The Great Pi Conspiracy, Part 2

Squaring the circle: It looks cool, but is it art?
Squaring the circle: It looks cool, but is it art?
Squaring the circle: It looks cool, but is it art?

Is bad geometry killing free energy?

Researcher Mark Wollum claims that the conventional value of pi is off by one-tenth of one percent. That doesn’t sound like much. But if Wollum is right, it could turn out to be a very expensive mistake.

[Read Mark’s original article from last February, “The Great Pi Conspiracy”]

He says that even such a seemingly negligible error could wreck rotational energy models. And as former Jane’s Defense Editor Nick Cook reports in The Hunt for Zero Point, rotational energy is a key factor in the antigravity and free energy systems that were under development in Germany during World War II and later seized by US forces…then buried under layers of classification and hidden from the world.

Conspiracy researchers have often claimed that secret societies are concealing esoteric information relating to the Golden Ratio (the number phi, 1.61803398875). Dan Brown’s The Da Vinci Code played with that idea.

So what if the big secret is that the true value of pi is based on the golden ratio, as Wollum asserts? And what if that secret information is necessary for advanced work in free energy, antigravity, space exploration, maybe even time travel?

Preposterous? Maybe. Take a look at the final proof offered in this article and see if you can show why it’s wrong.

Kevin Barrett, Veterans Today Editor

The Great Pi Conspiracy II: Important Proof Update, Answering Critics; by Mark and Scott Wollum

Our first article detailed this author’s quest for the true number pi, after learning of online claims from multiple sources (Jain, Stefanides, the now defunct, facebook/therealnumberpi) that we have been using the wrong number for the mathematical constant pi. We offered empirical evidence (by describing our technique for measuring the circumference and diameter of a circular disk to calculate pi) and multiple mathematical approaches to attempt to prove that the true number pi does indeed vary from the accepted textbook value by approximately 1/10th percent. The value obtained empirically from this author varied from 3.143 to 3.145, careful measurements at 3.144, and supported claims that the true value of pi (for linear/circumference functions) is four divided by the square root of the mathematical constant known as the ‘golden ratio’, or more precisely, 3.144605511029693144…, and not 3.141592654 as taught by academia.

In this article we will:

1) provide a brief review of the issue, and talk about the implications of a conspiracy;

2) answer some of the criticism received thus far;

3) This author will critique our (his) former work, and most importantly, withdraw The Previous Final Proof offered in our last article due to serious deficiencies found by this author.

4) In its place we will provide the Revised Final Proof, which thus far appears to this author to satisfy the necessary elements of a more formal proof.

This revised proof uses only basic algebra and trigonometry, and uses the definition of pi to arrive at a unique solution. It should be easier to understand and accept compared to our previous offerings.

5) We provide some references to others who have been pursuing this subject long before coming into this author’s awareness, particularly Mr. Jain and Mr. Stefanides, (sorry for mis-spelling your name last time around- comments were closed before I could respond) and those at The Real Number Pi from facebook, and the apparently now defunct website This list isn’t meant to exclude anyone, but contains some of which this author is aware. We don’t know whether our specific approaches have been duplicated; the important thing is to get the word out. I think you’ll find these prior works (Jain, Stefanides) quite detailed, insightful, and fascinating.

One of the main reasons for even putting my name to the first article was to pay tribute to my late brother Scott, who was a true math genius and had a way of re-defining issues which enabled simple solutions to seemingly complex problems. So I again pay tribute to my brother Scott and all those seeking the truth who have the insight to recognize belief systems and challenge their own assumptions when confronted with conflicting evidence. And we challenge those in academia to have the courage and show enough intelligence to challenge their own belief systems in pursuit of knowledge, in spite of severe political pressure to maintain the status quo.

And we thank Dr. Barrett for the courage to publish this, in spite of his continual doubts about the validity of this proposed new pi. Dr. Barrett’s love of truth, had cost him his job and livelihood when he was politically hounded out of academia for his personal political views (which had little to do with his actual course curriculum addressing Arab Studies). And he continues to challenge his silently-complicit peers to explain how steel-framed high rises can collapse at or near free-fall acceleration without controlled demolition. As of this date, not one of his peers have accepted a $2000 stipend to confront him in a 90-minute debate to tell him why he is wrong.

Brief Review- Why is Pi Important?

Pi the mathematical constant defines the ratio of the circumference of a circle divided by its diameter. Comparing this curved distance to its linear counterpart, it also defines the relationship between a circle and the square which circumscribes (surrounds) it, in which the circle and square intersect at only four points. The ratio of the circumference of any square divided by the circumference of it’s inscribed circle, is, by definition, 4/pi. We will see that this definition lends itself to an algebraic derivation of the true number pi.

Pi is arguably one of the most important physical constants in the universe.

Its value defines the relationship between matter and energy in our rotational models- everything from the cosmos to sub-atomic particles, because pi is used in energy calculations in anything with rotational energy. If pi isn’t correct, the mass/energy ratio isn’t correct.

The Pi Conspiracy

We propose that there are entities who know the true number pi, but for whatever reason, keep it’s value hidden. Are there secret societies which know this, and maintain this secrecy as a form of intellectual supremacy? We do not know, but we can assume that there are formal agencies which have to know the true number pi, but choose not to divulge this information. NASA for example, has to know the true number pi in order to accurately predict the behaviors of their satellites. So it would be a fair assumption that some level of conspiracy must exist to keep this knowledge hidden. Perhaps key personnel have gag orders or non-disclosure agreements. Perhaps the true number pi is stealthily programmed into their modeling software. Who knows?

Think about this for a moment. I am proposing a massive global conspiracy to keep hidden certain knowledge. Many people, perhaps even most, will defend the orthodox position out of good faith, simply believing in the infinite-sided polygon nonsense even in the absence of any proof or any supporting evidence whatsoever, apart from the fact that the textbook pi approximates the circumference of a circular disk to within three (and only three) significant digits.

I have been given suggestions that I should talk to professors at universities, or perhaps contact the people at CERN and ask for disclosure of the dimensions of their massive round collider. Again, think about this! Would anyone make these suggestions if they understood my accusations or believed there was any possibility that myself, Mr. Jain, Mr. Stefanides, or anyone else might actually be right? If this were true, wouldn’t there be someone, somewhere, at some university who understands that the very foundation of our math and physics models is built upon logical contradictions? Who could you trust for honest disclosure? That makes as much sense as asking the Pentagon- ‘Who was responsible for 9-11’?!

I’ve been through the university system as an undergraduate, (not to be confused with our state’s higher technical/vocational education system, which I found to be excellent), and can attest to the notion that the university’s main function is indoctrination into particular belief systems. If one has a good memory, one can be very successful academically by simply regurgitating all the answers that they want to hear. (My poor short-term memory made college difficult). There need be very little thinking or problem-solving ability. Indoctrination into various professions values quantity over quality. Sure, the engineers have to be able to solve some problems- those problems as defined by the establishment. But improper framing of problems already sets one up for failure, and can perpetuate dysfunction systems. The solutions are designed to benefit corporations and existing power structures, not necessarily humanity. Hence the push to receive corporate funding for state universities. Increased corporate funding of public education will surely destroy what is left of our academic institutions, as surely as corporate funding has greatly diminished the ability of our public media to serve the public interest.

Answering the Critics

Here are a few responses to some of the criticism/comments offered.

The limit of the infinite-sided polygon approach that we presently use to calculate the circumference of a circle just seems intuitive.

A true circle, as originally defined, is a collection of points on a given plane which are all equidistant from a single point on said plane. This is the definition I was taught several decades ago. As far as I know, no one has ever offered proof that the limits of an infinite-sided polygon (apparently the newer definition of a circle) are the same as a true circle. If there were such a proof, the mathematicians wouldn’t have had to change the definition of a circle to equal the limits of an infinite-sided polygon. That whole process itself is a logical contradiction, and this author claims it is, or should be, mathematically undefined.

Infinity doesn’t exist in reality because, by definition we can never attain it. But infinity exists as a limit because we can approach it. There are two infinities- positive and negative, that is, infinitely large and infinitely small. The perimeter of an equal-sided polygon equals the number of sides times the length of each side. With an infinite-sided polygon, we have an infinite number of sides whose lengths are infinitely small. The process of multiplying positive infinity by negative infinity is self-contradictory and undefined. How can this process have a physical correlate? How can an entity which is undefined even exist as a conceptual limit? It can’t. That makes as much sense as having a crooked straight line or a round square.

