, including some of those shown here:

If you know of any Golden Ratio, Phi or Fibonacci stamps that weren’t covered in this article, please let me know and I’ll include them, with a credit to you. Send your submissions via the Contact page.

References:

https://en.wikipedia.org/wiki/World_Stamp_Show-NY_2016

http://www.ny2016.org/

http://my-philately.blogspot.com/2007/12/golden-ratio.html

http://www.zazzle.com/golden+ratio+stamps

http://jeff560.tripod.com/stamps.html – Stamps on mathematicians and mathematics.

https://www.catawiki.com/catalog/100271-stamps

http://colnect.com/en/stamps/list/country/2618-Liechtenstein/series/191392-Mathematics

https://www.macaupost.gov.mo/Philately/XVersion/News.aspx?lang=en-us&pm=184

http://numismaster.com/ta/numis/Article.jsp?ad=article&ArticleId=6993

http://erikmccrea.tripod.com/zilchstadt.html

The post Golden Ratio, Phi and Fibonacci Commemorative Postage Stamps appeared first on The Golden Ratio: Phi, 1.618.

Stephen Silver Golden Ratio in Character Design video

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

The golden ratio is flawless because it gives the “small, medium and large” that design is built upon.

He describes the golden ratio is just perfect, flawless, ideal and brilliant because it gives you the “small, medium and large” components of a character or object. He notes, “that’s what we really build design on, because we’re so used to seeing it in the human face and everywhere we look in trees and animals.” It can be used for any kind of design, including:

• Character design
• Prop design
• Environmental design
• Building design
• Architectural design
• Clothing design, and
• Graphic design

The golden ratio stops him “from drawing things and making things so even.” He’s coined this term “avoiding the ladder” to describe the tendency to create identical spacing in design and composition. “That’s what you want to avoid in your design,” he says, “so you don’t make everything feel so ordinary.” Without this “it becomes almost too rhythmical when it’s not supposed to be. We want to have more versatility,” he says.

Stephen illustrates the concept by showing in his video how the golden ratio can be found in the body sections of insects and features of the human face and human body. These are the same principles shown on GoldenNumber.net since its inception in 1997.

Cartoon characters designed by famous artists with the golden ratio in mind.

Stephen illustrates in his video how he applied the golden ratio in his design of characters such as Kim Possible.

He notes too how golden section appears in the Fred Flintstone character created by Ed Benedict. Note in the photo on the right where he illustrates above the head that a nose that was instead 50% of the width would not seem as natural:

Stephen comments that many other designers use the golden ratio, some by direct application of a golden mean gauge and others by their innate sense of design in nature. Maintaining those natural proportions is the difference that makes them good designers. He shows several characters by well known designer Tom Oreb (Sleeping Beauty and others) and by famous artist/designer Ronald Searle, whose characters reveal the same proportions. Stephen says, “It works! It works, and it’s a beautiful thing!”

Create a rough version of your composition first to get the feeling and use the golden ratio to refine it.

Stephen goes to the drawing board to create a rough design like these:

He then applies his secret weapon (and other talents) to refine an amazing array of unique characters and animals for TV, animations and movies, such as these:

In designing a character, Stephen says that the most important thing is to first get your idea out, to think about who your character is. Get a loose concept first to find the feeling of the character. Then begin applying the golden ratio to make adjustments to the various proportions of the character to make it feel more ideal. He illustrates this concept with two characters he creates in the video. He starts with some rough first passes to get the overall shape and feel. He then refines their proportions to golden ratios in various features. Note in the illustrations below that the original sketch lines are slightly off from the final drawing. The refinements below include the waistline and length of the pants, the length of the goatee and the position of the pectorals. He does all of this quickly and simply, using the golden mean gauge he created, at right, for this purpose.

He says, “It’s not that if you don’t do it this way that all your drawings are going to fail,” but “it’s a great weapon to have on your side.”

Disney uses Phi in their animations and even in their logo.

Stephen was not the only one at Disney who thinks that the golden ratio, represented by the Greek letter Phi, is something special. Did you ever notice that the design of the Disney logo itself uses an upper case Phi (Φ) to dot the I and a lower case phi (φ) for its unusually styled Y? Now you’re in on the secret!

Learning more and getting the tools you need to design is easy.

Stephen’s website and courses have more information on all of this. He sells a signed copy of his gauge for $10 plus shipping to anywhere in the world. He concludes, “It really works, it’s cool and it’s a beautiful thing.”

For those of you who work with digital images, you may find that my PhiMatrix golden ratio design software can help you to accomplish the very same things, whether that be in characters, logos, user interfaces, products or any other kind of design.

Stephen Silver YouTube video “Art Lesson – How to use the Golden Section for Character Design” below:

References:

Stephen Silver Silvertoons website

Stephen Silver Golden Mean Gauge site

Stephen Silver Classes at Schoolism

Stephen Silver Apps

Stephen Silver Facebook Page

Stephen Silver Drawing Academy

Stephen Silver Newsletter

The post The Golden Ratio in Character Design, Cartoons and Caricatures appeared first on The Golden Ratio: Phi, 1.618.

Evidence of the Golden Ratio in the Great Pyramid complex.

There are many pyramid theories and questions as to who built the pyramids in ancient Egypt. It’s commonly known though in Egyptology that the proportions of the Great Pyramid of Egypt are within inches of a golden ratio-based pyramid. This is discussed in detail in my article Phi, Pi and the Great Pyramid.

Some say, however, that this single piece of evidence is just a simple coincidence. The primary rationale given to deny this claim is that there is no written historical evidence that the ancient Egyptians had any knowledge of the golden ratio.

That’s a reasonable objection, but what if the evidence showed that the application of the golden ratio at Giza was not limited to the Great Pyramid? That evidence is presented in this article.

The golden ratio connection to the Great Pyramid of Khufu

Let’s quickly review of the evidence on this wonder of ancient architecture. A golden ratio pyramid is based on a triangle whose three sides represent the mathematical relationship that defines the golden ratio. This triangle, known as a Kepler triangle, has base of length 1 and a hypotenuse whose length is Phi. The height of this right triangle is the square root of Phi. Using the Pythagorean theorem, (a²+b²=c²), this triangle represents one of the golden ratio’s unique properties:

1 + Phi = Phi ²

The geometry of the Great Pyramid of Giza varies by less than 0.025% from that of a perfect golden triangle pyramid. This is a difference of just inches in its height of 480.6′. Could the pyramid be this close to the golden ratio in its design, but with no intent of this by its builders? That seems rather amazing, if not unlikely. If the builders thought the golden ratio was important enough to use for this critical measure, would they use it only in this one instance?

What if the golden ratio appears in other aspects in the pyramid complex?

I began to investigate this question after receiving a comment from Pat Miller, a visitor to this site. Pat observed,

The first time I saw the 6 smaller pyramids of Giza, the image of the scale rulers placed next to artifacts came to mind. Three points denote a circle, so 9 points could denote 3 circles. Has anyone, or could someone, calculate their sizes and spacial relationships?”

