Here’s a shocking blow to the world of mathematics! A growing number of sources claim that the established value of Pi (π), 3.14159…, is incorrect. They propose instead an alternate value, claiming that Pi is 3.1446…, derived from 4 divided by the square root of the golden ratio and shown as π=4/√φ.

It seems we live in a world where truths are continually challenged, making it difficult to know what is a new discovery and what’s just fake news, click bait or a conspiracy theory.

As a recognized researcher and author on the golden ratio, these claims have naturally drawn my attention and merit a detailed response.

Among the proponents of this alternate value, Harry Lear is frequently referenced for his geometric constructions and physical measurements, which he asserts support this value at his site “Measuring Pi Squaring Phi.

This article critiques Lear’s claims, highlighting specific flaws in his assumptions and methodologies. The goal is to provide a clear, evidence-based affirmation that the traditional value of Pi remains accurate and more reliable than any proposed alternatives based on geometric constructions or physical measurements. On that note, see also my articles on a simple’s proofs of Pi’s true value via its area or circumference.

For ease of reading, the article begins with a concise explanation of all the key issues and findings. An appendix follows this for readers who wish to delve into the more detailed analysis.

Flawed assumptions in Lear’s geometric proofs for Pi is 3.1446

Lear’s foundational claim that Pi equals 4/√φ relies heavily on his first geometric proof, which he titles “Geometric Proof 1 for True Value of Pi.” While this construction is intricate and appears methodical, it contains a critical logical flaw that invalidates the entire argument.

Copyright 2015, Harry E. Lear, Jr. at https://measuringpisquaringphi.com/geometric-proofs-of-pi/

In this proof, Lear constructs a circle, a line, and several geometric shapes, arriving at a line segment (LI) with a length of 2√φ (approximately 2.544039). Up to this point, the construction is mathematically valid. However, his next step introduces a critical failure. Lear states:

“Draw 4 identical circles, each with a given 2-unit circumference, tangent to each other with midpoints on the line segment LI: Circles O1, O2, O3, O4.”

The issue? Lear simply “assumes into existence” these four circles without validating their properties.

Based on the traditional value of Pi, four circles with a 2-unit circumference have diameters that sum to 2.546479…, but Lear assumes with no validation whatsoever that they all fit on line LI with 2.544039… as its length.

This step amounts to forcing his desired conclusion of π=4/√φ into the proof with no validation. How? The length of LI is 2√φ, so its quarter length is √φ/2. When Lear assumes into existence a circle with a circumference of 2 and a diameter of √φ/2, the ratio of circumference to diameter is 2/(√φ/2), which is magically 4/√φ!

This assumption is neither proven nor supported mathematically. It’s a mathematical sleight of hand that leads the reader to assume that the 2-unit circumference circle introduced in the first step must be the same 2-unit circumference circle placed on line LI. Mathematics tells us that four circles whose diameters fit on line LI must each have a circumference of 1.998083…, not 2. Ah! It’s not the same circle after all!

To illustrate how logically flawed this step is, one could just as easily “prove” that Pi equals 22/7 by constructing a line segment of length 28/11 (approximately 2.545455) and making a similar unvalidated claim that the diameters of these four 2-unit circumference circles fit to it.

Ah, and why stop there? Let’s construct a line of length 8/3 (approximately 2.666667) and we can plop these four 2-unit circumference circles on it to prove that Pi equals 3! This is great fun, but do you see the problem?

Without a mathematical validation to support this critical assumption, the entire proof collapses. Every subsequent step in Lear’s Proof 1 builds on this flawed assumption, rendering the entire proof invalid. All of Lear’s other geometric proofs also depend on this initial flawed assumption, making them equally flawed and unreliable as well.

Mathematicians have rigorously tested and validated numerous methods for calculating Pi—such as infinite series, calculus, trigonometric approaches, and Monte Carlo simulations—over centuries. These methods consistently yield a reliable value for Pi, free from the forced outcomes or limitations of unvalidated assumptions in geometric constructions.

Unreliability of physical measurements to prove Pi is 3.1446

Lear also attempts to support his claims through the physical measurements of a wooden disk. While his work may seem impressive, it too is flawed because of the inherent limitations of the materials and tools used.

CNC Tolerances and Material Limitations

A Format 4 CNC machine, known for its precision, cut the wooden disk in Lear’s experiments. However, even the most advanced machines have tolerances, and wood as a material introduces additional challenges:

• CNC Accuracy: I spoke with the technician at the CNC manufacturer. The machine itself has a tolerance of ±0.35mm, which introduces variability. He explained wood components are typically cut slightly oversized—by about 1mm—to allow for final finishing by the customer. Even with careful sanding, achieving exact dimensions is challenging. This alone could overstate Lear’s results for Pi.
• Material Properties: Lear’s choice of birch wood further complicates matters. Despite drying and precise cutting, the disk’s dimensions change with environmental humidity. Birch wood exhibits tangential shrinkage of 9.5% and radial shrinkage of 5.5%. A humidity change from 20% to 80% could cause the diameter to vary by up to 7–8mm tangentially and 4–5mm radially, significantly altering the circumference.

These factors alone make wood an unsuitable material for such precise measurements.

Measuring Tape Tolerances and Manual Errors

Lear used a high-precision Pi Tape with a tolerance of ±0.4mm to measure the disk’s circumference. While this tool is precise for many applications, it is inadequate for precisely measuring a difference of less than 1mm in diameter—the amount needed to challenge the established value of Pi. This is especially true when it is just one of several tools and processes that each contributes to overall tolerance errors.

