There are a growing number of claims on the Internet that the value of Pi as 3.14159… is wrong, and that its true value is 3.1446…, which is 4 divided by the square root of the golden ratio, i.e., 4/√φ.
As I’ve been researching the golden ratio since the 1990s, I first heard about this years ago but found the proofs difficult to engage with, and the claim unlikely. It was only in 2022 though that I was finally motivated to investigate this claim in depth, after being contacted by one of the advocates for this new value for Pi.
To see which is the true value of pi, I developed a simple test that requires neither advanced mathematics or geometric constructions. I used the online graphing program Desmos, and then created a YouTube video to present the results in an engaging way.
Here’s a link to the solution on Desmos:
https://www.desmos.com/calculator/0quf8l0zqa
Here’s a link to the presentation on YouTube:
The value of Pi has been investigated since at least the time of Archimedes, who in about 250 BC estimated its area by using polygons that approached the circle’s circumference from both the inside and outside, finding its value to be between about 3.140845 and 3.142857. Zu Chongzhi, a 5th century Chinese mathematician and astronomer, calculated the value of Pi with accuracy to 7 decimal places, between about 3.1415926 and 3.1415927, using a similar method. His calculation remained the world’s most accurate for nearly 1,000 years. Isaac Newton calculated pi to 16 digits of accuracy a year after his invention of calculus in 1665. Godfrey Leibniz and many other found ways to approximate pi using infinite series. (See a complete history of the calculation of Pi.)
The work of all these various methods converged on the same value for pi, but this new claim says that there is an error of approximation in all of them, and that the true value of pi is revealed in elegant geometric constructions which show its value to be 4/√φ. Examples of such sites include Measuring Pi Squaring Phi, Proof of Pi, True Value of Pi and a MathForums site topic on this.
One note of interest on the Desmos model in the video: The blue line “complete coverage area” formula used in the Desmos model in the video has adds a full unit square that extends one unit beyond the radius along each axis. In this way, NO part of the circle’s circumference ever touches the border that defines the “complete coverage area.” The circle of any radius thus fits entirely in the complete coverage area. The circle’s area (a) is less than the complete coverage area (b), which is less than the area defined by π=4/√φ (c). If a < b < c, then how can a = c?
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I appreciate and encourage discussion on this topic so that all can learn and know the truth. As the owner and moderator of this site though, it is my responsibility and right to assure that the comments approved for display here add value to the quality of this post and this site in general.
Please keep the discussion here on topic and focus any comments posted here on the logical and mathematical validity of the method I’ve provided in the YouTube video and Desmos graph above.
I will generally limit discussion on this page to that topic, so that this single page does not duplicate the many years and countless pages of discussion on the value of Pi on other sites intended for that purpose.
I’ll be happy to post links here to such discussion sites that are provided to me where a broader discussions can be had. Here is such an example: MathForums site topic on value of Pi.
The transcription of the script for this video is presented below.
Video Script
Hello everyone,
It’s me, Pi, and I need your help. I’m having a bit of an identity crisis. I don’t know if you’ve heard, but there are some serious allegations being made against me. Now don’t think I’m just being irrational. Here’s what they’re saying:
- My true value is not 3.14159…, but instead 3.1446…., which is 4 divided by the square root of the golden ratio
- This new value is revealed in elegant geometric constructions.
- Archimedes, Newton, Leibniz and all the others who used mathematics to reveal my value all got it wrong because of their errors of approximation.
- Anyone who insists that I’m 3.14159… is in ignorance of the truth and standing in the way of progress.
Well, I want everyone to know the truth. Even if you love me as I am, I want YOU to be able to prove my value to yourself, and I want you to be able to show the truth to even your most mathematically-challenged friends.
So, I’ve created an incredibly simple solution that I think will show which of us is the true Pi.
We’re not going to use advanced mathematics. We’re not going to construct elegant geometries. They’re too time-consuming and too difficult to prove.
Here’s my plan: Let’s just draw a circle on a grid and then count the number of full squares that we need to paint over to completely cover the circle.
This might not be so quick and easy either, but you have more computing power at your fingertips than any of the great mathematicians of the past, so let’s use it.
I’ve created a model to do this test, quickly and easily. It’s available to you in a free online graphing program called Desmos, where you can see how it works and verify the results for yourself.
- The first formula creates the quarter circle in red, based on the circle’s formula, x²+y²=r².
- The next two formulas create a blue line that shows the full squares required to completely cover the quarter circle.
- This summation formula counts the number of full squares enclosed by the
- blue line.
- Next there’s a slider to change the radius r.
So what do we find?
- A quarter circle with a radius of 2 can be covered by painting 6 squares.
- A radius of 5 can be covered by 26 squares.