If the concept of infinity is difficult to grasp (it is for me), let’s put it aside and simply consider a polygon with a very large number of equal sides. These can exist. It doesn’t matter how large- 10 thousand, 10 million, 10 billion, or 10 trillion times 10 trillion. The perimeter of any polygon, no matter how many sides it has, will ALWAYS be smaller than the perimeter of its circumscribed circle. And the perimeter of such a polygon, as you increase the number of sides, will never approach the perimeter of its circumscribed circle because THEY ARE TWO DIFFERENT MATHEMATICAL ENTITIES! Our Proofs demonstrate this. The same thing can be said for a circle with a circumscribed polygon. The circumscribed polygon will always have a circumference which is greater than its inscribed circle, no matter how many sides it has. And we are contending that it makes no sense to talk about the limits of an infinite-sided polygon because such an entity is, or should be, undefined.

I don’t know at what point the definition of a circle was changed, or what was the rational for doing so. Was the definition always the limits of an infinite-sided polygon, and I was transported to an alternate reality? I doubt it. Some reading this may be old enough to know the answer. Maybe the mathematicians needed a quick fix for contradictions posed by their illogical definition, or needed a way to help perpetuate deception, or maybe a legal way to avoid being charged with fraud. Who knows? But I will re-state emphatically- A true circle is a collection of points on a given plane which are equidistant from a given point on that plane! That is not the same as the limits of an infinite-sided polygon, which is a logical contradiction, undefined, and therefore cannot possibly have a physical correlate!

Several critics have offered these:

Perfect circles don’t exist in reality. Or the reason I can’t use the textbook pi to accurately predict the circumference of a circular disk is because that is the difference between theoretical science and applied science (that last one from someone with a PHD degree, who offered no reasonable explanation why the two pi numbers, the measured VS the theoretical, should be so far apart).

A true circle is a mathematical concept. But the first day in basic geometry class we are taught that these concepts have physical representations. How accurate can these representations be? If we grind a round disk on a spinning lathe, all points along the edge will be the same distance from the center of the disk, as accurately and as smooth as the machining process allows. We can measure the circumference of a disk accurately to four significant digits using equipment which costs $20. And we don’t even need a perfect circular disk. The circumference of a circle equals pi times the diameter. Being a linear function (no exponents), all we have to know is the average diameter. We can take multiple diameter readings, and simply use the average diameter multiplied by pi to accurately predict the circumference of our disk. This is basic math and basic mechanics. This evidence alone was enough to convince this author that there could indeed be a serious problem with the textbook value of pi. And it astounds me that some people with advanced degrees don’t get this.

If a theoretical pi value cannot survive empirical measurement to at least 4 significant digits, it cannot possibly be correct.

A better argument would have been that there must have been undetectable slippage when rolling the disk to obtain it’s circumference, or perhaps I just measured wrong. But those arguments weren’t used. Perhaps people didn’t want to encourage others to actually try measuring the circumference and diameters of a round disk, because of the discrepancies which would be revealed. I encourage others to try this. See first article for more details.

The left Riemann summation of the function of a circle with n = 10, 000 yields a pi number which is already smaller than our proposed new pi. So the new proposed pi cannot be correct.

I found this to be the most interesting and perhaps revealing comment.

The Riemann summation is a method of approximating the area under a mathematical curve by dividing the area up into very small rectangles (‘n’ number of rectangles), and adding those areas up to approximate the area. The left Riemann sum will over-approximate the area. (There are some good online references if one wants to go further into this.)

This line of argument isn’t really relevant to our claims, as we are referring to the pi constant which defines the circumference of a circle to it’s diameter. We said nothing about area calculations. But this may be an important topic.

Using an online reference to calculate the Left Riemann sum, with n = 10,000, does indeed seem to produce a smaller pi value than what we propose, if you take the formula for the area of a circle as pi times radius squared. But think about what we are saying. We are proposing a radical mis-understanding of circle geometry, and need to re-evaluate all of our assumptions about circle geometry in light of conflicting evidence. Unproven assumptions need to be challenged!

Everything needs to be logically consistent. All of our assumptions need to be re-examined.

In this article we are offering a newer math proof (at least new to this author) which seemingly provides a solid basis to discredit the old transcendental pi number for circumference functions. (Following in the heals of other authors)! The new proposed pi also seems to be supported empirically.

We propose there are at least two other logical possibilities which need to be considered regarding the left Riemann summation:

1) Either there is a problem with the algorithms used for calculations of the left Riemann sum, or

2) We need to re-think the derivation of the formula for the area of a circle.

3) Can you think of any other logical possibilities? I presently cannot, unless someone can point out any possible conceptual flaws in the Riemann Sum. (Wouldn’t it be ironic if the dysfunctional transcendental pi was perpetuated by a faulty formula for the area of a circle?)

In the online reference for the Riemann sum, using n = 10,000, they actually listed all 10,000 values, but they were listed in such a way that it was very difficult to correlate the input numbers with their respective output values (making it difficult to be confident in the result). Also, since some of the terms become very small, any calculation of this sum needs to carry enough digits to maintain a given degree of accuracy. The online calculator might have been adequate for this, but we are not sure.

Keep in mind that if the old pi is wrong as we are asserting, then all the preprogrammed functions for pi and trig functions which depend on it, and any algorithms which use these values, in every calculator and software program in the world, are fundamentally flawed!

If you eliminate the impossible, whatever is left has to be probable.

Special note to any of my associates who may dismiss this out of hand before even hearing the full argument. (And to protect the innocent, their name does NOT rhyme with ‘gets her’.)

Dismissing anything out of hand before hearing a full argument, no matter how preposterous it may sound, is a sign of ignorance and inability to challenge potential belief systems.

This story is far from finished, only beginning. We expect these new findings to overturn centuries of flawed concepts, and negate any theorems which are dependent upon the old transcendental pi number. For a $10,000 grant, I would gladly maybe spend another year staring at circles, squares and triangles, to further explore these issues. (Please contact the editor with any offers!). Absent any grant, I can’t afford the time, and there are many others who have much more skill at this than I do. And they will find the answers, if they only know to look for them!

Author’s Critique of His Own Previous Work from the first article, The Great Pi Conspiracy

We had offered three of our own different math approaches to try to resolve these issues. I had so many failed attempts during this process, which at first appeared good to me, but later revealed some flaw (always found by this author). Without much exaggeration, this process may have happened 30 or 40 times. It happened so often, I wondered whether pi might be something else entirely, and not related to the Golden Ratio, wondering if those claims were put forth to further conceal the truth. But in all of our explorations, 4/pi and T, the square root of the Golden Ratio were algebraically and geometrically indistinguishable from one another.

In the last article, my first two approaches used somewhat unconventional techniques, which although logical, I didn’t think they would necessarily convince the academics. So the final proof was offered. One of the fundamental criteria for a formal proof is that it must demonstrate one and only one solution, if indeed only one solution exists. On these grounds we are not sure about the first two, but the previous final proof does not appear to pass this test. In haste to be done with this project, it was not adequately reviewed by this author. In that last proof I had identified an exception to my argument, but explained it as having changed the definition of the symbol pi used in the proof. This was not adequately explained. Taking another look at it after publication, I had reservations about its validity. As it turns out, you will see that any value for ‘S’ within a reasonable range will yield the same equation, so we did not demonstrate a unique solution, in the confusing format in which the proof was presented. Comments had been closed by the time I realized a problem. Attempting to clean this up resulted in many more failures, but continual new insights which kept us going. I also didn’t want to disappoint my late brother, in whose joint name I offered these works.

In the process of trying to salvage the previous final proof, I was able to find a different approach in what I believe is more definitive, as it appears to have all the elements required of a formal proof. This one should be much easier to understand.

Here we use the definition of pi using a circle’s relationship to its circumscribed square, derive a value for pi and demonstrate only one solution. We use only basic algebra and trigonometry, with nothing unconventional.

It is surprisingly, if not embarrassingly simple. And I welcome any critique which can point out any logical flaws in it.

Revised (New) Final Proof


We will prove the new pi by proving that triangles 1 and 2 in Fig. 1 are the same triangle. We do this by deriving an equation for triangle 1, using the definition of pi to describe the relationship between circle B (the circle defined by diameter B) and the square defined by side C. We then show that this equation (equation 2) is valid only for the set of triangles that have the same internal angles as this triangle 1. We refine the equation (equation 3) to describe one specific triangle in that set, in which A, the hypotenuse, equals 16/pi. We then derive the same equation for triangle 2 which defines the relationship between circle B and square C in triangle 2, proving that triangles 1 and 2 have the same internal angles. And since triangles 1 and 2 also have the same length hypotenuse, they must in fact be equal triangles, giving us the lengths of all three sides. This proves that pi is expressed in a Kepler triangle, whose hypotenuse/base must equal the Golden ratio, or 16/pi squared. Hence, pi must equal 4 divided by the square root of the golden ratio.