I have yet to find a relationship based on those circles. The question though inspired me to look at the relationships of the pyramids and the topography of the site.

The Golden Ratio in the three small pyramids with the Tomb of Queen Hetepheres

I used PhiMatrix, with its pixel-level accuracy, to see what ratios appeared in the site’s topography. Every line in the PhiMatrix grid is in a golden ratio proportion to the ones on either side of it.

I first placed the PhiMatrix grid over the “scale rulers,” the three small pyramids next to the Great Pyramid. Here I found a golden ratio in the placement and dimension of the middle pyramid in relation to the perimeters of the two outside pyramids:

The Pyramids of Queens reveals yet another Golden Ratio

There are three other small pyramids, which include the Pyramids of Queens. These are next to the Pyramid of Menakaure, the smallest of the three large pyramids. I measured these next. Although irregularly shaped, the distance between the corner points of their bases revealed the very same golden ratio relationship that appeared in the three small pyramids next to Khufu’s pyramid. Click for larger image:

The Golden Ratio in the pyramids of Khufu and Khafre

I then used PhiMatrix to investigate the relative positions of the two largest pyramids of the complex. Here I found yet another very surprising relationship. Create a rectangle with a perimeter that outlines the bases of the two larger pyramids, and the relationships are found:

• The base of the middle pyramid is a golden ratio of the width of that perimeter rectangle.
• The base of the middle pyramid in relation to the height of this perimeter rectangle is also very close to the golden ratio. It measures at 1.608, a variance of only 0.06%.
• Taken together, this results in a simple, yet elegant, golden ratio geometry. This geometry defines a relationship between the pyramids of Khufu and Khafre.

Click on images below for the high resolution images:

The Golden Ratio in the pyramids of Khafre and Menkaure

I next investigated the topography of the smaller pyramid in relation to the middle pyramid. Here too, I found a relationship. Create another rectangle with a perimeter that outlines the bases of the middle and smaller pyramids. The base of the smaller pyramid is Phi² (a golden ratio of a golden ratio) of the width of that perimeter rectangle.

The red line closest to the center represents the primary golden ratio of the width of the perimeter rectangle. The red line bordering the right side of the small pyramid is the golden ratio of that primary golden ratio.

Analysis of the Giza site plan by others

After first publishing this article, John Gury alerted me to some similar analysis of the Giza complex site. One of the plans expressed a concept similar to the Khufu-Khafre relationship I show above. This was by Jiri Mruzek from his site at http://www.vejprty.com/gizaplan.htm. Note that design is based on a square. The center of the square defines the middle pyramid. Each arm of the cross is a golden rectangle:

The apex points of the Khafre and Menkaure pyramids define the corners a Golden Rectangle!

Another site layout, proposed by Chris Tedder, provided a far simpler and more elegant approach yet. It’s based on two golden rectangles (one in portrait orientation, one in landscape orientation) whose corners align with the apex of each pyramid. I’ve included the original illustration below on the left. I then recreated it with the highest resolution Giza site map that fit on a 1920×1080 screen, and rotated it 90 degrees. The overlay of the PhiMatrix grid reveals how simple and precise this relationship is. I also applied this approach to a satellite image from Google Maps. I used PhiMatrix to create a crosshair (in green) on each pyramid to eliminate the perspective distortion and show the true position of its apex. I set the grid transparency low so as to not obscure the actual lines of the pyramid bases. The bases of the pyramids are not as affected by perspective distortion as are their apex points. Click on the images for full resolution.

I then combined the concepts of both Mruzek and Tedder to create this view of the site plan. The interesting addition is that the apex of Menakaure (the small pyrmid) is fairly closely aligned to the third golden ratio line of the square, as indicated by φ³. Click HERE on on image below for full resolution:

Validation of the results based on surveys of the Giza site

Seeking to validate these results with other measurement, my investigation took me to an analysis by Glen Dash based on very precise site surveys of the Giza site. Glen combined the results of Flinders Petrie, who surveyed the Pyramids in 1880 and 1881, and those of David Goodman and Mark Lehner, resurveyed the Giza Plateau in 1984 and 1985. Glen’s analysis provides the locations of the apex and base of each of the three pyramids to within ±10 centimeters, or about 4 inches. I used these measures to calculate the ratios presented above. Click on images below for full resolution:

This analysis shows a very close alignment of the pyramids to these golden ratios:

• The golden rectangle formed by the apex points of Khafre and Menakure has sides with a ratio of 1.613.
• The golden rectangle formed by the apex points of Khufu and Menakure has sides with a ratio of 1.618.
• The ratio of the base of Khufu to that of Khafre averages to a ratio of 1.610.

A progression of Golden Ratios

The analysis shows that the Giza site thus shows the applications of several basic golden ratio concepts:

• The bases of Khufu and Khafre express the concept of dividing a line at its golden ratio point.
• The apexes of Khafre and Menakure express the concept of the golden rectangle.
• The pyramid triangle of Khufu expresses the concept of the golden rectangle, aka the Kepler triangle.

Golden ratios beget golden ratios

One of the unique properties of the golden ratio is that when you DO use it in design, it creates multiple golden ratio relationships. This is illustrated by the expanded PhiMatrix grid below. Each line is in golden ratio proportion to the ones on either side of it, so imagine how many golden ratios there are to be found. It should be no surprise that the Giza site, if in fact based on the golden ratio, would produce multiple site layout variations that all contain golden ratios.

Multiple golden ratios in the design of the Great Pyramid complex

Could these golden ratio relationships be the basis for the sizes and positions of the pyramids at Giza? The ancient Egyptians did not reveal that to us in the writings they left behind. Perhaps, however, the evidence is beginning to speak for itself.

This analysis now provides us with at least eight instances of golden ratio proportions in the topography of the Giza site. This is in addition to the geometry of the Great Pyramid itself, which is within 0.25% of the proportions of a perfect golden ratio pyramid. The chance that golden ratios would appear nine or more times in the positions and sizes of these pyramids by coincidence alone seems quite unlikely.

If the SETI program received a message from space of these mathematical relationships, I suspect they’d be proclaiming the discovery of extra-terrestrial intelligence. When found in the site plan of Giza, should we chalk it up to chance and coincidence, or conclude that its design was based on these mathematical concepts?

This combined look at the evidences for golden ratios in the site plan of the Pyramids of Giza gives far more reason yet to believe that it does in fact exist, and was included in the design plan of the builders.

Which specific site design was used by the Egyptians in reality is anybody’s guess. The fact is that reasonably accurate plans can revealing golden ratios can be developed with reasonably simple approaches. This, in turn, should provide compelling evidence that the original site plans were based on a similar or related methodology.

Golden ratios are a simple and common outcome of many geometric constructions

Appearances of the golden ratio in nature and design should not be a surprise. The ratio appears in a variety of places in nature, and in features of the human face. This can lead to it being perceived as natural or aesthetically pleasing. This in turn can lead to its use in man’s own creative endeavors. It’s also very commonly found in a variety of simple geometric constructions, such as those shown below (AC/AB=1.618):

There is thus no need to assume that any use of the golden ratio requires a knowledge of irrational numbers or advanced mathematics. It’s really quite easy to construct with just a length of rope and a straight edge. It sometimes shows up even when you don’t even plan it.