- A radius of 10 can be covered by 90 squares.
- A radius of 20 can be covered by 335 squares.
So where am I going with this? Well, the difference between me and this new Pi in town is less than 0.1%, 1 part in 1,000.
This means we’re going to need a radius greater than 1000 to get meaningful results. So now that you see how the model works, let’s increase the radius to 2000.
- This quarter circle can be completely covered by painting 3,143,587 squares.
- Let’s add two more formulas.
- This quarter circle’s area as πr²/4 with Pi = 3.141592… is slightly less than this, at about 3,141,592.
- Now let’s calculate the area of the quarter circle as πr²/4 again, but this time using the claimed Pi value of 3.144605…, or 4 divided by the square root of the golden ratio, which is the square root of 5 plus 1 divided by 2.
- The quarter circle’s area according to new Pi is slightly more that the complete coverage area, at about 3,144,605.
So what does this tell us?
This simple model shows us that the area of a circle calculated by this new value for Pi is larger than the area that more than completely covers the circle, which of course is larger than the area of the circle itself.
So what do YOU think? Is it mathematically possible for the area calculated by new Pi to be equal to the area of the circle, or is this new value for Pi of 3.1446… just a false imposter?
I’d love to hear your comments, and to have you share this simple solution with others so that everyone knows the true value of Pi. Thanks for participating in this investigation with me. You’re now in my circle of friends, and I’m infinitely grateful for your support. If you’re ever in the area, stop by and say Pi, or however many of my digits you’ve memorized.
And for the math lovers out there, here’s a little bonus exercise for you. A radius of 2,000 can only reveal so much. What do you discover about the complete coverage area and its relationship to Pi when the radius is 20,000, 200,000, 2,000,000 or more?
And here’s a twist on the coverage formula that increases its border area beyond the circle’s circumference even further. With a radius of 20,000, by how much can you increase the border of the complete coverage area and still have it be less than the area value given if Pi were 4/√φ?
Have fun, and keep seeking and sharing the truth.
My question is does the Geometrical Pi ratio differs from the Mathematical one??? Please respond
I don’t see how it could. A circle is very clearly and simply defined, as is its circumference, diameter, radius and area.
What reason could there be for there to be a different answer for the ratio of the circumfernce and area to the diameter? The mathematics is just an expression of the geometry, not something different or inconsistent.
The use of mathematics and geometry to determine the ratio of the circumference/area to the circle doesn’t change anything about the circle, so it can’t change anything about the ratios of its elements.
Hi Gary!
Let me ask you a question: Why do you keep making things complicated that are, in fact, very simple.
Basically, any method you use that yields the π value of 3.1415926 shows that you are indeed working with straight segments of line. With a Polygone.
When you use a direct method, as I did, measuring π in an exact way, it means in only one step and not infinite ones, you obtain 3.1446.
This again doesn’t mean that 3.1415926 is wrong, it is just te closest value you can obtain approaching the circle with straight lines. What the value 3.1446 really means is, that, evidently, we can’t measure space in the same way with a straight line or with a circular one.
And this should be the point of investigation and not to find out which value wrong.
Personally, I thought my method was quite simple and elegant. It completely avoids the “issue,” if there is one, between straight and curved lines by focusing on the area rather than the circumference. Watch the video and you see that with one simple two-part area formula and a summation formula, I showed this:
For any circle with a radius over about 1400 (to provide enough resolution), if you paint over all the full squares required to COMPLETELY COVER the circle, the number of full squares required is LESS THAN the area that πr² using π=4/√φ says is the area of the circle.
In my example on the video with a radius of 2000, if the entire circle can be COMPLETELY covered by painting 3,143,587 full squares on the XY grid, how could the area of the circle possibly be 3,144,605, especially since the area of the circle is less than the number of full squares required to cover it?
I like where gary is going why do I see something different and everyone else doesn’t?
Don’t know whether to laugh or cry.
Discussions about Giza on Academia site often involve “xxx version of π” because it makes their theories work.
Tried to follow some of the “proofs’ you link to, but will need more time to pick apart. Anything based on the Great Pyramid is ignoring the fact that it’s dimensions were rounded to the nearest cubit. It’s not a perfect Kepler Triangle.
This may be relevant: See Section 5 A new method using Thoth’s Constant, here: https://doi.org/10.5281/zenodo.8105721
People dismissing Archimedes’ method as applied in modern maths, fail to understand the concept of limits. Our polygons have infinite numbers of sides, any “curvature” will be indistinguishable from a straight line. and certainly not enough to change the value of π.