Brief Proof Outline:

The figures should be fairly self-explanatory.

Figures 1 through 6: definitions needed to understand the proof;

Figures 7 through 13: body of the proof; the derivation of pi.

Figure 15: a geometric interpretation of the squared circle using the new pi;


1 comp

2 comp

3 comp

4 comp

5 comp

6 comp

Derivation of Pi:

7 comp

8 comp

9 comp

10 comp

11 comp

12 comp

13 comp


This solution for pi is derived from the definition of pi (using a circle’s relationship to its circumscribed square) and defines only one unique solution.

Since the old transcendental pi is inconsistent with this derivation, the old transcendental pi is inconsistent with its own definition and cannot possibly be correct.

Since the old transcendental pi is inconsistent with its own definition, it cannot possibly be correct.

Can you find any flaws here? I haven’t yet.

The new pi based on the square root of the Golden Ratio is the number you will get if you try to calculate pi to four significant digits by measuring the circumference and diameter of a circular disc. This is the pi that is needed for physics equations which define energy in rotational systems.

Note: I have not tried (and am not able) to fix all of the problems with our erroneous concepts regarding circle geometry. I believe this material, along with the work of others (some I have referenced) should provide a reason and framework for others to investigate further, and help alert others to adopt the new pi. I realize this information may spark awareness of other contradictions. But to sort it all out, we must re-examine our current concepts, and question that which is based on unproven assumptions and logical contradictions. It is possible, if not likely, that this new awareness will only come from a bottom-up grassroots movement, as the elite don’t appear willing to even look at this issue.

Special note to Mr. Jain and Mr. Stefanides, feel free to check in, would love to hear from you. If you can provide more links we will try to include them. I encourage everyone to support their work. Your works have been paramount to increasing awareness of this very important topic. Someone from facebook had checked in last time, and stated that this idea for revising our pi value was given to us several decades ago by contact with human beings of extraterrestrial origin. We find this idea fascinating. If any ET humans are around, please get in touch, would love to meet for coffee.

To summarize, I am asking the reader to consider:

1) that our entire concept of circle geometry is built upon logical contradictions and unproven assumptions.

2) that pi, one of the most important physical constants in the universe, is actually,…um…a physical constant.

3) that physical constants have representations in the physical world.

4) that representations of physical constants can be approximated with physical measurements with a given degree of accuracy.

5) that physical measurements of pi, and relatively simple mathematical proofs, are inconsistent with the textbook pi (referenced in these articles as ‘the old transcendental pi’).

6) that the existence of these contradictions necessarily implies a conspiracy to keep this knowledge hidden.

Do you want our ET friends to think we’re all just that bloody stupid?! It’s a conspiracy! That’s my story and I’m stickin’ to it…

In memory of my late brother, Scott, Wollum.

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  1. My “Case 2”, in my other post of today below, should have been something like:
    A = 16/pi, B = 3
    (4)(B)/(A)(C) = (circle B)/(square C)
    (pi)(B)/(4)(C) = (circle B)/(square C)

    because, of course, with B = 6 then B > A, which can’t make a right triangle with A = 16/pi as hypotenuse because the B leg would be longer than it.

    That minor typo of mine changes nothing about my proof that Mark’s Equation 3 is valid for multiple different right triangles with 16/pi as hypotenuse, and therefor that Mark’s “proof” fails (see my below post).

    In fact, my “Case 2” isn’t needed, because my “Case 1” with B = 5 suffices to demonstrate a different right triangle, that satisfies his Equation 3, than his Triangle 1 with B = 4.

    Also, in fact, his Equation 3 is valid for ALL right triangles, because we can simply change our unit of measurement to be whatever would make a hypotenuse equal 16/pi, which would adjust the measurements of the other sides to be in terms of this unit, and his Equation 3 is still valid.

  2. Bogus!

    In the below, what I represent with algebra is also necessarily the nature of the geometric case. Anyone can easily construct geometrically many different right triangles that satisfy Mark’s Equation 3.

    Equation 3 in Mark’s terms:
    (4)(B)/(A)(C) = (circle B)/(square C)

    Proof (by demonstration) that Equation 3 is valid for multiple different right triangles, each with different interior angles, with hypotenuse A=16/pi:

    Case 1:
    A = 16/pi, B = 5
    (4)(B)/(A)(C) = (circle B)/(square C)
    (pi)(B)/(4)(C) = (circle B)/(square C)

    Case 2:
    A = 16/pi, B = 6
    (4)(B)/(A)(C) = (circle B)/(square C)
    (pi)(B)/(4)(C) = (circle B)/(square C)

    The fact that Equation 3 is valid for many different right triangles completely breaks the “proof”, because the “proof” entirely depends upon equating Triangle 1 and Triangle 2 via the incorrect assertion that Equation 3 is valid only for a particular triangle. Whether you look at it geometrically or algebraically of informally, this is simply the nature of it.

    Triangle 1:
    A = 16/pi, B = 4
    (4)(B)/(A)(C) = (circle B)/(square C)
    (pi)(B)/(4)(C) = (circle B)/(square C)

    Triangle 2:
    A = 16/pi, C = pi
    (4)(B)/(A)(C) = (circle B)/(square C)
    (pi)(B)/(4)(C) = (circle B)/(square C)

    The fact that when hypotenuse A = 16/pi
    (4)(B)/(A)(C) = (circle B)/(square C)
    (pi)(B)/(4)(C) = (circle B)/(square C)
    means nothing about anything else.
    Which means that Mark’s Triangles 1 & 2 are not proven equal.

    • Equation 3 can easily be derived by alternate routes. E.g.:
      A = 16/pi
      (circle B) / (square C) = (pi)(B) / (4)(C) — Axiomatic
      (circle B) / (square C) = (4)(pi)(B) / (4)(4)(C)
      (circle B) / (square C) = ((4)(B)/C) (pi/16)
      (circle B) / (square C) = ((4)(B)/C) (1/A)
      (circle B) / (square C) = (4)(B) / (A)(C)

      Mark’s latest unclear comments about geometric constructions, and whatever he is imagining that his vague method has that somehow supports his “proof”, do not change the simple facts above.

      jviaene, can you understand this?

      By the way: Seventeen minus seven divided by five equals forty-two plus negative-eighteen divided by eight, because I say that if you position geometric constructions corresponding to this in a certain way then it looks that way!

    • Hi Derick,

      Case 1 indeed shows a right triangle which obeys the formula Mark had concluded to be unique. This is at least good progress on the math side. The triangle you found does not meet the 4/Pi ratio for A/B, but in Figure 13 Mark concluded that triangle 1 and 2 must be the same based on a shared hypotenusa and circle-to-square relationship. There is no mention of the 4/Pi ratio for A/B in Figure 13.

      @Mark, if are you still looking into replies, you could still be on the right track. Perhaps the wording of Figure 13 is incomplete or maybe you can refine to turn this into a formal proof which excludes all but 1 magic triangle, the one Plato called the most beautiful triangle in his Timaeus, but for which Kepler seems to have taken credit for it in the history books.

  3. Derrick,
    Thanks for response. I still believe you are misunderstanding the logic of the proof. In your initial comments it looked like you were trying to separate the algebra from their geometric representations. This would not make sense. This is not a pure algebraic proof. Pi is not just a number, it describes relationship between the square and the circle. I’m still reviewing some of your comments, but see my next post, in which I try to explain it more clearly.

    • Let’s try this. – There is a reason I did not use subscripts in my equations, they shouldn’t be needed.
      See if you follow this reasoning- We will consider only right triangle 1 with
      hypotenuse = 16/pi
      vertical leg B = 4,
      base C numerically undefined (no need for Pythagorean expression)
      We construct some geometric figures defined by this triangle, and derive equation 3 based on these constructions. Keep in mind that this equation defines relationships among various elements of the geometric constructions. And that there is only one way to do these same constructions in any given triangle. We then argue that any changes to the interior angles of this triangle will alter those relationships and render our equation 3 invalid for this triangle,. This means that no other right triangle with hypotenuse = 16/pi can be used to generate equation 3, when you compare similar geometric constructions. This means there is only one right triangle with hypotenuse = 16/pi, in which the product of its hypotenuse A times its base C, (which defines a rectangle area) divided by 4 times the vertical leg B, is equal to 4 times the base C divided by pi times the vertical leg B. Therefore, there can be ONLY ONE right triangle with hypotenuse = 16/pi with which we can use the same geometric constructions to derive the same relationships that we see in equation 3. If we find any other right triangle with hypotenuse = 16/pi with which we can construct equation 3 based on the same geometric constructions, then this triangle HAS to be the same triangle as triangle 1.