Manual measurements also introduce additional errors. Absolutely perfect alignment and tightening of the tape around the disk are necessary for an accurate measurement. Any slack, misalignment, or surface variations will distort the measurement. These errors will always tend towards overstatement, as you can’t pull the tape tighter than perfectly.

Disk diameter inconsistencies

You can visually detect the variances in the diameter of Lear’s birch disk in his YouTube video, shown from 3:10 to 4:21 in Pi Video Diam Brand. Despite the claim that “the radius stays constant,” it’s quite clear that the disk’s radius is inconsistent and exceeds the measuring pin position many times. In particular, see the video at 3:26, 3:47, 3:59 and 4:03. Travel either over or under the exact circumference line will both increase the path and overstate the circumference and the measured value of Pi. With disk radius variations so large as to be visible to the eye, this experiment doesn’t come anywhere near close enough to the precision of 0.1% required to determine a new value for Pi.

The HDTV Simulation: An Illustrative Example

To visualize the impracticality of Lear’s measurements, consider this:

Imagine displaying a 1,000mm semicircle on a 4K HDTV screen. In the image below, one semicircle represents the traditional value of Pi (3.14159…), and the other represents Lear’s proposed value (3.1446…). Even on a high-resolution display, without the physical variations of a birch disk, the difference between these lines is nearly imperceptible to the naked eye.

Display the image below or the related YouTube video on a 55″, 65″ or 75″ HDTV and select the semicircle that corresponds to the size of your TV. That circle will have a physical width of about 1000mm, which is about 39.37″ The semicircle consists of two semicircles. The inner one in red shows the size based on traditional Pi. The outer one in blue shows the size based on Pi as 3.1446, 4/√φ. The difference between these two lines is virtually imperceptible to the naked eye.

Now ask yourself: Could you accurately measure this difference with a tape measure on a wooden disk? The answer is clear—such precision is not achievable with Lear’s tools and methods.

Conclusion

While Harry Lear’s exploration of an alternate value for Pi is creative, methodical and likely sincere, his geometric proofs rely on flawed assumptions, and his physical measurements are undermined by material variability, tool tolerances, and methodological inaccuracies.

By contrast, the traditional mathematical methods for calculating Pi have stood the test of time, providing consistent and reliable results. The accepted value of Pi, 3.14159…, remains the most mathematically sound and scientifically verified.

Based on my review of other such proofs, I can state confidently that examination of every proof that claims to present a new value for Pi will contain errors in assumptions and/or methods that lead to invalid conclusions. I encourage anyone aware of or involved in these claims to stop spreading the misinformation as it is doing far more harm than good in misleading the public and in wasting the time and lives of those engaged in advocating for these proofs.

The hope of this article is to shed light on these misconceptions and reinforce the importance of rigorous proof and precision in mathematics.

Appendix: Detailed Analysis of Lear’s Claims

For those interested in a detailed examination of Lear’s proofs and measurements, please refer to the content below.

Introduction

The website Measuring Pi Squaring Phi with Harry Lear was created in 2017 by Harry Lear to promote this claim using geometric proofs and physical experimental evidence. It’s one of the more frequently referenced sites on this topic, with much of his work presented on a page titled Geometric Proofs of Pi.

As one who has studied the Golden Ratio since the late 1990s, I was intrigued to the investigate Lear’s claims and the evidence he provides, to assess their validity and to determine whether they offer any compelling reasons to reconsider the accepted value of Pi.

Overview

In this article, I critique the claims made by Harry Lear that the true value of Pi (π) is 3.1446… rather than the widely accepted 3.14159… value for Pi. Lear suggests this alternate value is derived from the formula 4/√φ, based on his geometric constructions and physical measurements. I review Lear’s geometric proofs and find significant flaws in their logic, which undermines their validity. The critical error lies in assuming unproven relationships between geometric figures, leading to an unsupported conclusion about Pi’s value.

Lear’s physical measurements of a CNC-cut wooden disk with a Pi Tape measuring tool are also analyzed. I identify substantial challenges and inaccuracies inherent in these methods, such as the tolerances of measuring instruments and material inconsistencies, which make it virtually impossible to achieve the precision required to redefine a mathematical constant like Pi. Through this analysis, I show that the traditional value of Pi, 3.14159…, remains the most reliable and mathematically sound constant, as corroborated by centuries of geometric proofs and physical applications.

This article aims to clarify the misconceptions surrounding these alternate claims and reinforce the importance of rigorous mathematical proof in establishing fundamental constants.

Review of Proof 1 for Pi is 3.1446

Upon reviewing Lear’s Proof 1, I found a critical flaw in its logic that undermines its conclusion that π = 4/√φ.

Proof 1 starts with a line with a length of 2, formed by the single rotation of a circle with a circumference of 2. This works, but the circle is an unnecessary assumption and an eventual diversion, as we’ll discuss later.

The proof then constructs various geometric shapes based on this line, leading to a segment (LI) that represents a circle’s diameter and has a length of 2√φ. The following steps are used:

• The line of length 2 is used to forms lines YH, UD and TW.
• From there, right triangle AJB is constucted with sides of 2 and 1, with a hypotenuse of √5.
• This hypotenuse is used to create another right triangle and other line segments with golden ratio proportions.
• Other valid steps of geometric construction lead to the creation of line segment LI (the diameter of the yellow circle) with a length of 2√φ.

The Critical Failure Point

At this point, the proof fails. Lear asserts that there are four circles, each with a 2-unit circumference and a diameter of √φ/2, perfectly fitting along line LI, saying:

• “Draw 4 identical circle O’s, each with given 2-unit circumference, tangent to each other with midpoints on the line segment LI: Circles O1, O2, O3, O4.”

However, this assumption is not proven.