Methodology of the analysis

The topographies shown above are based on maps of the Great Pyramid Complex site provided by ZeeMaps.com. In my first pass at this analysis, I used satellite images from Google Earth and Google Maps. The same relationships were evident, but there are shadows and parallax distortion in these images. This made it clear that a topographical map of the site would provide more accurate results yet. Click on images for higher resolution.

ZeeMaps uses Google Maps to display its maps. My analysis of course assumes that the topography provided in these maps is accurate. If so, this provides compelling evidence that the ancient Egyptians may have used the golden ratio in their construction of the pyramids at Giza. This could have been with intent. It also could have resulted from the use of some simple geometries in laying out the site.

One thing that seems certain: With all the technology, precision and resources that went into the construction of these pyramids, the builders must have had something very specific in mind in determining their sizes, positions and geometries. The sides of the square base of the Great Pyramid align to within 0.7 degrees of the four compass points based on true north. Its base was squared to within 0.2 degrees. It’s unreasonable to assume that the relative sizes and positions of the three large pyramids and six small pyramids were just an afterthought left to chance.

Other nearby ratios do not fit the topographical evidence as well

To minimize any doubt that the proportions found in the topography represent ratios other than the golden ratio, consider the images below. The grid lines on these images show the positions of the base of the pyramids if ratios of 1.600 and 1.625 had been used rather than 1.618. The alignment of the pyramids is clearly closer to the golden ratio. Click on images for higher resolution.

Other theories on the placement of the pyramids – The Orion Connection

It has been proposed by Graham Hancock and Robert Bauval that the positions of the three large pyramids correspond to the positions of the three stars in the belt of the constellation Orion. This is a rather brilliant observation, and is supported by the alignment of the air shaft of the Great Pyramid to the belt of the constellation Orion, as it appeared around 2500 BC. I tested this alignment of the stars by overlaying an image of the constellation to a topography of the pyramid complex. Click on the images for higher resolution.

The relative distances between the pyramids matched quite perfectly to that of the stars. The angle of the lines formed by their positions was slightly off by my analysis. I aligned two stars of the belt to the two smaller pyramids. When this was done, the third star was positioned slightly to the left of the apex of the large pyramid. By my measurement, the angle formed by the three stars is about 172.6°, while that of the three pyramids is about 175.2°. The alignment of the apex of each pyramids to the stars is thus very close, but could have been even closer yet if this were the intention of the builders.

The relative brightness of the stars in Orion do not correlate to the relative sizes of the pyramids

I also noted that while the stars are of roughly equal brightness in the sky, the relative sizes of the pyramids are very different. The center star in the belt is the dimmest of the three, but the southernmost pyramid is by far the smallest of the three. If the pyramids were meant to be a terrestrial representation of the night sky, what was the basis for their very different sizes?

The Orion Connection explains the positions of the pyramids, but not their sizes

Even if the connection to Orion is true (Hancock and Bauval offer a compelling body of other evidence that it is), this defines the only the position of the apex of each pyramid. It does not explain their relative sizes. Could the golden ratio be the key factor yet that led the builders to the positions and dimensions of the pyramids? Based on my measurements of the pyramids presented above, the application of the golden ratio to the topography of the site provides a even higher degree of accuracy than the stars of Orion’s belt. It appears to better explain both the placement and the dimensions of the pyramids.

The Great Sphinx

There’s one more monument at the Giza site that I’d be remiss to exclude from this analysis: The Great Sphinx. The satellite and aerial images of the Sphinx on Google Maps and Google Earth are too blurry to use with any reliability. Look, however, at the topographical map of the Sphinx on ZeeMaps. I think you’ll be find the images rather interesting as well. Click on any of them for higher resolution.

One more step towards a deeper understanding of a great mystery

I find these results intriguing and compelling. There are admittedly many questions that remain. If the golden ratio were applied to the pyramids with intent, there are many alternative approaches to their design that could have been included in the site plan and the sizes and proportions of the pyramids. Still, it seems that the relationships shown are fundamental enough to not be deemed arbitary. They are also quite precise in describing the dimensions and proportions that exist. This should make them unlikely to be the result of coincidence or chance.

I hope these discoveries and analyses provides others with an insight to build upon in doing more research yet. There is still much to understand about the history, mathematics, design and purpose of the building of the pyramids. I hope too that those with access to the site of the Great Pyramid complex will take more accurate measures yet to confirm or deny the proportions that the maps appear to reveal.

You, as visitors to this site, are my peer review group. You can use PhiMatrix for free to examine this evidence for yourself. For those of you who use Autocad, there are site details available on the Internet that can be used for more detailed analysis yet. I invite your analysis and comments.

Gary Meisner
www.goldennumber.net

References:

ZeeMaps.com maps of the Great Pyramid of Giza Complex, with editable version.

Google Maps Satellite imagery of the Great Pyramid of Giza Complex

PhiMatrix Design and Analysis Software

Graham Hancock and Robert Bauval on the Great Pyramid and the Orion Connection

Giza Site Plan analysis by Jiri Mruzek

Giza Site Plan analysis by Glen Dash

Wikipedia on the Great Pyramid of Giza

Wikipedia on the Pyramid of Khafre

Wikipedia on the Pyramid of Menkaure

Wikipedia on the Great Pyramid Complex

The post Golden ratios in Great Pyramid of Giza site topography appeared first on The Golden Ratio: Phi, 1.618.

In 2013, I reported that Michelangelo used the golden ratio in his painting “The Creation of Adam” in the Sistine Chapel. This Divine Proportion appears at the point at which Adam’s finger is touched by the finger of God, as God breathes life into Adam.

In July 2015, Clinical Anatomy published an article by a team from Brazil titled “More than a neuroanatomical representation in The Creation of Adam by Michelangelo Buonarroti, a representation of the Golden Ratio.” (PDF here.) The article claimed, incorrectly, that there was no previous association of Michelangelo’s works with the golden ratio. It identified the very same golden ratio touch point as the one reported here two years earlier.

It also claimed, however, a new finding. It claimed that the fingers touched at the golden ratio point of the length of all the paintings on the ceiling of the Sistine Chapel. My analysis of this claim revealed it to be inaccurate. (See separate article Discovery of the Golden Ratio in Michelangelo’s “The Creation of Adam.”) It motivated though me to take a much deeper look at Michelangelo’s composition of his paintings in the Sistine Chapel. This led to much richer evidence yet of his use of the golden ratio.

Over two dozen uses of the golden ratio appear in the Sistine Chapel.

This expanded analysis led to the discovery of more than two dozen examples of the golden ratio. This provides compelling evidence that this “Divine proportion” was an essential guide in the composition of this work. The analysis was performed using PhiMatrix design and analysis software to find golden ratios, or any custom ratio, with pixel-level accuracy.

