Kevin Barrett

The Great Pi Conspiracy

Posted by Kevin Barrett on February 5, 2015

The Real Value of Pi With Mathematical Proof, by Mark and Scott Wollum

Introduction by Kevin Barrett, Veterans Today Editor

In Yann Martel’s novel Life of Pi, the narrator – a person named Pi – recounts the story of his life: “A story that will make you believe in God.” At a key point, the story splits in two. The main version of the story is fantastic; a lesser, alternate version is horrific and tawdry. Choose the better story, the fantastic story, the narrator suggests; choose the story with God in it.

The problem is, the horrific and tawdry story seems more plausible.

Leo Strauss would have enjoyed Life of Pi. He would have advised his neoconservative cult followers to hypnotize the masses with the fantastic story, while reserving the horrific and tawdry truth for themselves.

I think the neocons are wrong: The truth is fantastic, and God is truth. But that’s a long story, and this is a short introduction to a very long article.

Paralleling the two versions of the Life of Pi, there are two stories about pi the number. According to the boring conventional story, pi is 3.14159… This number is derived from imagining a circle as an infinitely-sided polygon.

But there is another story about pi, the story told in this article. It is a fantastic story – wild, hard to believe, full of bizarre implications like something out of a Dan Brown novel. If Mark Wollum is right, the truth about pi has been intentionally hidden from humanity for centuries or millennia. Perhaps it has been an “eyes-only” secret reserved for 33rd degree masons.

Will the whole history of science and mathematics have to be re-written?

I have been arguing about this with Mark for more than a year. The traditional infinitely-sided polygon derivation of pi seems obvious to me. Mark thinks I must have smoked too much pot in high school – he says there is NO WAY a circle is the same as an infinitely sided polygon. I don’t know what he’s been smoking, but whatever it is, it’s giving him some amazing geometrical visions.

So you, dear reader, will have to decide. Has Mark Wollum, a very smart guy and one of the best 9/11 truth activists in Wisconsin, independently discovered one of the most closely guarded secrets of all time? If not, can you prove him wrong?

-KB

The Great Pi ConspiracyThe Real Value of Pi With Mathematical Proof, by Mark and Scott Wollum

This article presents the author’s quest for the true value of pi, with compelling empirical evidence and multiple mathematical proofs that the real value of pi differs from the textbook number by approximately one-tenth percent, and is based on the mathematical constant known as the ‘Golden Ratio’.

Contents

Introduction

The Debate Begins

Scientific Inquiry- Testing a Hypothesis; Measuring the Circumference of a disc to Calculate Pi.

Evidence the Old Pi might Be Wrong

Questions Raised in Considering a New Pi (including problems with the infinite-sided polygon approach)

Finally- The Real Pi?

Mathematical Derivations of the Real Value of Pi

Definitions

Derivation of An 8th-Degree Polynomial for Pi

Derivation of a 4th-degree Polynomial for Pi

The Simpler Solution

Interlude

The Final Proof to Lay the Old Pi to Rest (a more conventional geometric approach)

Summary

Dedication to My (late) Brother, Scott Wollum, the Real Mathematician

Introduction

On September 5, 1962 a 20-pound chunk of metal fell from the sky and landed in the streets of Manitowoc, Wisconsin. This was determined to be a piece of satellite known in the West as Sputnik IV, a Russian satellite launched 2 years earlier. The official story claims that the satellite had failed a re-entry attempt 5 days into its voyage, instead entering into a higher but gradually decaying orbit. A replica of the metal object can be seen in Manitowoc’s Rahr-West art museum, which, coincidently, was adjacent to where it landed. The original piece was returned to Russia.

Manitowoc happens to be the town in which I was raised with my late twin brother Scott. Interestingly he had an affinity for astronomy, was very gifted academically and was brilliant at mathematics and problem-solving. We were quite young when this incident occurred, but we had seen the replica many times, which sits in a glass case at the museum.

The placing of a satellite in orbit is the culmination of centuries of advancement in science and mathematics. The mathematical constant pi, the ratio of a circle’s circumference divided by its diameter, is fundamental to understanding the behaviors of orbiting objects. In fact, knowledge of the value pi is so fundamental to our physical constructs, it’s value is paramount to understanding how the universe operates. Pi is an important factor in energy calculations of anything with rotational momentum. Its value is therefore required to understand the behavior of everything from the cosmos to sub-atomic particles. It is used extensively in physics, electromagnetics, and trigonometry, and is arguably one of the most important physical constants.