With this statement, Lear is “assuming into existence” four circles with a circumference of 2 whose diameter is √φ/2, which is 1/4 the 2√φ length of line LI.

So, out of nowhere, and with no prior proof, we now have four hypothetical circles, each with a 2-unit circumference and a diameter of √φ/2.

What’s the ratio of that circumference to the diameter? It’s 2:√φ/2, which is none other than 4/√φ, which is 3.1446…, as his presumed value for Pi. With that flaw, and this fly now in the ointment, Lear extrapolates this unproven value for Pi to draw larger circles as his “proof” of this alternate value of Pi.

Implications of flawed logic on results

With the logic and math being used here, it would have been just as meaningful, and false, to prove that Pi = 22/7 by constructing a line LI of length 28/11 and stating “Draw 4 identical circle O’s, each with a given 2-unit circumference, circumference, tangent to each other with midpoints on the line segment LI: Circles O1, O2, O3, O4.” The ratio of the circumference to diameter in this case is 2 : 7/11, and voilà we’ve just proved that Pi = 2 · 11/7 = 22/7!

There’s simply no reason to believe that the circle with 2″ circumference that Lear introduces in the first step of the “proof” to simply roll and create a 2″ line is the same circle that now perfectly fits four-abreast on line LI with a quarter length of √φ/2, or 0.636009…, as its diameter.

Until proven differently, we must assume that the correct value for Pi is still 3.14159…, and that a circle with the diameter of √φ/2 specified by line segment LI has a circumference of Pi times diameter, or π · √φ/2. This circumference is 1.99808…, not the originating circle with circumference 2, as the proof wants you to believe.

Mathematics claims demand mathematical validation

When drawn by hand in pencil, as it is on Lear’s proof, the eye can never see the difference. The “proof” is unfortunately a bit of a “mathemagical” sleight of hand, like the magician who makes a quarter “appear” from one hand while making you think it was the very same quarter that was in the other. The trick: It’s not the same quarter, nor is this the same circle.

I do not doubt Lear’s intentions, nor do I intend any disrespect. I appreciate the creativity, detailed analysis and dedication in his attempt to prove 4/√φ as the value of Pi. This, however, is simply not a proof. It’s an unsupported proposition and claim based on faulty logic.

Let’s proceed to the other proofs.

Evaluating Further Proofs

The same logical flaws appear in Lear’s other proofs. In Proof 2, he assumes π = 3.1446… based on the flawed reasoning of Proof 1, making any conclusions equally unreliable. Proofs 4, 6, and 7 also build upon these incorrect assumptions. For example, Proof 6 attempts to use physical measurements to support the value of 3.1446…, but without clear evidence or accurate methodology, these measurements can’t be trusted. The details of these proofs are as follows:

• Proof 2 – Geometric Proof 2 is essentially the same construction as that of Proof 1, appearing to be just a refinement with clearer documentation of the steps. In step 9 though, Lear says “From Proof 1, let π=3.144605512…,” and uses this to calculate the length of line segment AI with two of the same circles on it from Proof 1, concluding that they also confirm the value of 3.144605512… for π. Since Proof 1 has been shown to be flawed and false, any conclusions drawn from Proof 2 are also flawed and unreliable.
• Proof 4 – No Proof 3 is listed on the site, so we proceed to Proof 4. Step 9 says “From previous Proofs 1, 2, 3 and 5, π=4/√φ,” so unfortunately this too is based on the flawed and false Proof 1, and its conclusions are flawed and unreliable.
• Proof 6 – No Proof 5 is listed on this page, so we proceed to Proof 6. In the upper left corner it says, “From Proof 3: Killing Pi, the physical measurement of Pi = 3.1446… NOT 3.1416…” In the lower left, upper right and lower right of the proof, it says “Given physical measurment: Pi = 3.1446…” and then proceeds with calculations that purport to confirm the “squaring of the circle” whose diameters are derived from a golden ratio-based Kepler triangle. So here again there is no “proof” of anything, but rather just a construction that assumes Pi = 3.1446… from a physical measurement that is not clearly referenced.
• Proof 7 – Proof 7 is listed as a “Physical and Geometric Proof.” It’s based on a geometric construction from Proofs 1, 2 and 5 that has been drawn on a 40″ x 60″ foam poster board. The circle drawn on the board has a diameter of 1,000mm, and has been cut with a rotary circle cutter and beam compass. Measurements are made with a Starrett Engineering Tape Measure. In this proof, the measurements supporting Pi = 3.1446… derived from the previous flawed geometric constructions are claimed to be confirmed by physical measurement.

Physical measurements as a proof of Pi’s value

Let’s now investigate the evidence that Lear provides on the measurements he took to establish the true value of Pi. Lear continued his physical measurement experiments in 2019 on a page called “Pi Measurement.” Here he had a CNC machine “cut a 1.0000 meter diameter circle” on a wood panel.

Lear’s initial findings were based on measurements taken from a foam board disk made with a rotary circle cutter. This is clearly not at the level of that of the wood panel cut by a CNC and measured by the PiTape. It would appear that Lear recognized the obvious limitations of using a foam poster board and rotary circle cutter in getting the accuracy that is critical to this investigation, and invested in better components and processes. Accordingly, I’ll review just the evidence provided on the CNC wood panel.

Challenges of physical measurements of Pi

To evaluate Lear’s claim that Pi equals 3.1446… based on physical measurements, we first need to understand the inherent challenges and limitations involved in measuring a circle’s circumference and diameter in the real world.

The value of Pi is based on a perfect circle, whose circumference consists of all points equidistant from a fixed center point. A point in geometry is a location. It has no size, i.e. no width, no length and no depth.