    • Of all the people who dismiss Mark’s challenging proposal, nobody seems to be able to pinpoint exactly where an error would have been made if any. Given the seemingly (and indeed enbarrassingly) simple diagrams and algebraic derivations, I suppose the more math minded among us here would have found it in the mean-time? Well done Mark and bro.

  4. Derick-
    Thanks for detailed feedback. I just discovered your input here, and started reviewing. I want to start off by saying I think you are misunderstanding my use of set theory in this geometric proof. This is not a pure algebraic proof. There is no need to prove that triangles 1 and 2 are orthogonal, we can define them that way. There is no question that they exist. There is no reason we have to to consider non-orthogonol triangles in our set comparisons. Our equations demonstrate that triangles 1 and 2 have to have the same internal angles, and therefore, have to be the same triangle.

    • I’ll drop the non-orthogonal triangles critique, because it is true that you can define them to be orthogonal (as I’ve mentioned). The primary reason I critiqued this is because I thought it would give insight into the fact that, even when defined to be orthogonal, Equation 3 and its counterpart for Triangle 2 still do not prove that they have the same internal angles. I’ll reply to your Nov. 19 post about these equations, and please read my Nov. 2 post that addresses them more.

  5. The proof does appear to be valid. We just need to clarify the set logic to illustrate this. A more detailed presentation would make this more clear.
    From equation 3 for triangle 2:
    (A)(C) = square c
    (4)(B) circle B
    we can derive another, call it equation 3b:
    (A)(C) = (4)(C) since in triangle 2, C=pi.
    (4)(B) (C)(B)

    (sorry this didn’t display correctly, the division line didn’t appear. It should read A times C divided by 4 times B, etc)
    Both these equations apply to triangle 2.
    If we change the interior angles by changing A/C,
    equation 3 cannot be valid.
    If we change the interior angles by changing B/C,
    equation 3b above cannot be valid, and therefore, neither can equation 3.
    ANY changes to the interior angles renders equations 3 and 3b invalid.
    Since any changes to the interior angles of triangle 2 with hypotenuse = 16/pi will render the equation 3 invalid, there can only be one specific right triangle with hypotenuse = 16/pi which satisfies our equation.
    Since triangles 1 and 2 both have hypotenuse = 16/pi, and both satisfy equation 3, they have to be the same triangle.

    • (I don’t know why you say “clarify the set logic” when the rest of your post does not mention any sets nor set operators.)

      Let’s start with the definitions:

      A = 16/pi
      B₁ = 4
      C₁ = sqrt( A^2 – B₁^2 )
      B₂ = sqrt( A^2 – C₂^2 )
      C₂ = pi

      and the derived equations:

      (A)(C₁) / (4)(B₁) = (square C₁) / (circle B₁) [1]
      (A)(C₂) / (4)(B₂) = (square C₂) / (circle B₂) [2]
      (A)(C₂) / (4)(B₂) = (4)(C₂) / (C₂)(B₂) [3]

      If we change the interior angles by changing A/C₂,
      we must change A and/or C₂, which contradicts the definitions, which is why this “proof branch” is invalid. If we change only C₂ (C₂≠pi), obviously equations 1,2 above would in fact still be valid, which contradicts your statement about what you called “equation 3” (but which in fact is not the Equation 3 from your “proof” (equation 1 above) but is what I call its counterpart (equation 2 above)).

      If we change the interior angles by changing B₂/C₂,
      we must change B₂ and/or C₂, which contradicts the definitions, which is why this “proof branch” is invalid. Obviously, equations 1,2 above would still be valid. If we change only B₂, obviously equation 3 above would also still be valid, which contradicts your statement about what you called “equation 3b” (equation 3 above).

      Any changes to the interior angles will of course contradict the definitions that include the Pythagorean Theorem equations. But this does not necessarily “render equations 3 and 3b invalid” (equations 2,3 above), as I just proved.

    • Anyway, such playing with the interior angles is not very relevant for considering if the two right triangles have the same interior angles.

      There are two specific right triangles that satisfy the definitions and equations, one for:
      A = 16/pi
      B₁ = 4
      C₁ = sqrt( A^2 – B₁^2 )

      and another for:
      A = 16/pi
      B₂ = sqrt( A^2 – C₂^2 )
      C₂ = pi

      Triangles 1 & 2 do not each satisfy what you call “equation 3” (equation 2 above), only Triangle 2 does, because Triangle 1 is defined in terms of different sides (that must be represented by distinct symbols, until proven equal). It’s like you’re trying to say “x = 1 satisfies y = z/2”, which is nonsense of course.

      Triangle 1 satisfies Equation 3 from your “proof” (equation 1 above) but not necessarily its counterpart (equation 2 above). Triangle 2 satisfies the counterpart from your “proof” (equation 2 above) but not necessarily Equation 3 from your “proof” (equation 1 above). This is the crux and flaw in your “proof” like you almost realized in your Oct. 31 post.

      I hope this helps you see that the two separate orthogonal triangles do not necessarily have to be the same, and nothing else attempts to prove that they are, let alone succeeds. Therefore it still has not been proven.

      Nothing you have done or said proves that B₁=B₂ and C₁=C₂! They very well could be unequal and all your definitions (including orthogonality) and your pointless derived equations would still be valid! Which means you did not prove a new value of Pi!

  6. In order to prove that the value of our old π is correct and contrary as is asserted in this interesting

    article, or part 1 of this article, and given that the author has always used multiples of the golden ratio Ф

    as components of the sides of his ‘kepler triangle’ due to the unique and peculiar behavior of

    Ф (Phi ~ 1.618033..), or its inverse ρ (phi ~ 0.618033..)

    when these reciprocal values are raised to powers, they simplify themselves by assuming a sum which is a

    proportion of itself plus a constant, a unique property possessed by no other numbers.

    For example

    Ф² == 1Ф + 1 and Ф³ == 2Ф + 1

    where it can be shown that the constant values associated with any power expression of Ф are always consecutive

    numbers of the Fibonacci sequence and at the same time their ratio happen to be an approximation of Ф itself,

    the higher the power the closer the approximation. I wonder if there also is a similar property that holds for
    π itself such that powers of π such as

    πᵐ == mπ + n

    but this is besides the point.

    • The two unique constants Ф and ρ are closely related to the natural exponential function, trigonometric functions, to geometry and specifically on the pentagon, and a host of other natural phenomena and physics such that these constants are rightly called the constants of harmony and the base of the mathematics of nature.

      Hence it is natural to use a relationship that ties the constants π and Ф together in order to help us derive the value of π.

      Ф as derived from platonic geometry and its value is fixed and can be calculated numerically as:

      Ф == ( 1/2 )( √5 + 1 ).

      When analyzing a pentagon, we can deduce the easily provable trigonometric identity that:

      cosine( 36° ) == ( 1/4 )( √5 + 1 ), which is ( Ф/2 ),

      and this is all that is needed to prove the value of π because the inverse cosine then is our original angle of 36°.

    • Expressing this angle in the units of radians, so that we bring π in the relationship, where we re-write the angle as

      36° == ( π/5 ) radians.

      Hence given that cosine (36°) == (1/4)(√5 + 1) == (Ф/2)

      π/5 == cos¯¹(Ф/2)


      π == 5 cos¯¹(Ф/2) = 3.1415926535….

      which is the value of π derived from its relation to Ф.

      The author of the article here did not pay attention that the triangle he has used does not conform to a kepler triangle because if a kepler triangle is given by the following proportions:

      ā, ā √Ф, ā Ф

      then the only problem with his triangle is that it does not have a constant multiplier ā for all the 3 sides, such that it is as:

      ā, b √Ф, c Ф

    • But you use the cos¯¹ that is defined in terms of the standard value of Pi. If Pi has a different value then cos¯¹ would need to be adjusted, and so cos¯¹(Ф/2) would give π/5 in terms of the different Pi. Your example is circular reasoning and so doesn’t prove what the value of Pi is.

  7. Problem with the proof?
    In trying to clarify the presentation, I’ve been looking at a more detailed set analysis and there may be a logical problem. The issue involves thinking that the set of right triangles defined by equation 3 must be within the set of right triangles defined by equation 2. This does not necessarily appear to be the case. This is really the crux of the proof. I am still reviewing this along with other possible interpretations, and other equations. Jury is out.

    • Neither Equation 3 or 2 defines right triangles. Maybe what you mean are the sets of right triangles that satisfy Eq. 3 or 2. The set that satisfies Eq.3 is in fact within the set that satisfies Eq.2, and that is necessarily the case. Equations 2 & 3 are in fact completely satisfied by larger sets of triangles that go beyond orthogonality, and the complete set for Eq.3 is in fact within the complete set for Eq.2. This is simply the result of your choices of defined and undefined sides.