The Debate Begins

I was therefore a bit taken aback while researching alternative energy when I saw references which indicated we were using the wrong value of pi. There were claims being made that the traditional value of pi, 3.14159… was wrong, and the real value is 3.144605511…., or 4 divided by the square root of a mathematical constant known as the ‘golden ratio‘, generally referred to by the Greek letter ‘Phi’. The golden ratio = square root of 5, +1, all divided by 2. (More on this later). Four divided by square root of Phi = 3.144605511….

Upon further investigation, I found some interesting sites. There is a mathematician Jain who promulgates this new idea for pi who has written several books on subjects involving the golden ratio, and has just released a new book on the topic of pi. He expects that this revelation of new pi is expected to spark international math conferences to hash this all out, as this new knowledge reverses a couple thousand years of erroneous thinking and many flawed math concepts and physical models which all depend on this ratio pi.

There was a facebook page that discussed this topic (I can no longer find the link to this, but I did save a screen capture of the page.) The page had been terminated with the statement, “page frozen until further internal computations are made. So for now this will only be work in progress”. It also states, “3.1446055110297 (author’s note- the new pi) can be useful only under certain circumstances, which may be elaborated once (and only if) sufficient proof can be produced”.

These are very odd and nonsensical statements. How many values for pi do they want to entertain?

There had been discussion on this topic at ‘abovetopsecret.com’. Another site called ‘vortexspace.org‘ also espouses the new pi. (When last checked prior to release of this article, that site appeared to be down and inaccessible.) The problem I had digesting this material, was that most of the time, the things proffered as proof were merely geometrical expressions of this new pi based on the golden ratio. They looked as though they were begging the question. There were several sites with reference to a geometrical depiction of pithat was offered in 2006 by Bengt Erlandsen. Another site used this depiction and offered a new formula for the square root of Phi, supposedly based on this depiction. That formula is square root of Phi equals the area of a circle multiplied by 16, divided by the circumference squared. However the derivation of this formula was not explicit, perhaps buried somehow in Erlandsen’s depiction (?).

Now there was one reference, with a link posted at ATS to a paper written by a Greek engineer Stephanides, offering something that looked like it might be a real proof. The paper was from a symposium in 2008 on symmetry, held in Budapest, and offered some insightful precursor material, followed by his proof. This consisted of a rather elaborate (and very ingenious) geometric diagram, and followed logic that I had difficulty digesting. (My proofs ultimately were derived after seeing a hint provided by Jain, and from hints received from my late brother during a night-time dream visit. More on that later.)

Scientific Inquiry- Testing a Hypothesis

Measuring the Circumference of a Disk to Calculate Pi

But wait. Could any of this really be true? Could we have been using the wrong value of pi all these centuries and not have realized it? Could there be an easier way to verify one way or the other? One thinks of mathematics as being a pure science, one which needs no supporting empirical evidence, as everything is manipulated according to our created definitions and uniform rules of logic. Science however, must be supported empirically. The two purported values of pi are close, but not that close. They are off by the 4^th digit, 3.14159… VS 3.1446055…. It is a small difference, but not that small, roughly 0.1%. It’s rather curious that among the debates, I had not seen one person suggest just trying to measure the thing. Since pi is supposed to represent the distance around a circle of given diameter, with some diligence, one should be able to measure pi to 4 significant digits. This can be done with under $20 worth of equipment, a digital caliper which reads to thousandths of an inch (or hundredths of a millimeter), a tape measure, and an accurate round disc. Measure the circumference of the disc accurately to 4 places, place a reference mark along the edge of the disc, then roll the disc along a flat surface, carefully documenting starting and ending points along the flat surface. If you roll a 5-inch disc 3 times, measure this distance, divide by 3, then divide by the diameter of the disc, this number should represent pi.

Evidence Old Pi Might Be Wrong

Turns out this is easier said than done, but with with some diligence one can do it. ( I used an optical disc which was ~4.7 inches diameter, but not truly round, off by a few thousandths, so I had to try to find an average circumference). The difference between the 2 corresponding values of pi would be roughly 1/16^th inch (actually, roughly 3/64th of an inch or 1 millimeter) rolling such a disc 3 times. The three problems encountered are finding a true circular disc, accurately measuring the circumference, and rolling the disc with no slippage. Error is easily introduced if the disc deviates from the vertical plane while being rolled. My measurements did vary, but my most diligent attempts turned out considerably higher. than the old pi. This was enough to convince me that the old pi might be wrong. and gave me enough reason to continue investigation. During the course of developing mathematical proofs, I repeated this measurement one more time with the optical disc, as carefully as I could, and got 3.1448 (measuring to 4 significant figures, the 5th digit can only be an approximation). With a 10-inch brake disc and 12-inch caliper, my initial tries at 5-place precision were unsuccessful. Although I had a well-machined disc to within about 1/1000^th inch at 10.124”, the thicker disc seemed to provide other problems with reproducibility, along with variations in various tape measures themselves. My numbers varied between 3.143 and 3.145. still far from the old pi at 3.14159….