Taking measurements of a circular disk in the physical world introduces these problems:

• If the disk is at all out of round, it will give a false measurement of the circumference.
• Any variances or imperfections along the circumference in the radius of the disk, whether less than or greater than the intended radius, will overstate the measure of the circumference. As illustration, “walking down into a valley and back up” or “walking up over a hill and back down” both require more steps than walking directly on a straight path.
• A measuring tape can only produce a perfect measure only if it has no thickness and is sitting exactly and directly atop the circumference. Wrapping a tape around a disk can only tend to overstate the circumference.
• A measuring tape cannot be wrapped tighter than the disc, and anything less than perfect alignment, perfect contact and perfect tightness will also overstate the circumference.

So any physical measurement is going to tend to overstate the circumference, with almost no way possible to understate it. Any physical measurement that shows the true and perfect value of Pi will require that every component and process used to be perfect as well.

Additional issues of tolerance and precision

We then must consider the issue of tolerances and precision:

• No object in the physical world will have the exact measurements that were specified for its construction. That is why manufacturers specify tolerances, such as ±0.2mm.
• High-precision manufacturing must be done with high precision components and high-precision measuring devices. A reasonable, reliable approach would be to use high-precision steel milled with high precision equipment and measured with high precision lasers. By contrast, it is not possible to achieve the same accuracy or precision when using materials like foam poster boards and wood panels, measured by hand with measuring tapes.

Challenges of using wood disks to measure Pi

Lear’s Pi Measurement page describes the experimental evidence he gathered to test his proposition that the true value of Pi is 4/√φ. Its states:

• The circular disk he used a wood disk that was cut to a 1,000.000 diameter by a $100,000 Austrian built Felder Profit H08 Format 4 CNC machine.
• He used an NIST-certified Pi Tape Corp tape measure with a cost of about $400, capable of measuring to 4,000mm.

The first thing we must do is to evaluate the level of accuracy that is possible with these components and processes. In this process we will find that the while this disk may have been specified to have a diameter of 1,000.000 mm, its actual dimension cannot be that precise and accurate. This will have a direct impact on the resulting measurement of the circumference, and conclusions on the value of Pi.

In Lear’s Pi Video Diam Brand YouTube video, he states from 3:10 to 4:21 that the video is showing that the measuring pin aligns to the circumference of the 1000 mm disk. If you watch carefully though, it’s clear that radius of the disk is extremely inconsistent. This will make any measurements taken to determine the value of Pi extremely unreliable:

Tolerance issues with precision measuring tapes

PiTape.com lists its “Precision Outside Circumference Tapes” with item #CR4MM for a circumference of 3000mm-4000mm for $488. It’s important to note from the product page that this tape has a tolerance of ±0.4mm. The standard thickness of the tape is 0.25mm.

Tolerance issues on components made with CNC equipment

The tolerances for the Felder Profit H08 Format 4 CNC machine were not readily available online, so I called their technical support line and learned the following from one of their technicians:

• The tolerance of components cut with this CNC is based on the cutting bit and how it is used..
• Wood parts will have a slight spring, and with a brand new bit will have a “give” to it.
• On wood parts, the first couple of passes must be made with a large bit, and then more passes are made with smaller bits. The Profit H08 CNC has an 8 position tool holder. A separate CNC program must be written for the pass made by each bit size, and this is critical to the results. The results thus depend on programming, and the tooling and bits used, and may not give clean edges.
• The CNC generally gets the measurements of the part to about 98% of the desired specification, leaving it oversized, with about 1mm for finishing touches. The bit will leave have small ridges that then must be finished with sanding to get to the desired dimension.
• The expected precision tolerance if all the processes are done absolutely correctly is ±0.35mm. This is confirmed by the “Conditions and Guidelines” chart he provided, which says that for workpieces larger than 400mm show that an accuracy of ±0.35mm must be expected.

The circumference of a 1000mm diameter circle with traditional Pi is approximately 3,141,592.65. If the circumference is 3,144,605.51mm, as based on 4/√φ, the diameter based on traditional Pi would be approximately 1,000.959mm.

So the determination of the value of true Pi is based on a difference in its continuous diameter of only 0.959mm. That difference can appear anywhere on the circumference and lead to an error in measurement. The tolerance errors of the disc and the measuring tape is already at 0.35mm and 0.40mm, respectively, before we even introduce the other potential errors due associated with the measurement process.

Then consider what the CNC technician said: “The CNC generally gets the measurements of the part to about 98% of the desired specification, leaving it oversized, with about 1mm for finishing touches. The bit will leave have small ridges that then must be finished with sanding to get to the desired dimension.” This means that the 1000 mm disk delivered to Lear was likely somewhat over its 1000 mm nominal size. This fact alone would lead to measurements greater the true value of Pi.

Do you see the many problems and challenges here with using physical measures to determine the value of Pi?

Seeing the challenge with an HDTV simulation of a 1000mm disc

To illustrate the challenge of accurately measuring a 1000mm disc, I’ve created an image in 4K resolution (3840×2160) showing a 1000mm semicircle. Display the image or the related YouTube video on a 55″, 65″ or 75″ HDTV and select the semicircle that corresponds to the size of your TV. That circle will have a physical width of about 1000mm, which is about 39.37″ The semicircle is comprised of two semicircles. The inner one in red shows the size based on traditional Pi. The outer one in blue shows the size based on Pi as 3.1446, 4/√φ. You may not even be able to distinguish the two separate lines.

Ask yourself this: Could you take a measuring tape and measure the red line and the blue line accurately enough to declare that one value of Pi is true and the other is false?