Perhaps the most stunning example, as shown in my 2013 article, is the point at which the fingers of God and Adam touch in “The Creation of Adam. This is found at the golden ratio of both their horizontal and vertical dimensions. Click on the image for full resolution:

The same golden ratio point appears if you instead use the inside or outside borders of the painting, rather than the borders of God and Adam. Click on images for higher resolution.

Should it be suggested that this touch point does not actually represent the golden ratio, see the illustration below. The green line shows the difference created by using a ratio of 1.6 instead of 1.618.

This evidence of its use is rather conclusive in itself, and yet it’s just a subtle hint to the rest of the story that unfolds.

The fingers and hands also touch at Golden Ratio points in other paintings

Michelangelo repeated this theme of the characters touching the golden ratio point in other paintings of the Sistine Chapel, as illustrated in the five paintings below. The green lines show the golden ratio of the height and/or width of each painting.

The last of the series of nine biblical narration paintings on the center ceiling is of Noah’s disgrace. In it, the fingers of two of Noah’s sons point directly to the golden ratio points from the painting’s sides. It’s done as if to show the viewer exactly where they are, and the fact that Michelangelo had indeed applied the Divine proportion. After all, this incredible collection of paintings was created for the Vatican, the seat of Peter and successor Popes of the Roman Catholic Church. If he wanted to give evidence to his use of the Divine proportion, he would have wanted it to be clear and undeniable. He may have done so with a bit of wit:

In this and other paintings, the hands are positioned as though grasping this golden proportion. It can be viewed as a visual metaphor of human desire to grasp the Divine. Click on any image for full resolution.

Golden Ratios in the positions of the subjects within the paintings

Further analysis of the paintings reveals other likely applications of the golden ratio in their composition. The images below illustrate that Michelangelo appears to have used the golden ratio in the positioning of the primary characters within each painting. Click on any image for full resolution.

Admittedly, the more you look at the detailed elements of any painting, the more chance you have of finding some golden ratios. These examples, however, use the primary elements of the composition. The best evidence is the many obvious examples of its use based solely on the height and width of the paintings. These demonstrate that Michelangelo’s use of the Divine proportion was quite deliberate and intentional.

Golden Ratios in the proportions of the paintings to the frames

Each of the paintings on the center section of the ceiling is framed by two larger ceiling sections that separate the paintings. For seven of the nine paintings, the width of the painting is a golden ratio of a golden ratio. This simply means that you take the golden ratio of a dimension (red grid) and then in turn take the golden ratio of that dimension (blue grid), as illustrated below:

The green lines in the three large images below illustrate the application of this concept. The red arrows show the second golden ratio point in relation to the width of the framed area. Click on any image for full resolution.

The image below shows how this pattern of golden ratios of golden ratios was repeated for the paintings and framing sections across the ceiling. Click on the image for full resolution.

Note the alternating pattern:

• A large painting, whose dimensions are based on the outside border of the separating section.
• A small painting, whose dimensions are based on the inside border of the separating section.

The last two paintings of the biblical narrative, on the left side in this image, are close to these same golden ratios but do not use it so precisely as the others.

Golden Rectangles in the Ceiling Paintings

In addition to the above evidence, the dimensions of two of the five small paintings inset on the ceiling approximate Golden Rectangles with an accuracy of 1/100th. This is shown by the green rectangles:

“The Drunkenness of Noah” – Aspect Ratio of 1.6

The Sacrifice of Noah – Aspect Ratio of 1.6

Also, the positions of the numerous human figures sitting along the opposite sides of the ceiling are based on the golden ratio of the width of the center section of the ceiling.

The tablets of the ancestors of Jesus are Golden Rectangles

If there remains any doubt that Michelangelo used the Divine Proportion, look to the paintings of the tablets listing the ancestors of Jesus on the side walls of the Chapel. The height to width of the name plates form a golden rectangle, as shown in green in the images below. Most of the name plates are within a pixel or two of being golden rectangles. The average height to width ratio of of all the paintings is 1.62, accurate to within 1/1000th of the golden ratio of 1.618. Click on any image for full resolution.

Michelangelo was in concert with other Renaissance greats of Luca Pacioli, Leonardo da Vinci and others

Michelangelo painted the Sistine Chapel between 1508 and 1512. The book “The Divine Proportion” (De Divina Proportione), written by Luca Pacioli and illustrated by Leonardo da Vinci, was composed between 1496 and 1498, and first printed in 1509. Knowledge of the golden ratio clearly existed before Pacioli began to write the book. There’s evidence of this in the earlier paintings of Leonardo, Botticelli, Raphael and other Renaissance painters. Leonardo da Vinci used this Divine proportion in a number of his paintings, including The Annunciation in about 1473, The Last Supper in 1495 and Salvator Mundi around 1500.

Luca Pacioli, author of “De Divina Proportione,” begins the book with this introduction:

“A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.”

Dr. Mario Livio, author of The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number, describes “De Divina Proportion” (on pages 132-133) as follows:

“Pacioli raves ceaselessly about the properties of the Golden Ratio”

“there is no doubt that Pacioli had a great interest in the arts and that his intention in The Divine Proportion was partly to perfect their mathematical basis.”

“His opening statement on the book’s first page expresses his desire to reveal to artists, through the Golden Ratio, the “secret” of harmonic forms.”

Kenneth Clark, author of “Civilisation” observed:

“This union of art and mathematics is far from our own way of thinking, but it was fundamental to the Renaissance.”

The golden ratio, known then as the Divine proportion, was a highly regarded tool for artistic composition and expression in the Renaissance art. It was used by the masters of the Renaissance to bring both mathematical and visual harmony to the elements within a work of art. It was thought to embody an essence of the Divine. Consider these quotes of Michelangelo’s contemporaries:

“Without mathematics there is no art.” Luca Pacioli, author of “De Divina Proportione”

“Where the spirit does not work with the hand there is no art.” Leonardo da Vinci

Michelangelo’s painting of the Vatican’s Sistine Chapel had incredible religious significance. It really should be no surprise that Michelangelo would have used the Divine proportion extensively in the creation of this work. In retrospect, it would be much more of a surprise if he had not.

References:

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

https://en.wikipedia.org/wiki/Michelangelo

https://en.wikipedia.org/wiki/Sistine_Chapel_ceiling

https://en.wikipedia.org/wiki/Gallery_of_Sistine_Chapel_ceiling

https://en.wikipedia.org/wiki/The_Creation_of_Adam

Design and Analysis by PhiMatrix software

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Google’s design follows in the footsteps of Leonardo da Vinci and other masters

When Luca Pacioli published “The Divine Proportion” in 1509 (with illustrations by Leonardo da Vinci), he described his work on this “golden ratio” of 1.618 as a “very delicate, subtle and admirable teaching” that would “delight in diverse questions touching on a very secret science.” Johannes Kepler later called it “a precious jewel” of geometry. The designers at Google have apparently found its value too, as we see when we study and appreciate the underlying design of Google’s new logo, iconic G, the microphone icon and even the layout of the Google search page.