Attempts to reproduce measurements prior to publications of this article again highlighted difficulties encountered, with variable results. It is possible to reproduce consistent and accurate results with proper technique and diligence. The subject of how to do this could be it’s own article, and this article needs to be kept to a palatable length. But a few points need to be made. If you want to try this, here’s my advice:

1. If you use a standard optical disc, it must be a CD, NOT a DVD. The CD’s (at least the one’s I have) have a square edge, my DVD’s don’t. The CD’s I used had a pretty uniform diameter, didn’t vary more than a thousandths of an inch in a given individual disc. Different discs seem to vary by a couple thousandths.

2. When you roll it, it must be kept perfectly vertical. The rolling surface I used was wooden with a piece of electrical tape along the rolling surface to add a little more friction. I had to fashion a jig to keep the disc vertical. Rolling must be done with no slippage. Do not use much downward pressure as you roll the disc, as the thin disc is easily distorted, and doing so will affect accuracy.

3. Accuracy of measuring equipment can vary. I have several stainless-steel 6-inch calipers which are all consistent and give precise reproducible measurements. I had some difficulty getting consistent measurements with my 12-inch calipers. I suggest using a good 6-inch caliper with an optical CD, as they are inexpensive and accurate to four decimal places.

4. The accuracy of tape measures I have found can vary by 1/32nd of an inch among various models, and even within the same unit, depending on start and stop points. But this accuracy is enough to refute the old pi. If you want consistent 4-place accuracy, you may have to check your tape against others, or roll the disc over a longer distance, up to 8 feet.

Only one of my tape measures was graduated in 1/32ths of an inch, and I had to interpolate between lines. When I was able to get consistent rolls of 3 rotations, the same measured distance using five different tape measures revealed the following calculated pi values:

1) 3.144

2) 3.144

3) 3.143

4) 3.143

5) 3.145

You will need to check measurements for yourself. If my numbers aren’t reproducible, they won’t mean anything. If people want to obscure the truth, they will just post bogus numbers. If you are good with detail work, have a steady hand and a good eye, know how to use linear measuring equipment, you should be able to duplicate my results. I contend that these results call into serious question the validity of the old pi, and the hypothesis that the circle equals an infinitely-sided polygon must be re-examined.

If the old pi cannot be supported empirically it must be discarded.

Questions Raised in Considering a New Pi

Could this really be true? Could we have been using the wrong value for pi for 2000 years? If so, what would be the implications?

The true value of pi is paramount to understanding the interactions of all our constructs of reality, since they all involve rotational energy.

So if you really wanted to interfere with a culture’s technological advancement, just give them the wrong value of pi. If we assume (and we do) that there can only be 1 value of pi (this assumption might be up for debate, as problems can arise when attempting to define the length of an arc), and we have been using the wrong value, then several questions come to mind.

1). How could we be using the wrong value for pi all these centuries and not realize it?

2). NASA needs the true value in order to understand the behaviors of their orbiting projects. If the rest of us were using the wrong value, why would they not inform us?

3). How could this go un-noticed by experts, and would any of them be aware of the true value?

4). If the old value of pi is wrong, then the method used for its derivation must be flawed. Could this old derivation really be flawed? Could every scientific calculator and trigonometry table in the world be fundamentally flawed?

1). An accurate value of pi is not needed for most daily applications. Three-place accuracy will usually be sufficient. In the mid 1970’s we were still using slide rules for engineering, and most only had 3-place accuracy. Most mechanical or electrical systems have enough tolerances built into them, and problems/discrepancies can be accounted for by other elements in a system, generally un-noticed. I tried to explain to a friend of mine that my pi measurements didn’t match the academic number. His response was, “I’d rather trust the math”. The hardest things to see can be those for which you aren’t looking.

2). We know NASA needs the true value in order to successfully predict the behaviors of orbiting bodies. One tenth of a percent is large when dealing with astronomical distances. Would they have any reason to keep its true value hidden? There were rumors online that NASA had to use a slightly higher version of pi to successfully land on the moon. Jainclaims to have received a communication from a worker from NASA who confirmed that he had to use a higher value for pi when doing some precision mechanical work. Did NASA want to make sure they maintained advantage over competitors in the space race? Does knowledge of true pi have profound implication for the understanding of other energy sources or the very fabric of reality? Perhaps the key to a unified-field theory? Is this one of the elite’s guarded secrets?