Conclusion on Harry Lear’s determination of the value of Pi

With respect to the geometric “proofs” of 4/√φ as the value of Pi, we find:

• Proof 1 contains a critical flaw that introduces that desired value into the construction as a mathematically unsupported and unproven assumption.
• All the other proofs either flow from the flawed Proof 1 or by interjecting an assumption that π=3.14460… There is nothing in the geometric proofs to support this value for Pi.

With respect to the physical measurements to support 4/√φ as the value of Pi, every aspect of taking such a measure can only lead to overstating the value of Pi:

• Achieving any measurement of a disc’s circumference and diameter would have to be absolutely perfect in every aspect of the construction of the disc and measurement of the disc to produce a value for Pi that agrees with its known mathematical value.
• Any variances in the disk’s diameter will overstate the distance travelled around it.
• A wood disk produced by a CNC machine is by design of the process intended to leave the disc slightly over the specified diameter to allow for fine finishing.
• Whatever the results of the measurements of a disk, they cannot be more accurate than the tolerances for manufacturing of the disc and the measuring tape.

In conclusion, the methodologies used by Lear are subject to flaws in logic and shortcomings in the capabilities of physical objects and measurement equipment.

Given the flaws in both the proofs and the physical measurements, there is nothing here that can support, by reason or evidence, that the value of Pi is anything other than the value of 3.14159… that has been corroborated by every mathematically-based proof and process ever used in modern times.

As shown by the Youtube video referenced at the beginning of this article, it is very clear that a circle can be completely covered by a quantity of squares that is less than the area it would have if the value for Pi were 4/√φ. That value of 4/√φ thus overstates the true value of Pi, and can only be false.

References

Many of the claims for π=4/√φ seem to be driven by claims of Billy Meier, but as shown in these articles even those original claims have since been refuted and reversed.

https://netpic.isgreat.ca/upload/2024/02/10/20240210115628-7310302b.pdf
https://netpic.isgreat.ca/upload/2024/02/10/20240210115627-dcb2c1fe.pdf
https://forum.futureofmankind.co.uk/d/70-pi-number/5
https://forum.futureofmankind.co.uk/u/Hush
https://www.futureofmankind.co.uk/Billy_Meier/Contact_Report_712
https://www.futureofmankind.co.uk/Billy_Meier/Contact_Report_722
https://forum.futureofmankind.co.uk/assets/files/2024-02-17/1708193263-857759-pi-e.png
https://theyflyblog.com/wp-content/uploads/2023/08/Important-information-for-the-Circle-Number-Pi-Christian-Frehner.pdf

The post Pi is 3.1446 per “Measuring Pi Squaring Phi” by Harry Lear—Reviewed appeared first on The Golden Ratio: Phi, 1.618.

With the growing number of claims on the Internet that Pi is not 3.14159… but rather 3.1446… (4/√φ), I set out to find the simplest possible way to determine which value is correct. As someone known for my work with Phi (φ), this claim, which links Pi to Phi, naturally deserved my attention.

While dozens of mathematicians, dating back to Archimedes, have already provided rigorous proofs for the value of Pi, these methods are often inaccessible to the average person. This makes it challenging for people to independently verify claims, especially when they involve complex mathematical concepts. After all, who among us—whether walking down the street or sitting in a high school math class—can confidently tackle Pi using calculus, infinite series, geometric approximations, Monte Carlo methods or Fourier series? And let’s be honest, who doesn’t love the appeal of a good conspiracy theory?

You’ll find countless claims on the Internet that attempt to “prove” Pi equals 3.14460… using intricate geometric constructions. However, for the average person, validating these claims is no easier than tackling the traditional mathematical methods. Carefully working through all the steps of these constructions, and identifying the flawed assumptions or logical errors, can take hours—if not days.

Pi = 3.14159… vs Pi = 3.1446… – A simple solution

In my other article on this topic, I used an incredibly simple approach to test the value of Pi: I drew a circle on a grid and counted how many squares it took to completely cover the circle.

This simple method only became possible with the availability of personal computing power, so it was never used by mathematicians of earlier eras.

In this model, the area based on the true value of Pi must be less than the total number of squares needed to completely cover the circle. So, an area based on any claimed value of Pi that exceeds the number of squares required to completely cover the circle must be false.

Since the difference between 3.14159… and 3.14460… is less than 1 part in 1,000, you only need a circle with a radius slightly larger than 1,000 to prove which value is correct.

You can explore the details in my article, Pi = 3.14159… vs Pi = 3.1446… – A Simple Solution, where I also provide a Desmos model and a YouTube video demonstrating how it works.

This method clearly proved that the traditional value of Pi is correct. However, even after this conclusive result, some supporters of the Pi = 4/√φ theory didn’t give up. They then claimed that there are actually two values for Pi: one for the area of a circle and another for its circumference. Let’s take a closer look at why that idea doesn’t hold up either.

Proving Pi’s relationship to Circumference and Area

Let’s tackle this from two angles: first by understanding it conceptually, and then by proving it through empirical methods.

Here’s what we already know about circles:

• The radius of a circle (r) is half its diameter (D), so 2r = D.
• The area of a circle (A) is Pi times the square of the radius, or A = πr².
• The circumference of a circle (C) is Pi times its diameter, or C = πD, which can also be written as C = 2πr.

But why exactly does the area of a circle equal πr²? Why must this be the same Pi?

The answer is actually quite simple. The area of any shape is calculated by multiplying its height by its width. If you imagine cutting a circle into very fine slices, you can rearrange these slices to form a shape that looks like a rectangle. From there, the area becomes as straightforward as multiplying height times width, just as with any other geometric shape.

This concept is beautifully demonstrated in this YouTube video by MathematicsOnline:

Area is simply Height times Width!