This is the kind of thoughtful design work that follows in the footsteps of Leonardo, Michelangelo, Raphael, Botticelli, Seurat, Le Corbusier and other masters of design, and that would make Pacioli proud.

Here’s a version without the arrows for a clearer view:

There’s a good chance you’ve used Google to search the Internet lately, maybe even in getting to this site. With over a billion people visiting per month, Google obviously puts in a lot of thought to create the best possible experience in using their search page. As simple as it is, the design of the logo and its search page has evolved considerably over the years. Consider how Google looked in 1998:

Logos and trademarks are critically important to a company’s image and brand recognition. On September 1, 2015, Google announced the new design for its logo and other trademarks. Their ongoing refinements of the logo and related design elements have led to the use the Golden Ratio (GR) in its design.

Golden ratios in the new logo and symbol are revealed by graphic analysis

For those not familiar with the golden ratio, it’s a simple as this:

• Divide a line into two segments, a and b.
• The ratio of the longer segment a to the shorter segment b must be the same as the ratio of the original line (a+b) to the longer segment a.
• That unique dividing point is the “Golden Ratio.”

The grid lines shown on logos in this article represent the golden ratio(s) of the height or width shown by the grid:

Let’s now see how the Golden Ratio appears in this new design, aided by the pixel-level accuracy of PhiMatrix design and analysis software:

• The height of the lower case letters is a GR of the height of the capital “G.”
• The upper point on the right side of the “e” is a GR of the height of the lower case letters.

Next, take the height of the letters from the top of the upper case “G” to the bottom loop of the lower case “g”. Note alignment with the golden ratio lines at:

• The upper line of the crossbar on the upper case “G.”
• The inside edge of the lower case “o”, “g” and “e”.
• The pointed tip on the bottom curve of the lower case “e”.

The new Google G symbol

The new Google “G” symbol also closely aligns with the Golden Ratio in it’s design:

• Horizontally – The bottom of the crossbar of the “G” and the inside point of the green arc align with the GR lines horizontally.
• Vertically – The inside points of the red and green circles fall at vertical GR lines of from the center of the “G” to the right edge of the “G”.

The new Google microphone symbol

My analysis reveals that new Google microphone symbol has a surprising number of golden ratios that were taken into account in its simple but very elegant design:

• The blue microphone is a GR of the inside and outside diameters of the circular arc the surrounds it. (Images 1 and 2)
• The width of the microphone stand is a GR of the width of the inner diameter of the circular arc. (Image 3)
• The point at which the color changes from gold to red appears at a GR of the height of the circular arc from the base of the stand. (Image 4)

These are quite intricate design decisions for such a small icon. Think though how it ever so subtly relates it to the logo above it. It creates visual harmony by using same proportions for the design of all the elements of the composition. These same techniques appear in the composition of paintings and buildings by recognized masters of design, back to the Renaissance and before.

Golden Ratios throughout the new font created for letters in the logo

We also find the golden ratio in use in the width of the letters and other fine detail of the new font that Google calls Product Sans. Look at the width of the letters in relationship to each other.

• The width of the lower case “g” is a GR of the upper case “G”.

• The width of the lower case “l” is in GR proportion to the lower case “g”:

• The points on the lower arc of the lower case “g” are also based on Golden Ratios:

• The diameter of the hole inside the “o” is a GR of the height of the “o”. That in turn makes the line thickness of the “o” a GR of the radius of the hole.

• The capital G for the Google icon was designed with exactly the same approach:

The PhiMatrix grids used above make this kind of design easy to do. For those just itching for a little math at this point, here’s how this works:

• When the diameter/height of the outer circle for the “G” and “o” is the Golden Ratio of 1.618, the diameter/height of the hole inside the circle is 1.
• This leaves 0.618 remaining for the line thickness of the font above and below the hole.
• The line thickness of the font is thus half of that at 0.309.
• The radius of the hole is 0.5.
• The ratio of 0.5/0.309 is … 1.618 again!

It’s the unique mathematical properties of the Golden Ratio that makes this relationship happen … again and again and again …, and why it is so useful in achieving visual harmonies in design.

Google Search Page design layout

Google didn’t stop there. Look at the proportions of the Google logo and search bar:

• The center of the Search bar is positioned at the golden ratio of the distance from the top of the Google logo to the bottom of the “Google Search” and “I’m Feeling Lucky” buttons:

• The bottom of the Search bar is positioned at the golden ratio of the distance from the top of the Search bar to the bottom of the “Google Search” and “I’m Feeling Lucky” buttons:

And where are the letters positioned within the search box and buttons? At this point, it probably wouldn’t surprise you to see:

• “Ok Google” and the bottom of the microphone at the GR of the height of the search box, and
• “Google Search” and “I’m Feeling Lucky” at the GR of the height of their respective buttons.

The Google logo is positioned very closely to golden ratios of the distance from the top of the browser window to the various components of the search box:

• The top of the logo is at the GR from the top of the browser window to the top of the search bar.

• The top of the lower case letters of the logo is at the GR from the top of the browser window to the bottom of the search bar.

• The bottom of the lower case letters of the logo are within several pixels of the GR from top of the browser window to bottom of the “Google Search” and “I’m Feeling Lucky” buttons:

Graphic analysis and composition with pixel-level accuracy

The above analysis was done with PhiMatrix analysis software, which makes it easy to identify and apply Golden Ratios in design and composition with pixel-level accuracy. Golden ratios allow proportions to be kept in consistent visual and mathematical harmony, unlike any other ratio. Beyond its unique mathematical properties, the appearance of the golden ratio in nature and the human face leads most people to find it natural and pleasing.

This importance of this is highlighted by Darrin Crescenzi in his excellent article, “Why the Golden Ratio matters. In defense of using visual harmonies in Design.” Darrin is Design Director of Innovation at Interbrand New York, frequently quoted in FastCo Design as an authority on design and one of Fast Company magazine’s 100 Most Creative People in Business.

Google used the Golden Ratio in its previous logo design and search page layout

This is not the first time that Google has used the Golden Ratio in its design. The website design they just replaced had GR in both the logo and the layout of the page. In the previous logo:

• The height of the lower case letters were a GR of the height of the upper case “G”.
• The right point of the “e” was slightly higher at the midpoint of the lower case letter height.
• The crossbar of the upper case “G” though was notably lower, and positioned at the GR of the height of the lower case letters, rather than at the midpoint of the height as in the new logo.

The width of the crossbar of the upper case “G” is a golden ratio of the width of the “G”.

The lower portion of the lower case “g” is a golden ratio of the height of the lower case letter as well!

The design of the Google Search Engine page layout embodied the Golden Ratio

The previous Google search engine home page also based its layout on the golden ratio in a slightly different way, as shown in the images below.