My relevant ruler and compass:
https://www.linkedin.com/pulse/polygonal-method-compared-triangular-quadrature-panagiotis-stefanides/?trackingId=ioa5z4BoRz6lLiTGKr1i9Q%3D%3D
— http://www.stefanides.gr/Html/piquad.html
P.C.Stefanides
http://www.stefanides.gr
Gottfried. You don’t say John Kepler do you.
What a strange story. What would Lenny Euler think. What do they think about his identity, it could cause a crisis.
I recently learned that if you set all the digits so far calculated of either pi or e in 9 point type it would reach to any of the inner planets.
We’ve been studying phi since before we could read. Maths predates writing systems by millennia. I’ve been looking at 16th century Hermeticism and the likes of Dee in particular. I haven’t seen this claim made by any one of that era or earlier.
It’s fascinating that this stuff gains traction.
Hi Gary,
You’ve posted a very clear and straightforward argument for the current value of pi. Thanks for taking the effort to do that as it took some work, Makes a lot of sense.
Is there any reason why there’s not been a physical measurement of pi using modern laboratory technology such as lasers, etc….
That would certainly resolve the issue.
The Value of Pi is an interesting topic. I do not usually comment on such matters online but have studied this in my own way and have applied a system where I could prove a more elegant answer. Firstly this comment is too short to explain it all in detail (a book will be released in 2025) which will.
So I will explain briefly, why the context of a circle is important in finding the correct answer to this connundrum. It is very clear that when we measure a circle ie on paper it has different properties to a circle created within nature, or a standing wave, for example. To measure distance or length, or to measure a wave will create different results. Also in a classical sense what we observe and measure are always approximatations as such.
Let’s just say that my personal opinion on this is that they are both approximations and do not truly define the underlying reality of what pi actually is and how pi functions within the time/frequency domain. For what it is worth I will say that Pi is 3.142696805 (pythagorus pi) and this is correct for 9 decimal places (discreet) , and in so the most accurate Pi that can be used.
So where does this new value of PI leave Euler’s Identity?
There is no such thing as “the ratio of the circumference of a circle to its diameter” which is a 2d concept found nowhere else but in our math books! The correct definition is “Area ratio to its volume” which makes the SQUARE AREA of a circle in 2d be
8/√(2(1+√5)) = 3.1446055110296.
Pi area is pure pseudo area because it is the same “square area” on both a circle and a square! Cred to Jain 108 who also calculated “e” based on the volume of the area which is based on this equation:
e = (7^7/6^6)-(6^6/5^5) = 2.7182621608246
Peace!
No such thing? Found nowhere but in our math books??? The most fundamentally and universally accepted definition for the value of Pi is “ratio of the circumference of a circle to its diameter.”
Take a string and wrap it around the edge a coffee cup, plate or tire. The ratio of the length around the object to the length of its diameter will be the same. That’s Pi, right there in your kitchen and garage.
Space.com shows the Earth’s equatorial circumference and diameter to be 40,075 km and 12,756 km. Their ratio is 3.14165…, within the accuracy for Pi you can expect with 5 decimal places of measurement.
It’s easy to prove that the circle’s area is based on Pi as well, so it’s not a “pseudo area”. See:
https://www.geeksforgeeks.org/how-to-prove-that-the-area-of-a-circle-is-pi-r-squared/
A sphere is just a circle rotated on its diameter, so all these relationships hold between 2D and 3D.
As the YouTube video in this article clearly proves, the claim that Pi is 3.1446… is false.
I can’t get your equation for e to work. I get 2.721464602 on my TI-30Xa and my iPhone. Could you share some of the 5 constituent values with us so that I can check my calculators? 7^7 = 823543, 6^6 = 46656, and 5^5 = 3125. 17.6513846 and 14.92992 are the quotients.
Thanks.
try it in 3 dimensions instead of 2
How or why would that make any difference? A sphere is just a circle rotated on its diameter, so all these relationships hold in 2D and 3D.
>>>>……………… What do you discover about the complete coverage area and its relationship to Pi when the radius is 20,000, 200,000, 2,000,000 or more?………… ….<<<
pls re [ relevant – microcosmos and megacosmos]f:
https://youtu.be/ZXKQFXlWvE0?t=2
If you use the Desmos model, it clearly shows that increasing the radius beyond 20,000 just results in proof of the traditional value of Pi, but to more places of accuracy. At a redius of 2,000,000 the complete coverage area is 3.1415946526×10 to the 12th, which varies only slightly from the value of Pi at 3.1415926536×10 to the 12th. This shows that 4/√φ CANNOT be the true value of Pi.
Pi is contained in base 10 décimal !
80/8.1=pi² about ! =9.876543209876543210
sqrt(9.87)
sqrt(9.8765432)=3.1426 which would be not bad but it’s not a proove just a determined number !