It’s easy to see how the rectangle we use to calculate the area of the circle is created. The height of the rectangle is simply the circle’s radius (r). The length of the rectangle comes from half of the circle’s circumference, since we divide the circle into slices with half on the top and half on the bottom. Since the circumference is 2πr, half of that is πr.

So, the area of the rectangle becomes height times width, or r times πr, which equals πr².

Conceptually, this shows that the same radius (r) is used to calculate both the area and the circumference of a circle. Therefore, the value of Pi that determines the circumference is the same value of Pi used to calculate the area.

Since Pi is used in both circumference and area calculations, it’s essential that the same value is applied consistently to both, as they are interconnected in a circle’s geometry.

That should be enough to demonstrate that there can only be one value for Pi, but let’s take it a step further and prove it empirically.

Emperical proof for π=3.14159… for the circle’s circumference

I’m still working on a way to demonstrate this using a Desmos graphing calculator model, but for now, let’s explore how we can get to the same solution and proof using an Excel model.

The model is quite simple. If you’re familiar with the Pythagorean theorem (a² + b² = c²) for right triangles, you’ll easily understand it. You can visually check the accuracy of each cell in the Excel spreadsheet by looking at the formulas or using a basic calculator.

This model uses the Pythagorean theorem to calculate the x and y coordinates for every point along the circle’s circumference, based on the equation of a circle: x² + y² = r².

It then calculates the length of the hypotenuse (the line that follows the curve of the circle) for each point on the circumference, where the y-value is an integer.

Excel’s 15 places of accuracy is more than enough

I can understand that some might say, “But that’s just an estimate, not the exact value of Pi.” And technically, you’re right—it’s an estimate. But let me reassure you that this method is far more accurate than what’s necessary to address the Pi debate.

The traditional Pi value of 3.14159… and the alternative claim of 3.14460… only start to differ noticeably at the 3rd decimal place. However, with Excel, we calculate the hypotenuse lengths to 15 decimal places, which provides an extremely precise estimate. This gives us a level of precision far beyond what’s needed to determine an accurate estimate for the circumference.

Additionally, we can make these estimates as precise as we want by using larger radii. The larger the radius, the smaller the difference between the arc of the circle and the hypotenuse used to measure it. Below are the results of the model using radii of 200, 2000, and 20,000:

As you can see, even with the “low resolution” version using a radius of 200, the Pi estimate of 3.14148… differs from the traditional value by only 0.0001. When we increase the radius to 20,000 for higher accuracy, the Pi estimate of 3.14159… varies by only 0.0000001.

The Excel model used a radius of only 20,000 to keep the file size small. If you copy and paste the last row to extend it to a radius of 1,000,000, you’ll find that it produces as estimate for Pi of 3.14159265329582, which varies from the traditional value of Pi by only 0.00000000029398!

This level of precision offers conclusive empirical evidence that the value of Pi as 3.14460… simply cannot be correct.

The Excel model used can be downloaded here:

Pi_circumference_estimate_Excel_model_Gary_Meisner_2024.xlsx

You can also view a screen shot of the model and the formulas used here:

Moving beyond the Pi = 4/√φ myth

If you’ve been wondering whether the traditional value of Pi, 3.14159…, could be incorrect, I hope you find the simple proofs presented here helpful, easy and conclusive. You can validate it yourself: for the circumference, use the model described above, and for the area, check out the Desmos model in my article at Pi = 3.14159 vs Pi = 3.1446 – A Simple Solution. It’s important to note that neither model uses Pi in the calculations—both are based purely on applying the Pythagorean theorem to the circle’s formula.

In every proof I’ve examined for π = 4/√φ, there have been flaws—usually due to incorrect assumptions or logical errors. If you’ve been involved in supporting or developing proofs for Pi = 4/√φ, please know that revisiting these ideas is part of the scientific process. There’s no harm in learning and improving—math is about exploring and refining our understanding.

The most common mistake I’ve seen is an unsupported assumption that equates properties of squares and circles, which simply isn’t valid. The difference between 3.14159… and 3.14460… is less than 0.1%, and it’s not something you can detect with the naked eye. This makes it easy to be misled by geometric constructions, which is why verifying with proper mathematics is so important.

I understand that many people working on these proofs might not have a deep mathematical background, and that’s okay—it’s a learning process. I encourage you to review your work carefully, validate your results with well-established methods, and continue expanding your understanding of math. It’s crucial to approach this with an open mind and a commitment to accuracy.

If you still believe that π=4/φ, it’s critical to realize this:

You can’t just do a geometric construction or even a physical measurement that you claim proves your case. You have to explain why every other method ever used to calculate the value for Pi failed. Calculus, infinite series, Monte Carlo simulations and other mathematical methods are used in countless industries for countless applications and all produce accurate results wherever they area applied. How can these methods get it right for every other problem to which they are applied, yet fail for something as simple as the value of Pi? How can that make any sense? Move beyond the concerns about the accuracy of Archimedes and his polygons and take on Newton and calculus and the fact that we can now create very simple computer programs and spreadsheet models that show the value of Pi to 15 digits within a few minutes from our home computers. How can you show the failings in all of these other methods?

As for those influenced by claims that physical measurements show Pi to be something other than 3.14159…, it’s important to recognize that such small differences require extremely precise equipment. Basic measurement tools used in these “proofs” aren’t sufficient to challenge centuries of work, especially when multiple mathematical approaches all confirm the same value for Pi.

It’s time to move beyond the idea that π = 4/√φ and direct our energy toward projects that deepen our understanding and lead to meaningful discoveries.

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

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

Can beauty be defined by science and the golden ratio?