From the top of the browser window to the top of the logo to the top of the search box:

From the top of the logo to top of the search box to the bottom of the buttons:From the top of the search box to top of the buttons to the button of the buttons:

It may be just a tool, but the Golden Ratio has broad appeal and application

No matter what kind of design you do, the golden ratio offers a reliable way to achieve a natural balance and aesthetically pleasing appearance in design. It’s much more than just a more natural and accurate alternative to the rule of thirds. It’s a mathematically unique system of ratios within ratios that can keep an entire composition in visual harmony. See many other examples of the golden ratio in design of art, architecture, logos and products.

It’s just one of many tools that good designers use to achieve great composition, but no designer should be without a knowledge of its concepts and application. At least that’s what the graphic design team at Google seems to think, and more than a few great masters of design through the centuries.

Author background

Since the 1990’s, Gary Meisner has been researching the appearances of the golden ratio in the beauty and nature, and its application in the design arts. He created www.goldennumber.net in 2001, which reaches 1 million people per year and shares contributions from dozens of others from a variety of backgrounds. He is the developer of PhiMatrix Golden Ratio Design and Analysis software, which is used in over 70 countries. He has connected with thousands of other people with common interests in the topic, sharing with them and learning from them. His interviews on the topic have included the LA Times,  a TV educational documentary and radio.

References:

Google Official Blog Page on Logo Redesign

Evolving the Google Identity

Google Official UK Blog Page on Logo Redesign

Google unveils new logo at turning point in company’s history by The Guardian

Can you design a better Google logo? by the Guardian

Short link to this post: http://goo.gl/91fCk3

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Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance and lived from 1483 to 1520. He is recognized as one of the three great masters of that period, accompanied by Michelangelo and Leonardo da Vinci. His work is admired for its form, composition, and visual achievement of the ideal of human grandeur.

One of his most famous works is The School of Athens, a fresco in the Apostolic Palace in the Vatican. It captures the spirit of the Renaissance, and is revered as his masterpiece. It was painted between 1509 and 1511.

The School of Athens: Inspired by a union of art and mathematics

It was also in 1509 that Luca Pacioli published the book De Divina Proportione (The Divine Proportion), with illustrations by Leonardo da Vinci. MonaLisa.org reports that The School of Athens “incorporates many of the mathematical theories of Luca and Leonardo.” “Civilisation” author Kenneth Clark is quoted there saying “This union of art and mathematics is far from our own way of thinking, but it was fundamental to the Renaissance.”

Raphael made his use of the Golden Ratio obvious

If there’s any doubt that Raphael used the Golden Ratio in this painting’s composition, it can be eliminated with a good degree of confidence by the golden rectangle that was placed front and center in the painting. It’s as though Raphael made a small but undeniable statement to answer the question before it was asked. This small rectangle is about 18″ by 11.1″ and is a rather unusual feature. Perhaps it once bore the title or some description of the painting. Even the framed area immediately below it shows a framed that is in golden ratio proportion to the borders on its sides. Click on the images for full resolution:

No other ratio would accomplish the same result in this composition

The painting has thousands of intricate lines, so some might say that finding golden ratios within it would be a simple exercise in pattern recognition. There are two ways to overcome that objection. You can test it for yourself by setting the Line Ratio in PhiMatrix (no cost trial) to any other ratio to see if you get the same abundance and consistency of results. Further, the golden ratios are very evident on the major elements of the composition alone. Note in the first image below (click to enlarge) that simple golden ratios of the width and height of the painting define the positions of the large wall of the first arch, the top of stairs at the floor and the top of the second arch. Other golden ratios define other key elements of the composition in the other images.

The intricate application of the golden ratio is brilliant

To appreciate the intricacy and depth of Raphael’s planning and application of the dimensional proportions in this painting, consider the illustration below. This image includes four rectangles that overlay the painting:

• Each rectangle begins at the left side of the left column in the painting. This point represents the first architectural reference point of the actual school building as viewed through the arched portal of the fresco.
• Each extends to a prominent composition feature on the right side of the painting.
• Each is divided into one or more golden ratios (accurate to the pixel with PhiMatrix software).
• Each dividing line illustrates a golden ratio formed within another prominent feature of the composition.

Trace the four rectangles separately, each in its own color, following it until you see the golden ratios that are revealed. It may be easiest to look first at the rectangles as overlayed onto a black background, also shown below.

Raphael’s use of the golden ratio is not only clear and compelling, but quite brilliant in its execution. Do you agree? Your comments are welcome below.

References:

http://en.wikipedia.org/wiki/Raphael

http://en.wikipedia.org/wiki/The_School_of_Athens

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Golden ratio proportions in composition of Salvator Mundi starting with golden rectangle for the head.

See the YouTube video “Leonardo da Vinci’s “Salvator Mundi” and the Divine Proportion.“

2017 Update – Salvator Mundi, “the last Leonardo” sells for a record-breaking $450,312,500, decimating the prior world record for the most expensive work of art sold at auction. See Christie’s article and learn below some of what makes this piece so incredible and valuable.

In 2011, the discovery of a lost painting by Leonardo da Vinci was announced to the world. This painting, Salvator Mundi, had been in the art collection of King Charles I of England in 1649, was auctioned in 1763 and then lost for many years. Its recovery was led by Robert Simon, an art historian and private art dealer, and it was restored by Dianne Dwyer Modestini. Many unique qualities of this painting led experts to confirm that it is indeed an original work of Leonardo da Vinci, one of only fifteen now in existence.

Da Vinci and the Golden Ratio in Art Composition. A lesson for all artists and designers.

This article reveals new insights into da Vinci’s genius that can be found in this painting. Da Vinci created the illustrations for the book “The Divine Proportion” written by his contemporary, Luca Pacioli. He used the Divine proportion, also known as the golden ratio, in his composition of earlier paintings, including The Annunciation in about 1473 and The Last Supper in 1495. Salvator Mundi was created around 1500, and includes many more clear examples yet of da Vinci’s application of the golden ratio.

These findings can give artists and designers a deeper understanding of how to use the golden ratio in their own works. Da Vinci’s application of the golden ratio is subtle, yet much richer than simply applying the golden ratio to horizons or other dividing points. In this way he creates an overall balance and harmony among all the elements of the composition, similar to the way the golden ratio appears in the human face.

Corroboration by the Restorer of Salvator Mundi

I presented the findings of my work to Dianne Dwyer Modestini. As the art conservator who restored Salvator Mundi, she spent 18 months with this painting and likely spent more time gazing upon it than anyone since Leonardo da Vinci himself. She commented on the analysis, as featured in the video below in this article:

I found that Leonardo thought long and carefully about the Salvator Mundi, since he was trying to create a portrait of a divine being and used a number of devices to achieve the extraordinary presence and power of the image. I’m not surprised that he made extensive use of the golden ratio, the “divine proportion” and it is very interesting to see it mapped out in this way. – Dianne Dwyer Modestini, 2014

Composition analysis with graphics software, accurate to the pixel

The golden ratios presented in the illustrations in this article and the related video were created with PhiMatrix. As its grid is resized and lines are added, every line it creates is in a golden ratio proportion to the ones on either side it, accurate to the pixel.