      Did you instead mean to say Equation 3 and its counterpart in Fig.13? It is true that the right triangles that satisfy them do not necessarily have to be the same, and this is indeed part of the crux of your “proof”. The other part being the fact that there also are non-right triangles that satisfy the equations and the partial definitions.

      The crux of your “proof” is your attempt to constrain and identify two sets of beyond-orthogonal triangles with one particular orthogonal triangle, using algebra, but your algebra fails to do this.

      Even if you add the Pythagorean Theorem equations, you still can’t equate the two triangles based on internal angles because you still won’t know all the angles because you still won’t know if B1=B2 and C1=C2. Equation 3 and its counterpart still can’t do this for you, as I elaborate on in another new comment posted today. You can attempt to derive those equalities via derivations from the Pythagorean Theorem equations. Let us know if you succeed.

  8. Here is the main part of the email I sent Mark Wollum:

    (I’ll have to attempt to post this across multiple comments.)

    The “proof” is invalid because the two equations derived for the two triangles do not prove that the triangles have the same interior angles, and none of the algebra proves that the triangles are orthogonal triangles, and so it is invalid to say that the triangles’ sides can be identified with one orthogonal triangle, and so it is invalid to conclude that the sides make a Kepler Triangle. I.e. for a triangle to be a Kepler Triangle, it must be proven to be orthogonal and proven that the sides are in the required ratios, but neither of these requirements are met. I elaborate below.

    I refer to the “B” and “C” sides from Triangle 1 as “B1” and “C1”, and similarly to Triangle 2 having “B2” and “C2”. This is helpful because these separate entities should not be given same names that suggest equality that is yet to be proven.

    The two triangles are actually not defined to be orthogonal triangles, even though the pictures (Figures 7, 11 & 12) and some comments try to say that they are. In the algebraic calculations (Figures 8, 9, 11, & 13), there is nothing that makes the sides’ lengths make an orthogonal triangle, and indeed the equations (before the Kepler Triangle part) are valid for all possible triangles with the given lengths of sides whether orthogonal or not. …

    • … I.e. the derivations of Equation 3 and its counterpart for Triangle 2 are also valid when the angles between sides B1 and C1 and between sides B2 and C2 are something other than 90 degrees. All that these equations show are the scaling relationships between the various lengths if you define A=16/pi B1=4 C2=pi but leave the rest unknown. There are in fact infinitely many triangles that satisfy Equation 3, or its counterpart, with the same definitions for the sides but where C1 or B2 are given lengths that don’t make orthogonal triangles (in these cases the “B” and “C” squares don’t meet at a right angle but this is irrelevant).

      Fig.11 does not prove that “Equation 2 is valid only for that set of triangles that all have the same interior angles as Triangle 1”. This should be obvious based on my previous paragraph, but I’ll elaborate. Fig.11 only addresses the case where the triangle is orthogonal, but this is not sufficient. Because Equation 2 is derived from the definition A/B1=4/pi, then of course you can calculate a contradiction by scaling A in it but not B1. But this aspect actually isn’t relevant because Equation 2 applies to the case where A/B1=4/pi is fixed (i.e. (A)(Y) / (B1)(Y) = A/B1 = 4/pi) and where C1 is unknown and variable. In this case, we can scale C1, and thus change all the angles, and still have equality between both sides of the equation in Fig.11. …

    • … This actually proves the opposite of what Fig.11 claims to prove. Equation 2, and Equation 3 and its counterpart, are valid for infinitely many triangles that differ in their interior angles while still having the defined sides. This is an unavoidable result of not defining C1 and B2.

      (If instead, C1 and B2 are defined by the Pythagorean Theorem in terms of the other sides, thus constraining the triangles to be orthogonal, you’ll get equations that are difficult to work with in attempting to prove C1=C2 and B1=B2. I played with this for days recently and always failed.)

      It is a coincidence that Equation 3 looks the same as the counterpart equation in Fig.13. The “B” and “C” values in these are in fact referring to different entities and so those equations should be:
      (4)(B1) / (A)(C1) = (circle B1) / (square C1)
      (4)(B2) / (A)(C2) = (circle B2) / (square C2)
      These similar equations can be derived because both derivations involve A=16/pi and (circum square)/(circum circle) = 4/pi. This isn’t too surprising since all possible triangles satisfying the algebraic definitions of Triangle Sets 1 & 2 will have the same length for side “A” and the circumferences of the circles and squares will always be in linear ratios regardless of their sizes and orientations.

    • So, since those equations absolutely describe a variety of triangles that are not constrained to having the same interior angles, we cannot use the similarity of the equations to say that “Triangles” 1 & 2 describe identical triangles. We absolutely can have two (and more) non-identical triangles, that satisfy the algebraic definitions of “Triangles” 1 & 2 and satisfy Equation 3 and its similar counterpart, respectively, with the “pi” in the definitions still truly being whatever the unknown correct value of Pi is. E.g. simply define such triangles so that they have non-identical interior angles, and Equation 3 and its similar counterpart are still valid. E.g. using any approximation of Pi, we could physically construct many such non-identical triangles that demonstrate the large possible variations in the angles, and the triangles would still satisfy Equation 3 or its counterpart, where the small inaccuracies in the approximation of Pi and in physical measurements and fabrication are not enough to doubt the validity of the empirical satisfaction of those equations.

      The calculation involving the Kepler Triangle only shows what an “x” must be to satisfy a Kepler Triangle having sides of lengths 16/x, 4, and x. The “proof” fails to prove that any instance from the sets of triangles defined by the algebraic definitions of “Triangles” 1 & 2 makes a Kepler Triangle, and so it fails to prove that Pi must satisfy that “x” position in that Kepler Triangle equation.

    • What I’ve shown does not prove that Pi is not 4/sqrt(Phi), but it does prove that the given “proof” is not proof that it is.

      (The end of my email.)

  9. In each triangle 1 and 2, we define only one angle (90 degrees) and only two sides. This is enough to define unique triangles in each case, even though we don’t (yet) know the dimensions of the third side. This is basic geometry. (The only assumption needed for triangle 1 to exist is to assume the hypotenuse 16/pi is greater than side B which = 4. There is no question that 16/pi is greater than 4. The only assumption needed for triangle 2 to exist is to assume that 16/pi is greater than pi, the base C. There is no question that 16/pi is greater than pi). We show that triangles 1 and 2 have to have the same internal angles because each triangle conforms to the same format: (A)(C)/(4)(B) = square C/circle B.
    Think about this for awhile. This is intended for people who have understanding of basic geometry.

    • That “format” equation does not imply that the triangles have the same internal angles, even if the triangles are orthogonal. Ignoring your invalid symbolic conflation of the two different “B” sides and the two different “C” sides, we can still see that the “format” equation is the same as:
      (4)(C)/(pi)(B) = square C / circle B
      because A=16/pi. This equation, and any that can be derived from it by substitution or rearrangement, is rudimentary and has nothing to do with the interior angles.

      When we properly distinguish the symbols for the numeric lengths of the “B” and “C” sides, because they have not been proven to be equal preceding your attempt to do so with your “format” equation, we have:
      (4)(C1)/(pi)(B1) = square C1 / circle B1
      (4)(C2)/(pi)(B2) = square C2 / circle B2

      Does this make it easier to see that the “format” equations are valid whatever values of C1 and B2 would make orthogonal triangles? We still don’t know C1 nor B2, so we still don’t know if B1=B2 and C1=C2, which implies that we still don’t know if the internal angles are the same. If it turned out that the correct value of Pi makes C1 and B2 make different orthogonal triangles with different internal angles, the “format” equations would still be valid, which contradicts your reasoning, which is one way of seeing that it’s an invalid method of reasoning.

  10. Important! I just had a revelation. Some posters comments stating triangle 1 doesn’t, or hasn’t been shown to exist might make sense if they are construing the ‘B’ representing the base to imply the same numerical value for each triangle 1 and 2. This was not the intent! Read definitions carefully! We only defined two sides of each triangle. Those generic lettering labels were not meant to be numerical labels, only position labels, representing hypotenuse, vertical leg and base. I apologize if this was a source of confusion. This might have been more clear if triangle 2 was labeled as D, E, and F. The designation A, B, and C was meant to represent any triangle with the further refined definitions as stated. That is, in triangle 1, defining two sides and one angle defines one specific triangle, but C represents an unknown. And in triangle 2, B represents an unknown.. I could have shown the Pythagorean expression for the unknown sides. The proposed proof did not rely on stating those values. I’m not sure whether this was a source of confusion.