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

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

But science requires good evidence-backed findings

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

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

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

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

As described in the article on the DailyMail:

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

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

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

The 10 Most Beautiful Buildings in the World

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

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

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

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

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

Anthropomorphism – Making non-human things look human!

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

But do buildings have human features???

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

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

So where are the golden ratios in building design?

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

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

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

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

And what is a golden ratio anyway?

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

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

This concept is illustrated below:

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

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

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

The Golden Ratio in Logo Design

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

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

The Golden Ratio in Geometry

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

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

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

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

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

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

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

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

Sharing the true beauty of the Golden Ratio in design

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

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

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In his paper, De Groot examines the economic cycles of GDP growth in over two dozen countries. He found that their durations are grounded in golden ratio relationships. De Groot took an empirical approach to chart a pattern in the lengths of sub-cycles in GDP growth. Understanding the interrelationships between the length of economic sub-cycles then allows their cycles and their fluctuations to be detected more easily. With this information, the resulting signals of future changes in cyclical behavior can provide additional analytic tools to combat economic and societal distress.

Macro to micro economic Golden Ratio and Fibonacci patterns

To some degree, the appearance of the golden ratio in economic cycles should not be a surprise. The appearance of the golden ratio in the timing and price trends of financial markets has been studied and applied by investors for decades in the stock markets, commodities markets, Forex and elsewhere. Their application to the far more complex multi-structured cycles of entire economic cycles though has been much more challenging. Researchers have been seeking to understand these cycles for over one hundred years, so this is where De Groot’s work brings new insights and value.

Method of Analysis

To take on this task, De Groot used time series data of GDP growth for Europe and 25 OECD countries. De Groot relied primarily on Fourier analysis to detect the cycles. With this approach, a series of observations of a variable over a given period of time can be decomposed into cosine waves. When the data showed irregular trends though, this approach can become unusable. To overcome this, De Groot used a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. This makes the data less irregular while still preserving the cyclic behavior of the original data. After performing these two tests, De Groot used an algorithm to determine which cycles were prominent. He then lastly estimated a trend-cycle that described the amplitudes and phases of the economic cycles.

Results of Study

The models confirm that the cycles he found strongly correlate with the time series data. Almost all the estimates were statistically significant at a level of 5%. The detected cycles from the sample can describe swings in GDP growth rates of up to 5 percentage points. The models can thus be useful to predict future turning points of an economy as well. This is done by letting the model forecast a part of the data that is left out of the sample, but that is readily available. It is then possible to assess whether the model actually forecasts what happens.

His results indicate two to five cycles are present that in each economy. Cycles with a length of 5–6 years and 9–10 years appear most frequently. De Groot’s paper reveals that “A meta-analysis on the detected cycle lengths reveals that the ratio between the lengths of the shorter to the longer cycles in consecutive cycles often closely matches the golden ratio.” Interestingly, this finding opposes several existing theories about multi-cycle structures, which imply that the lengths of shorter cycles should be integer fractions of the lengths of longer cycles. De Groot’s paper thus provides a new direction for theory development regarding economic cycles and dynamic stability.

GDP Cycle connection to the Golden Ratio

While supported by complex mathematical analysis and statistical correlation analysis, the underlying data and concept can be easily understood. In his paper, De Groot identifies the length of two consecutive economic cycles for over two dozen countries:

De Groot then performs a harmonic regression and other processes and develops a table with the ratios between the lengths of the consecutive cycles:

The average of all the ratios in the above table is 0.619. A probability value for the test of the results against the golden ratio at 0.618 is calculated to be 0.94, which means that the hypothesis cannot be rejected statistically.

De Groot’s full paper can be found here:

https://www.sciencedirect.com/science/article/pii/S0040162521002250

Broader implications of de Groot’s research

With de Groot’s research, we now have further evidence that markets often move in patterns related to the golden ratio. Technical traders have been applying the golden ratio and the related Fibonacci sequence for decades to improve success rates in trading of stocks, commodities, currencies, and indices. De Groot’s work corroborates, with the tools and rigor of academic research, that these same trends can be found on a global macro level in the Gross Domestic Product measures for economic cycles of entire countries, and for the world at large. Given that the economic cycles of countries are certainly related to the activities of the financial markets within them, this correlation should not be a surprise. It’s fascinating though to see his research corroborate the appearance of the golden ratio in such a broader yet application.

Golden Ratios in the timing movements of stock markets

Technical analysts often use patterns based on the golden ratio and Fibonacci sequence to identify price and timing movements in individual stocks and in market indices. The patterns can be shown to occur with relative frequency when looking historically. The challenge is knowing which pattern will appear in future movements. Technical analysts thus use these Golden Ratio and Fibonacci patterns as just one of a number of analytic tools to improve their overall success rates in trades. Market analyst Bill Erman likened it to going down an interstate highway. The analytics can give you an idea of when the next interchange is likely to come up, but you need other information to know whether to go north or south.

Here is an example of golden ratio patterns appearing in the timing movements of highs and lows in the Dow Jones Industrial Average (DJIA):

Just as De Groot’s work reveals GDP economic cycles over extended periods of years with durations in golden ratio proportion, the above chart shows multiple cycles with a single year. In this year, the DJAI hit a major low in May between February and October (gold grid), another low in September between June and November (aqua grid) and yet a third in September between September and November (red grid).

Golden Ratios in the price movements of stock markets

The grid below shows the DJIA for the same period, and reveals that the price movements also followed golden ratio patterns. The interim highs and lows are in golden ratio proportions from the market high in February to the market low in October. The price points that define the high and lows are referred to as resistance or inflection points.

PhiMatrix Golden Ratio Design Analysis software was used to create the golden ratio grids in the above charts, and is considered an essential tool by some technical traders and teachers of market trading.