Da Vinci would have used a different tool of course, perhaps similar to golden ratio calipers used today for design and composition. These calipers exist in various forms, and are quite easy to create. All that is required is a couple of arms with a pivot point at the golden ratio of their length.

With this approach, we can explore the composition decisions that da Vinci likely made in the this creating Salvator Mundi, Christ as Savior of the World. We can appreciate it at a new level, one that reveals not just the genius of da Vinci’s artistic technique, but that of his design and composition as well.

Examples of da Vinci’s use of the Golden Ratio

Portraits have fewer distinct lines that paintings of landscapes and architecture, but there are some very interesting distinct golden ratios that point to its use in the overall composition. The dimensions of key elements of the painting are in golden ratio proportion to one another. Start with a golden rectangle based on the height of the head:

• The dimensions of the hand are based on a golden ratio of its width.
• The dimensions of the orb are based on a golden of its height.
• The dimensions of embroidered emblem are based on the golden ratio of its height and width

This is illustrated in these three images. Click on any for the full resolution view.

Further analysis reveals that the eyes of Christ are bordered by the two golden ratio lines from the width of the painting. When a center line is drawn, the golden ratios from the center to the sides of the canvas also align with the width of his mouth and hair. In the vertical dimensions, the most prominent elements of the painting are the head, garment neck line and hands. These also reveal golden ratios in their positioning. As added evidence that the Divine proportion was being applied, look also at the width of the cloth medallion in relation to the corner points of the hem and the width of the jewel in relation to the width of the medallion. The hand holding the orb also appears to be based on a Divine proportion. One of the most interesting application of the golden ratio is in the intricate design pattern of the embroidery. If you look at a single section of the repeating pattern, you find that it is based on two golden ratio patterns that are interwoven together. One is from the left side to the right for the width of the pattern. The other is from the center to the left and right side for half of the width of the pattern.

More detailed evidence is presented in our video

A more complete look at golden ratios in Salvator Mundi is presented in this YouTube video. As you watch the video, consider the detailed anatomical studies da Vinci did and imagine him planning and measuring every aspect of each element of this painting. Imagine him striving to capture an essence of the Divine on a two-dimensional, 26” x 18” board. Note the facial proportions, the positions of the hands and fingers, the design and draping of the robe, the intricate details in the patterns of the embroidery, and even the positions of the lights that reflect from the glass orb. Note how so many of them are based on the Divine Proportion that was described in the book by the same name (De Divina Proportione in Italian). In this we can see the genius of da Vinci.

It is recommended that you click on the YouTube button and watch it in full screen mode.
Select 720p from the settings (gear) icon to see it in full HD resolution.

Click on HERE to go directly to HD resolution on YouTube.

Pacioli and da Vinci in a collaboration of art, mathematics and the spirit

Pacioli said “Without mathematics there is no art.” Da Vinci said “Where the spirit does not work with the hand there is no art.” Their collaborative work, The Divine Proportion, opens with this statement of intent:

A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.

There’s no way to know with certainty if Da Vinci used every golden ratio shown in this analysis in his composition decisions of Salvator Mundi. Some examples are clearer and more undeniable than others. We can reasonably assume that he didn’t just paint Christ as a man in a tunic in the same way as that any other artist would. It’s more likely that he used the same depth of analysis and precision reflected in his other works and illustrations, and in doing so revealed some of the secret science that is touched upon in The Divine Proportion.

Others copied this masterpiece, but none compare to the original

To illustrate the genius of da Vinci in his work, you only need look at copies of Salvator Mundi created by lesser artists who had the opportunity to see this painting before it was lost. Interestingly, these copies were one of the clues that the lost original masterpiece existed:

Seeing what da Vinci saw in the Divine proportion.

I hope this analysis gives deeper insight into his genius, gives you a glimpse of what da Vinci saw, and an appreciation of some of the subtle, secret science of the Divine proportion.

For the artists among you, perhaps it will be an inspiration to apply some of these same concepts to your own works by using golden ratios within golden ratios, as is often found in the beauty of nature itself.

References:

CNN Interview with Robert Simon:

Interview on The Story with Dianne Dwyer Modestini:

http://en.wikipedia.org/wiki/Salvator_Mundi_(Leonardo)

http://robertsimonfineartleonardo.blogspot.com/

https://www.nyu.edu/alumni.magazine/issue17/17_square_art.html

http://robertsimonfineartleonardo.blogspot.com/2011/12/interviews-with-robert-simon-and-dianne.html

http://barnard.edu/headlines/close-masters

http://www.thestory.org/stories/discovering-leonardo-10312013

http://www.robertsimon.com/pdfs/ArtNews_Sept11.pdf

http://www.conservators-converse.org/2012/01/breaking-up-with-a-leonardo-is-hard-to-do/

http://www.cnn.com/2011/11/04/living/discovering-leonardo-salvator-mundi/

http://www.cnn.com/2011/11/10/living/restoring-leonardo-da-vinci/

http://www.theguardian.com/artanddesign/2011/jul/12/leonardo-portrait-of-christ-salvator-mundi

http://www.thehistoryblog.com/archives/11915

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Da Vinci’s illustrations appear in Pacioli’s book “The Divine Proportion”

Da Vinci created the illustrations for the book “De Divina Proportione” (The Divine Proportion) by Luca Pacioli. It was written in about 1497 and first published in 1509. Pacioli was a contemporary of Da Vinci’s, and the book contains dozens of beautiful illustrations of three-dimensional geometric solids and templates for script letters in calligraphy. The original manuscript can be viewed online at http://issuu.com/s.c.williams-library/docs/de_divina_proportione. As of this writing in 2014, no English translation is available.

In the book, Pacioli writes of mathematical and artistic proportion, especially the mathematics of the golden ratio and its application in art and architecture. Some geometric solids, such as dodecahedrons and icosahedrons, have inherent golden ratios in their dimensions and spatial positions of their intersecting lines. Other examples of golden ratios in the illustrations include the one architectural illustration in the book and the one script letter (G) that is not divided horizontally at its midpoint.

On the first page of De Divina Proportione, Pacioli states that his intent is to reveal to artists the secret of harmonic forms through the use of the Divine proportion, describing his writing as:

A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.

Golden ratios / Divine proportions appear in Renaissance art in paintings of religious importance

Da Vinci’s use of the Divine proportion is evident in some of his own works, even before his collaboration with Pacioli. The paintings I reviewed suggest that the use of the Divine proportion in paintings among Renaissance artists may have been more common in paintings of special religious significance.

It appears that Da Vinci applied Divine proportions in his rendition of “The Annunciation, painted in about 1472-1473. An exact determination is difficult with 100% certainty because various images available of this painting are cropped slightly differently. It appears to be the basis for the dimensions of the walls and entry way of the courtyard, as well other elements of the composition, as shown in the photos below. Other golden ratios can be found, but to avoid any perception that this is arbitrary those shown are based on very distinct features of the painting. Botticelli also used the golden ratio in several renditions he created of “The Annunciation” between 1485 and 1493. Michelangelo used it in “The Creation of Adam” in 1510.