  11. To Derick E. who provided thoughtful feedback per email:
    My initial response (still in process of reviewing and conceptualizing your points) is that while I disagree with some of your assertions, I agree there seemed to be some conceptual issues in the proof, until I realized I forgot to mention that triangles 1 and 2 are both defined within the squared circle. In this context, the proof should make more sense to you, feedback welcome. We may have to elaborate later if you have trouble with triangle construction. (circle = diameter 32/pi, square side = 8).

    Regardless of the squared circle geometry, there is no logical reason whatsoever why triangles 1 and 2 cannot exist, each with unknown third side. This is basic geometry. (All we need is for the hypotenuse to be greater than side B). And there is no reason we cannot define them that way, because we leave the third side undefined. You seem to misunderstand the entire proof. The equations don’t define triangles 1 and 2, but equations 2 and 3 define the relationship between the circle, the square, and the triangle. As you correctly note, the only requirement for equation 2 to be satisfied is that the ratio of A/B = 4/pi, and for equation 3 to be valid, A/B = 4/pi, AND B = 4. A reference angle is necessary to provide a logical basis to compare triangles 1 and 2, and we have this in the geometry of the squared circle. (continued next post)

    • response to Derick E (continued)
      But out of curiosity, what happens if we do not refer to the squared circle depiction with reference angle of 90 deg. for each triangle? Say we provide no reference angle at all, the only requirement for triangle 1 is A/B = 4/pi, and B = 4, to satisfy equation 3, then as you point out an infinite variation of internal angles would satisfy equation 3. But then wouldn’t equation 3 apply to any triangle A, B, C, in which those conditions are met, and more importantly, only when those conditions are met? (If you change the ratio of A/B, equation 2 and 3 would not be valid). Couldn’t we refer to triangle 2 with no defined angles, only defining A/C = 16/pi squared? Then we still satisfy equation 3, don’t we? Then we might conclude that in triangle 3, A/B must = 4/pi, but so what? With no reference angle, there would be an infinite number of possible solutions, no single unique solution, so this wouldn’t be helpful.
      What if we chose a reference angle in each triangle 1 and 2 of something other than 90 degrees, say 80 degrees? Then there should be only one unique solution for pi, if we conclude that triangles 1 and 2 are the same. But how would you solve for pi in that case without using trig tables? Is this possible? I would not know how to do that. Trig tables assume a value for pi.
      Using a reference angle of 90 degrees provides an easy solution, (it appears to me the only way to obtain a solution) and the geometry of the squared circle gives us a logical basis for doing so, and should leave no doubt as to the existence of those triangles, and the meaning of pi in those triangles. This is an important conceptual point. There is nothing in the geometry or the logic that would forbid us from defining reference angles of 90 degrees. They are there. Consider this response and see what you think.
      Thanks again for detailed feedback, still looking at this.

    • I am somewhat familiar with attempts at “squaring the circle”. Your article inspired me to review a couple of them, and I find that they fail to meet the strict requirements of formal proof. (Maybe secret societies have valid proofs, but I don’t know.) Because “proof” can be subtle and tricky, I’d have to see specifically what you’re talking about regarding Triangles 1 & 2 being defined within a squared circle, before commenting.

      It seems you are not understanding the separation and independence between your geometric pictures, your informal comments, and your algebra. Your “proof” depends entirely upon your algebra, and your pictures and informal statements, which are the only things that say the triangles are constrained to being orthogonal, are irrelevant to formal algebraic proof. This is just the way algebraic proof works.

      Algebraic proof is simply valid transformation of formulas, where the validity is determined by a strict system of rules. Anything that is not a formula in the system does not exist to it, and the only meaning that the formulas have is strictly limited based on the rules of the system. This is the crux and perpetual difficulty of formal proof — we trust it because of the strict limitations and rules (which make complete sense when you study the meta theory of formal calculation), but these limitations and rules often make it quite challenging to adequately capture and model every important aspect of the concepts that we desire it to.

    • … (There are formal proofs that are formally valid but still fail to adequately model what someone wanted them to.)

      With your proof, we’re assuming the standard system of rules. Under those rules, you can’t say that Equation 3 is the same equation as the counterpart in Fig.13, for the reason I described in my email to you. To do that, you’d first have to prove that B1=B2 and C1=C2 so that you could then substitute them for each other so that “Equation 3 = counterpart” could be derived as a theorem. You can’t attempt to derive such a theorem outside the rules of the formal system and then say that it’s “proof”.

      But even if you could equate those equations, it still would not mean that Triangles 1 & 2, in the algebra that matters, are orthogonal triangles, for the reasons I described in my email to you, and so the “proof” would still fail.

      The independence between your algebra and pictures/comments is absolutely critical. You might imagine how some separate geometric arrangements might give insight into what you’re trying to do, but until you capture every critical detail of that in the algebra, it doesn’t exist to the algebra.

      The way standard algebra works is that the entities are as vague and variable as possible and so range over the largest set of values, until you either add additional axioms (definitions) that constrain things or until you derive additional theorems using only the limited rules of the system. Anything else cannot said to be algebraic “proof”.

    • If you could write natural language prose that was as obviously air-tight in its meaning and context as algebra, then you could have a valid proof in that form, which has been done before, but this is very difficult to make convincing because it’s very difficult to do series of transformations and relations that preserve precise meaning.

  12. Wollum can’t substantiate his assertion that hypotenuse = 16 / pi
    Without this false claim there is absolutely no link between his triangle and pi.

    • Consider the squared circle depiction, circle with diameter 32/pi, and square with side = 8. Both have the same circumference. Both triangles 1 and 2 that I used in the proposed proof are defined by that depiction, which embodies the definition of pi. In that context there is nothing arbitrary about them. There’s the substantiation. I should have illustrated this for clarity.

  13. John K.
    Thanks for keeping an open mind and leaving a thoughtful response. By taking this to the academics, do you mean the PHD’s who don’t understand that the textbook value of pi can be debunked with $20 worth of measuring equipment? Or their students, who sometimes exhibit more common sense? Thanks for the great ideas, I’ll take them under advisement.

  14. When we strip away Wollum’s smoke and mirrors, what he actually does is
    1) define a “Kepler triangle” with sides pi, 4, 16 / pi and
    2) use the known ratio of the legs of a Kepler triangle to calculate pi (ie. pi = 4 / 1.272… = 3.144…)

    To see what is wrong with this methodology, let’s consider the Kepler triangle x, 5, 25 / x
    Solving for x we get x = 5 / 1.272… = 3.930…

    If we had been devious or ignorant enough to use pi rather than x above (ie. pi, 5, 25 / pi), would we have just proven pi = 3.930…? No. We can’t claim the shorter leg of our triangle is pi just because we’d like it to be so, nor can Wollum, even if he does so indirectly by claiming his hypotenuse is 16 / pi. There is nothing in his so called proof that requires the shorter length to be pi, and, until he provides that, all he has done is to calculate the shorter leg of a Kepler triangle whose longer leg is 4. (And claim he’s done something more.)

    • I don’t know, or much care, if Wollum believes his own nonsense or is a charlatan taking advantage of those who can’t figure this stuff out for themselves. I am however disappointed, disgusted and outraged that Dr Barrett would provide space – and lend his credibility – to this ridiculous rubbish.

      It is bad enough that someone with a PhD is apparently not proficient enough at high school level math and logic to debunk it himself. What’s worse is that he apparently did not feel he had a responsibility to run it past someone who could give an informed opinion.

      Not too long ago, Barrett had A. K. Dewdney on Truth Jihad radio to talk about global warming, which begs the question, why didn’t he ask Dewdney to comment on Wollum’s “proof”? Ironically, while Dewdney is eminently qualified to discuss the latter, I am not aware of any reason to give particular weight to his views on the former.

      I hope that Barrett will request and publish Dewdney’s opinion, however belatedly.

    • John K.
      The triangle you mention in your example has not been demonstrated to be a right triangle. A Kepler triangle must be a right triangle, and the triangles used in my proof were taken from (embedded within, defined by, constructed within) the squared circle depiction, based on its own geometry and definitions, so there is nothing arbitrary or ambiguous about using right triangles in my proof. They are defined by the squared circle depiction which is constructed by the definition of pi. I apologize for forgetting to mention that crucial part of the proof. This concept was emphasized with similar triangles used in Part I of this article.
      I don’t define a Kepler triangle. The math I present illustrates how the Kepler triangle is defined by the definition of Pi.

    • “The triangle you mention in your example has not been demonstrated to be a right triangle.”