For additional information on how the Golden Ratio and Fibonacci sequence are related to the price and timing movements of markets see these pages:

Stock Market Analysis, Phi and the Fibonacci Sequence

Foreign Exchange Markets (FOREX) and Trading

Stock market analysis with PhiMatrix software

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If We Built It Today – Code of the Parthenon

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

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

Gary Meisner’s perspectives on the Golden Ratio

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

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

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

What’s a classic Greek story without an Antagonist?

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

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

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

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

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

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

The Parthenon’s Golden Ratios revealed

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

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

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

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

Nashville – The Athens of the South

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

Where to watch the show

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

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

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

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

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

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

Thoughts on the Parthenon’s lasting impact

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

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

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

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

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

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

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

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

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

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

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

The Visual Spectrum of Light

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

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

Chakras and the Human Body

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

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

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

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

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

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

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

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

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

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

Testing the model for validity and accuracy

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

Analysis of alternative spacings

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

Analysis of correlation to stated color wavelengths

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

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

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

Analysis of spectrum images to actual spectral photos

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

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

Analysis of impact of individual body variations

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

Background

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

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

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

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

Conclusions

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

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

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

An invitation to continue the research

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

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

The post Light, the Human Body, Chakras and the Golden Ratio appeared first on The Golden Ratio: Phi, 1.618.

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

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

Occurrences of the Golden Ratio

Phyllotaxis

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

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

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

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

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

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

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

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

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

Meaningfulness of measures that find the golden ratio

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

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

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

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

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

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

Human Attractiveness

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

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

Marquardt:

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

Levin:

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

Schmid:

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

Meisner:

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

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

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

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

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

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

A Golden Universe

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

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

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

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

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

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

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

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

Uses of the Golden Ratio

The Pyramids of Egypt

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

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

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

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

The Golden Ratio in Art

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

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

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

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

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

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

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

https://youtu.be/nXDmAtTJ6JY

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

Impacts of the Golden Ratio in the future

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

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

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

Quantum Gravity Research (QGR) is a research organization which states its mission as:

“QGR’s mission is to discover the geometric first-principles unification of space, time, matter, energy, information and consciousness. Our team of physicists, mathematicians and chemists are hard at work on what we call “emergence theory,” a new quantum gravity theory (or “Theory of Everything”) that unifies space, time, energy, matter, information and consciousness.“

QGR produced a 2017 YouTube Video titled “What is Reality?” that explains how the golden ratio plays a significant part in this.

At the 20:42 mark of the video, this is explained, saying:

The golden ratio may be the fundamental constant of nature. It is a ratio that is fundamental to circumscribed equilateral triangles and it’s weirdly ubiquitous, appearing everywhere from quantum to celestial scales. Ever so interestingly, it appears in black holes.“

The golden ratio is the precise point where a black hole’s modified specific heat changes from positive to negative:“

M⁴ / J² = Φ

It is part of the equation for the lower bound on black hole entropy:“

8πSl^2P / ekA = Φ

The golden ratio even relates the loop quantum gravity parameter to black hole entropy:”

2^πγ = Φ

Why does this support the claim that the golden ratio is the fundamental constant of nature? Because a theory of everything must unite general relativity with quantum mechanics, and a black hole is where these two theories converge at their limits.”

In matrix mathematics, the highest probability non-trivial eigenvalues that show up in binary matrices are Φ and -1/Φ!“

So we now see a deep root between black holes and quantum mechanics because the golden ratio appears deeply in both black hole physics and in quantum mechanics.”

The golden ratio appears to startling accuracies in many other ways throughout the universe in scales both large and small. It is so prevalent that its existence simply cannot be looked at as coincidental.”

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

As someone who has researched and written about the golden ratio for decades, I have to say that this is more than even I expected. As this site at www.goldennumber.net has grown, my research and articles have shown it to appear in the following areas:

• Life, from the proportions of the DNA spiral to those that define the human face.
• Beauty, including human perceptions of beauty in the human face and in the aesthetics of harmony and innate natural appeal in any of the design arts, from ancient architecture to modern day logos.
• Mathematics, including many ways that it can be derived.
• Geometry, including many fascinating ways that it is found in a variety of geometric constructions.
• Cosmology and physics, including appearances in the solar system and even in the relative proportions of the Earth and Moon.
• Financial market price and timing trends, and gaming strategy.

The proposition by QGR that it is “so prevalent that its existence simply cannot be looked at as coincidental” is not surprising, but is certainly fascinating and validating.

This perhaps explains why it has captivated some of the greatest minds and best artists in history, including Plato, Euclid, Leonardo da Vinci, Michelangelo, Johannes Kepler, Georges Seurat, Le Corbusier and more.

I’d like to thank QGR for their work and discoveries in this area, and recognize and thank Scott Beach, an independent researcher and long time contributor to this site, for bringing the video by QGR to my attention.

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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.

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

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

What the expert sources say

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

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

Wikipedia’s article on the Golden Ratio says this:

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

Checking the dissenting sources

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

Dr. George Markowsky on the Parthenon

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Dr. Mario Livio on the Parthenon

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

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

Dr. Keith Devlin on the Parthenon

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

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

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

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

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

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

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

Flaws in the Markowsky analysis

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

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

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

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

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

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

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

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

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

What the evidence shows

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

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

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

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

Parthenon Golden Ratio Summary

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

Façade:

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

Floor Plan:

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

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

A geometric perspective

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

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

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

An application perspective

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

A historical perspective

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

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

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

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

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

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

A forensic analysis perspective

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

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

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

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

Myth or Misinformation?

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

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

I look forward to your comments below.

Update on new research

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

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

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