Da Vinci’s “The Last Supper” has many clear examples of Divine proportions

Perhaps one of the best illustrations of its use is in “The Last Supper,” painted between 1494 and 1498. Various design and architectural features show very clear golden ratios. Some believe that even the positions of the disciples around the table were placed in divine proportions to Jesus.

Da Vinci’s Mona Lisa and the Golden Ratio

One of Da Vinci’s most famous paintings is “La Jaconde,” the Mona Lisa. This painting was begun in about 1503 and work on it continued for years. It has similar composition to “Christ as Savior of the World.” The application of the Divine proportion in this painting is the most subject to interpretation and debate. Unlike “The Last Supper” and “The Annunciation,” the Mona Lisa has few straight lines or architectural elements to use as references points in making a determination. It’s possible to find golden ratios in the Mona Lisa, but they’re not as definitive. If you do a Google search for Mona Lisa golden ratio you’ll also find some very creative interpretations of golden ratios in the Mona Lisa. Many of these are not easily supported as representing Da Vinci’s intent because their starting points, position and size are somewhat arbitrary and inconsistent.

There are, however, observations that can be made about the Mona Lisa. The image on the left below shows that golden ratios from the sides of the canvas. This approach reveals no golden ratios from the canvas edges that align with key elements of the composition. There may still be golden ratios here though. The width of her face is very close to a golden ratio of the width of the canvas. This is illustrated by the yellow rectangle of the same dimensions. In the image on the right, we see that her eye is rather precisely aligned with the center of the canvas. Golden ratio lines from the center of the painting to the sides of the canvas align nicely with the width of her hair. There may also be golden ratios in the vertical dimensions of the painting. As with the painting of Christ above, the most prominent elements of the composition are her head, the garment neck line and her arm. These also show golden ratios in their positioning. So, while this is not as definitive as the straight lines of architectural elements in other paintings, a reasonable case can be made that the Mona Lisa also embodies intentional golden ratio proportions in its composition. Given Da Vinci’s prior usage, it would not be unusual or unexpected for him to have applied it here as well.

Analysis of Da Vinci’s Vitruvian Man

Another of Da Vinci’s most famous works is that of the Vitruvian Man, created around 1490. The official title of the drawing is “Le proporzioni del corpo umano secondo Vitruvio,” or “The proportions of the human body according to Vitruvius.”

Wikipedia states,

The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. Vitruvius determined that the ideal body should be eight heads high. Leonardo’s drawing is traditionally named in honor of the architect. This image demonstrates the blend of art and science during the Renaissance and provides the perfect example of Leonardo’s deep understanding of proportion.

The Wikipedia article describes how Vitruvio measured the entire human body in integer fractions of the height of a man. Examination of the Vitruvian man illustration shows that the guide lines drawn by Da Vinci on the body appear to be based on integer fractions of the height, which is also equal to its width. The illustration below shows the height/width divided into 4ths, 5ths, 8ths and 10ths. This aligns with the guide lines drawn horizontally at the collar bone, chest, genitals and knee. They align horizontally with the guide lines drawn at the wrist, elbow and shoulder.

Examination supports that Vitruvian Man was drawn to represent the fractional measures put forth by Vitruvius and is not based on the golden ratio. As discussed on the Great Pyramid page, however, irrational numbers like Phi can be closely approximated with integer fractions. So while Vitruvian Man may produce a very well proportioned human body, it seems unlikely that the human form would be based on a system of halfs, 4ths, 6ths, 7ths, 8ths and 10ths when we find constant rates of fractal expansion in the proportions of other living organisms. Other studies and approaches to measuring the human body express it just as well or better in golden ratio, which is better aligned to the growth relationships found in nature. See the Face, Body and Hand/Foot pages for illustrations.

The Vitruvian Man does have some dimensions that align with golden ratios. These are shown in the second drawing below and include the following points:

• In the distance from the Da Vinci’s guide line drawn at the hairline to the guide line at the foot, the following are all at golden ratio points:
  •  the navel, which is most often associated with the golden ratio of the total height and not the height of the hairline
  • the guidelines for the pectoral nipples
  • the guidelines for the collar bone
• In the distance from the Da Vinci’s guide line drawn at the elbow to the guideline at the fingertips
  • the base of the hand is at a golden ratio point.

Golden ratios are easy to identify and apply with simple tools

Divine proportions are quite easy for an artist to apply. All it takes is a simple two prong gauge that pivots at its golden ratio point. You take a measure on one side and then simply flip it around to get the golden ratio of that measure. Another popular design is the three prong gauge. In this case the golden ratio appears in a single line. Tools like this would provide very good estimates of golden ratio points, but are of course only as accurate as the precision of placement of their pivot points. They would not, in general, be as accurate as the pixel level analysis of those images as done in this article. This would explain some of the minor variations where the composition lines on a painting do not match the digital grid to the pixel. In most cases, however, the alignment is quite good and indicates the likeliness of the artist’s intent to apply a golden ratio.

Not every Da Vinci’s painting shows clear evidence of golden ratios

If you review all of Da Vinci’s paintings, you will likely not find clear evidence of the golden ratio in many of them. He may have used it in many more paintings than those shown above. Many paintings though do not have distinct reference lines like those in these paintings, so it is difficult to support. While golden ratios may exist in elements of his other paintings, finding them after the fact can be subject to creative interpretation, and thus can be arbitrary and incorrect. I’ve tried in this article to provide examples where its application seems so clear that most reasonable people would see its intentional application by Da Vinci as verifiable and true. History has shown though that there will continue to be differences in viewpoints. If you have doubts, my recommendation is to do the analysis for yourself with high resolution images and a tool like PhiMatrix that creates golden ratios, custom ratios and rulers with pixel-level accuracy.

References:

http://en.wikipedia.org/wiki/List_of_works_by_Leonardo_da_Vinci

http://en.wikipedia.org/wiki/De_divina_proportione

http://en.wikipedia.org/wiki/Luca_Pacioli

http://en.wikipedia.org/wiki/Vitruvian_Man

http://www.newworldencyclopedia.org/entry/Luca_Pacioli

http://bookishgirl.com.au/2012/02/15/art-and-maths-the-divine-proportion-of-luca-pacioli-with-drawings-by-leonardo-da-vinci

http://www.codicesillustres.com/catalogue/de_divina_proportione/

http://books.google.com/books?id=bUARfgWRH14C&pg=PA131#v=onepage&q&f=false

http://books.google.com/books?id=bUARfgWRH14C&printsec=frontcover#v=onepage&q&f=false

http://www.goldenmeancalipers.com/blog/page/2/

http://cdn5.fontspring.com/fnt_imgs/a1/f3ee/9718705fe92afbca1a1ec5b769/sd-720.png

http://www.theguardian.com/artanddesign/jonathanjonesblog/2012/dec/25/leonardo-da-vinci-angels-flying-machine

http://www.museumsinflorence.com/uffizi/15/15.html

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