      Even those who are not comfortable with equations should be able to understand the concept of scaling; if we shrink or enlarge a given Kepler triangle we will get another Kepler triangle. Scaling does not affect the angles or the ratio between sides.

      So, if we start with the Kepler triangle Wollum used, which has sides
      3.144, 4, 16 / 3.144
      and scale it (multiple each side) by 5 / 4, do we get the Kepler triangle he questioned?
      3.930, 5, 25 / 3.930
      Let’s see.

      3.144 * 5 / 4 = 3.930
      4 * 5 / 4 = 5
      (16 / 3.144) * 5 / 4 = 6.361 = 25 / 3.930

      Is Wollum too dim to have worked this out for himself? Or to have plugged my values into the Pythagorean theorem to verify that my triangle is a right triangle? Did he think I would not see through his BS?

      I think not. He is once again cynically trying to confuse the marks who have fallen for his scam. Which is underscored by the fact that as weak as what I quoted is, it is also the only valid statement in Wollum’s post; the rest is pure rubbish.

      I don’t like bullies, including those who try to take advantage of others who are less knowledgeable. Wollum should be ashamed of himself. He’s been caught with an ace up his sleeve and doesn’t even have the decency to admit it and apologize to his marks.

      Stay tuned.

  15. John K:
    If you take a circle of diameter = 1, circumscribe around it a polygon with 100 billion sides or any large number, (we propose) the perimeter of that polygon will be greater than 3.144, if you calculate that circumference using trig tables that don’t use a faulty pi. As you increase the number of sides toward infinity, you are approaching a mathematical number which has no physical correlate, because an infinite side polygon is physically undefined.

  16. Mr. Stefanides:
    Thank you for checking in through facebook.
    Mr. Stefanides is the Greek engineer who I referenced in my article as being one of the researchers who has been working for a long time on issues related to this new proposed pi. I believe we are allowed one link per comment, so you can find some of his work here (as I referenced in my article).

  17. IMPORTANT NOTE: Both right triangles 1 and 2 CAN exist, and DO exist within the depiction of the squared circle. The choice of 90 degrees for the designated angles was not arbitrary. That angle, and the defined triangles 1 and 2 both exist according to the dimensions defined in the squared circle depiction. Also in that depiction, the symbol π has no ambiguity. This is an important point and I should have pointed this out in the proof.

  18. Watch David Chandler’s video “Finding Pi by Archimedes’ Method”, (16:55). Chandler calculates a lower bound for pi using inscribed polygons, but a similar spreadsheet can be put together for circumscribed polygons showing that pi < 3.1416, which means pi < 3.144…

    Outline of a proof that circumscribed polygons define an upper bound for pi.

    Prove that pi < P / D.
    A circle with diameter D is fully enclosed by any circumscribed polygon, but the polygon's corners stick out beyond the circle. The circle's area is therefore less than the polygon's. Since a circle encloses the maximum area for a given perimeter (usually called its circumference) this implies that the circle's circumference (C) is shorter than the polygon's perimeter (P).
    C < P
    C / D < P / D
    pi < P / D

    Prove that P(2n) < P(n).
    Where P(x) denote
    s the perimeter of a circumscribed polygon with x sides. Given any circumscribed polygon with n sides, we can construct a circumscribed polygon with 2n sides by "snipping off" each corner with a line tangent to the circle. The corners removed consist of triangles where one side is the new side added to the 2n polygon while the other two are parts of P(n) that were removed. One side of any triangle is always shorter than the other two combined therefore
    P(2n) < P(n)

    Understanding limits.
    limit of P(n) as n goes to infinity = 0
    simply means we can get P(n) as close to C (and P(n) / D as close to pi) as we like.

  19. A quick note here please, hypotenuse of our triangle here works out by the pythagoras theorem to be so close to the product of φ and π.

    The hypotenuse is √(16+π²) = 5.0862171012540703991245083142357

    whereas φ x π = 5.0832036923152598158095090132422.

    This is a strange problem, pi of course is not what is claimed here but how to debunk the hypothesis here is hard.

    If you consider:
    A kepler triangle is the triangle that has sides:

    ‘a’, ‘a.Θ^½’, and ‘a.Θ’.

    The triangle being considered here has proportions:

    4.Θ^(-½), 4.Θ^(0), 4.Θ^(½).

    which is obtained by taking the kepler proportions and dividing by Θ^(½), it still posses its kepler properties and is a right angle triangle. So far so good.

    The moment we try to “demonstrate that π appears to be equivalent to 4/T” we send the whole thing awry, and the keyword here is ‘appears’ and as always appearances can be quite deceptive.
    Still thinking this one out.

    • Note that our usage of the symbol π has no relationship to the value that is associated with the ratio of a circle’s circumference to its diameter, we may as well replace it by the symbol x for all it matters, and if its value may happen to approach this value then it is only coincidental.

      Having said that I am still going to look further into this strange problem although I do not expect the premise of this article to be true.

  20. Excuse me for not studying this problem in depth as it is new information to me but still if our triangle can not satisfy the Pythagoras criteria then it is not a right angle triangle at all will not coincide with the centre of the circle and bisect the diammeter.

    I just glanced at part 1 of this article and I think the Thale’s theorem needs certain conditions to be satisfied before it can be true, and this condition is the Pythagoras theorem otherwise the vertices will not coincide. Also I suspect the angles of the triangle are more than 180°.

    I will try to think this problem in every conceivable way I can and I will really be astonished if the preposition you presented here is true.

  21. MW, although our triangle satisfies the ratio test, A/B = B/C, so that its sides are related to the golden proportion, it still fails to satisfy the criteria to be a Kepler triangle.

    A Kepler triangle is the triangle that has sides ‘a’, ‘a√Θ’, and ‘a.Θ’ where Θ is Phi.

    The basic relationship between the sides is 1, √Θ, Θ and any constant multiplier ‘a’ can scale this relationship.

    Our triangle with sides 16/π, 4 and π does NOT have a common constant multiplier such that if the 3 sides are divided by it would still retain the Kepler relationship of 1, √Θ, Θ
    and so basically fails to be a Kepler triangle.

    • To elucidate further I can have a triangle with sides 8, 4 and 2
      as A, B and C.

      It satisfies A/B = B/C and A.C = B² but it fails to satisfy either either the Kepler requirement as well as the Pythagoras theorem.

    • Aziz-
      OK, I consider your argument that our triangle cannot be a right triangle.
      I’ve tried to demonstrate that this triangle can exist as a right triangle, being derived from 2 right triangles that can exist.

      I am trying to demonstrate that π appears to be equivalent to 4/T, (where T = √φ) so that 16/π, 4, and π are equivalent to 4T, 4, and 4/T, respectively. Or if you prefer: 16/π = 4√φ, 4 = 4, and π = 4/√φ, giving us the Kepler Triangle.
      If this isn’t conclusive, at least I have been unable to find any evidence to decisively refute it. It all looks consistent. And the new proposed pi seems to have empirical evidence to support it. On that basis alone, I would conclude the textbook pi cannot be correct. If one has a lot of time, Jain and Stefanides have a lot more information to ponder. Glad to see someone taking time to think about this.- Mark

  22. Further, Kepler or not, the same triangle below Fig 13 with sides 16/π, 4 and π do not satisfy Pythagoras theorem.

    • Aziz-
      A Kepler triangle is a right triangle by definition, so it must also satisfy pythagoras. Perhaps you mean to state that 16/π, 4, and π cannot exist as a right triangle. We try to show that it can, by deriving it logically from triangle 1 and 2, each being defined by only two sides. One might contest this proof on conceptual grounds, but it cannot be disproved by assuming a value of pi which is conceived illogically (the textbook value) and has no demonstrable empirical evidence to support it. -MW

    • Yes tony
      where the proportions of a true kepler triangle are linear, the proportions given by MW are reciprocal.

  23. To make head or tail of this problem just consider Fig 6, and pay attention to (A)(C) = B².

    Then proceed to Fig 12 and confirm that the 3 sides of the triangle truly define the Kepler triangle.

    However the explanation below Fig 13 where it states:
    ‘this identifies a Kepler triangle in which A/C = Θ, the Golden Ratio’, we find an inconsistency in the form

    Θ = 16/π².

    This does not equate to the known value of Θ = 1/2 root 5 plus 1/2 which implies that in our need to seek a new value of π we also have to have a new value for Θ.

    Just look into that everyone please.

  24. I wish a team effort would also pay attention to another ‘constant’ related problem and help to solve it, here is the problem that has not found a solution yet:

    It is known that the continued fraction of Phi is [1,1,1,1,…].

    What constant, in closed form, has the continued fraction equal to the Fibonacci sequence ? I.e. [1,1,2,3,5,8,13……]

  25. To all who find this article interesting I suggest you would also find the following book VERY interesting.

    Secrets of the Aether – Unified Force Theory, Dark Matter and Consciousness.
    by David W. Thomson and Jim Bourassa.

    Third Edition 2007 -ISBN: 0-9768128-2-7
    2004,2005 – TXul-224-594
    © Quantum AetherDynamics Institute
    518 Illinois St., Alma, IL 62807

    Published by
    The Aenor Trust
    P.O. Box 4706
    Salem, OR 97302

    • “We show that the fundamental constants in physics are not just random
      values, but have an exact value based upon a quantum-scale, dynamic
      Aether (the Aether unit has a precise value equal to Coulomb’s constant
      times 16π²). The Aether Physics Model is stunning in that it
      mathematically predicts and explains the measured values of physics with
      striking precision.”

      Page xi

  26. to samvado from facebook:
    If the formula you use for the circumference of the polygon uses a trig function, that formula is potentially flawed as suggested by my argument. It would be helpful if you could briefly cite the formula used.-MW

  27. to Daniel N. from facebook:
    Pi is a ratio of two distances. Distances are mathematical concepts that have physical representations. Would you argue that you can’t accurately calculate the ratio of the length of a table top divided by the length of one of its legs, to four significant figures, because distances are mathematical concepts? The circular distance around a disk is easily measured by rolling the disk, provided there is no slippage. This can be done to four significant digits.
    If you do find an error in my proof, I don’t think it will be algebraic, it would likely be conceptual, as everything is explicitly defined, and the algebra is basic. Let me know if you find any problem. I can’t count the number of mistakes I’ve made trying to figure this out! But I’ve gotten pretty good at finding them.- MW

  28. DaveE:
    Thanks for the comments. After thinking about this, I’m wondering whether string theory was developed in the 1970’s in order to take our attention away from rotational models. There are many examples of rotation in the physical universe. But strings? You just have to scratch your head wondering where was the inspiration for that.

  29. This article reminds me of the “How many angels could dance on the head of a pin?” medieaval theological discussions.

    PI is an irrational number. That means it cannot be expressed as a ratio. Hence it is derived using infinite series, and infinite series give infinitely long answers. PI can be expressed to as great an accuracy as we require. I usually settle for 3.14159. That’s accurate enough for most uses.

  30. This is a good article but it needs some thinking. It is kind of like a puzzle which takes some time to sort it out like DavE said below.

    In one respect the author is correct to state that we can agree in any definition. For example we could agree to define Density as ‘the mass enclosed within a sphere of radius of 1m’ and then proceed to quote density as kg/m2 instead of kg/m3 because the surface area of the sphere is constant. Apparently the electrical density is defined as such anyway.

    Or we could agree to define velocity not as distance travelled per time but instead the time it takes to travel 1m. So instead of having velocity as meters per second we will have seconds per meter. How this will affect dimensional analysis is anyone’s guess but is worth a try.

    I am going to attempt to analyze this pi problem but specifically with an enclosed circle in an isosceles triangle whose base and height are unity. In this configuration the sum of its sides is 2 Phi.

    I believe involving the circumference of a circle and Phi may cause a new value of pi that is not transcendental.

  31. Figure 5. seems problematic.
    All it states is that
    [ A = (A + 1) / A ]
    and that
    [ A^2 – 1 = A ]
    which are as valid a math trick as is dividing by zero.

    • hebgb:
      The equations are correct in Figure 5, and I believe you stated them correctly,
      A^2 = A + 1, or A^2 – A – 1 = 0.
      A squared minus A minus 1 equals zero.
      This is the quadratic equation which yields the Golden Ratio. A unique feature of the Golden Ratio Phi is: Phi + 1 = Phi^2. Also, Phi – 1 = 1/Phi. The constant Phi has several interesting algebraic properties.
      Glad to see someone is actually reading the details.

    • Thank you for responding directly, Mark.
      So 1) The ratio can only work with these figures.
      And 2) The fact that it can only work with these figures is, in essence, your final proof.
      Am I correct in summarizing in this way?
      If so, is not this logic rather circular? (excuse the pun)
      Bonus question: Can this be shown to work with Tau also?

    • hebgb
      Your interpretation sounds close, but as stated doesn’t necessarily sound circular to me. The proof might sound circular if one tried to refute it by claiming that representing pi as an unknown might introduce potential error (food for thought). But I can’t see why the real pi wouldn’t have to conform to these equations. A more accurate interpretation might be that we’ve found 4/pi and tao to be algebraically identical, and geometrically indistinguishable from one another.

      Any proposed pi needs supporting math and empirical evidence (actual measurement) to support it. The old textbook pi has neither, in my estimation, for reasons cited. The new proposed pi has both. I would look at any proposal for pi which demonstrates both. Thanks for the thoughtful input.

  32. Close but not quiet. Pi as we know it only applies to static designs or geometry on paper. Even Einstein new that. When atoms or molecules undergo internal oscillation or are placed in motion, then Pi becomes a variable based on its oscillatory frequency or it rotational angular velocity. Under the velocity of gravity 9.81 mps it is as low as 3.132 or the square root of gravity; or you could say that when an atom is in motion gravity alters PI. The earth as a sphere resonates at 3.132 HZ. The square of this is 9.81 MPS or the velocity of gravity. Non perfect spheres radiate energy. Perfect spheres or atoms do not, until struck by an electron or photon. This is called the photoelectric effect. It’s in Einsteins paper. This effect wasn’t discovered until about 1913 so most math departments simply overlook it and dont properly teach it correctly.

  33. Reminds me of when in 1939 the tuning of the note ‘A above Middle C’ to 440 Hz was adopted in the world of music and A=440 became the international standard. Austrian genius Rudolph Steiner (1861-1925) was on to all of this. He said:
    “Music based on C=128hz (C note in concert A=432hz) will support humanity on its way towards spiritual freedom. The inner ear of the human being is built on C=128 hz.”

    The physical universe is built out of sound vibrations. It’s a concept that is held by some of the sharpest minds in the physics community. The 528 hz frequency is known as, the “528 Miracle,” because it has the remarkable capacity to heal and repair DNA within the body and is the exact frequency that has been used by genetic biochemists.

    Bottom line, their goal has clearly been to keep us as gullible, subservient and weak through multifarious means.

  34. Truth can have no approximations.

    Sorry to have to say this, but the True and correct pi has already been found, and has been available on the net over the last five years.

    The True pi is 22/7 or 3.142857142857…. (with this decimal expansion continued ad infinitum).
    The following Axiom (with a little elaboration after) will help the reader to better understand why it has to be so:

    ‘The intersection of 2 arches can never create a point ….
    only when 2 straight lines intersect will a point occur.’

    An arch, or curvature, is not only part of a circle but also part of any vibration, wavelength, oscillation, frequency, pulsation, quiver etc. etc., and which no mathematicians have ever taken into account or consideration when dealing with the measurements of a circles True circumference, and thus also the True length of a curvature. In other words, if, f.ex., 2 different frequencies were able to meet or “clash” in a ‘point’, the ‘Radio’ would never have been invented!

    To this absolute conclusive proof, the dear reader can verify for him/her-self by taking a look in this forum linked to, and read both page 48 and page 49 in the thread referred to:

    If the reader would like to know what the thread is mainly about, go to page 1 and directly to post #11.


  35. Satellites are falling out of the sky as their orbits degrade with this new information…

    I just read how pi was derived (the wrong way according to this article), and it makes more sense to me than the geometry in this article. Since pi is also the area of a circle with a radius of 1 (a unit circle), we should be able to use integral calculus to confirm the value is estimated to be 3.1416.

    A quick google search found several examples of this being calculated with different methods.

    As much as I would like to believe that this is the key to overunity, I think the methods in this article may be a dud. Bearden, Bedini, et. al. don’t rely on a redefinition of pi. They seem to mostly take advantage of nonlinear properties in the materials they use in their devices (Bedini’s batteries and Bearden’s Metglas cores).

    • Paul:
      Don’t confuse area measurements with circumference measurements. My argument is for the pi constant which defines the circumference of a circle, without making any argument for area calculations, as explained in the article. And to clarify Dr. Barrett’s introduction, I’m not making any claims for zero point energy. We can only speculate. I don’t know whether a change in pi would make any significant difference until we do a drastic overhaul of our present models. We need to start over from scratch, instead of trying to salvage models which are convoluted beyond sensibility and only have limited predictive ability.

  36. This breaks all of trigonometry and physics. My GPS stopped working, and my radio tuner just dials in static now.

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