3). Don’t get me started on the academics. If they are so morally and/or intellectually challenged as to acquiesce to absurd government propaganda regarding the collapses of the Twin Towers and building 7, then we can hardly look to them as pillars of academic truth. After all, they were challenged enough at applying high-school level physics to the collapses of the Twin Towers. How can you expect them to be adept at applying high-school level algebra and geometry to other real-life issues? And you can just ask Dr. Barrett what happens to intellectual honesty when politicians confront academics who lack cojones.

4). The old value of pi is believed to be a transcendental number, that is, non-algebraic, not able to be represented by any polynomial (see link for detailed definition). It is derived using a circle with an inscribed polygon. As the number of sides of the polygon is increased, it appears as though the circumference of the polygon approaches that of the circle, as the number of sides of the polygon approaches infinity. This idea is so thoroughly ingrained that I have seen this argued as being the actual definition of a circle. Well, a circle can be defined any way we want, but to be consistent with reality and the basic algebraic expression of a circle, one would have to say that a true circle is a collection of points on any given plane which are equi-distant from a given point on said plane. That is a true circle. From my recollection, that is what I was taught in high-school geometry class. I will attempt to prove that is not the same as the infinitely-sided polygon.

It has been argued that further evidence of the validity of this old method is the fact that attempting to approach the circle with a polygon circumscribed around the circle (on the outside of the circle) yields the same results. That is, the limits of an infinitely sided polygon circumscribed around the circle, equals the limits of an infinitely-sided polygon inscribed within the circle. It’s been decades since I studied calculus, so I’ll take someone’s claim that the 2 numbers are equal. That would make sense, they would have to be the same, as those two polygons have points in common. But I would argue that those limits do not equal the true circle. Some have argued that no matter how many times you divide the arc, there will always be a value of an arc that is longer than the subtending chord. (more on that in a minute). Think about this- you are looking at limits of adding up an infinite number of things which are becoming infinitely small, which, even if you could get to infinity, would never equal the circle. What could possibly be wrong with that logic?

People have argued that this approach is similar to a graph of y = 1/x, where the function will approach the x-axis as you approach infinity, (the x-axis being defined as an asymptote or a limit). Although the graph never reaches the x-axis, the difference between the graph and the x axis keeps reducing as you carry out calculations to more significant digits.

This is comparing apples to oranges. In y = 1/x, as the variable ‘x’ becomes infinitely large, the value of the function ‘y’ becomes infinitely small. The value for the circumference of the polygon is not simply an inverse function, but a different algebraic manipulation involving the limits of adding an infinite number of objects which are becoming infinitely smaller. In this case there are two variables which are dependent, the number of sides of the polygon and the lengths of each side. As the number of sides becomes (approaches) infinitely large, the lengths of each side become infinitely small. (I never could wrap my mind around that concept). Remember the old school days when the astute student’s inquiry questioning the logic of this approach were met with creative and unintelligible answers from the teacher? Yes, the good old days…..

The concept of infinity can be difficult to grasp. Infinity does not really exist, as you can never get there, you can only approach it. Approaching something that doesn’t exist is like chasing the pot of gold at the end of the rainbow. You have a direction and a boundary but you can never reach it. There are two infinities- infinitely large and infinitely small. One can get a sense of an infinitely large number as being really large, and an infinitely small number as being really small. The idea of approaching it only establishes upper and lower limits (boundaries) as you increase or decrease the value of a function.

In the case of the infinitely-sided polygon, you have a limit (the circumference of the polygon) which approaches opposing infinities at the same time. As the number of sides becomes infinitely large, the length of each side becomes infinitely small. Approaching both opposing infinities at the same time makes no conceptual sense. What would you have if you could get there? (You’d have a whole lot of nuthin’, baby, ain’t that somethin’?). With some work, that might give you a good blues song, but not a good circle.

Some people have criticized the infinite-sided polygon by arguing that the area under the arc never disappears. But in the context of infinite limits, the idea is that area should diminish to insignificance.

It is this author’s contention that the entire process is conceptually flawed, and should be classified as undefined and discarded. And even accepting this mythical polygon, there is no proof that this actually equals the traditional definition of a circle. It has only been assumed that this entity has something to do with reality.

Since they couldn’t prove that an infinite-sided polygon equals a true circle, they just changed the definition of a circle to equal the infinite-sided polygon.

You can’t make this stuff up.

If you are still skeptical, just remember that for two thousand years we have accepted the limit of the polygon as being the circle based on, as far as I know, no physical evidence whatsoever, only a belief that the two are the same. The fact that relatively simple algebraic derivations of pi exist, along with reproducible empirical evidence which contradicts the old pi, PROVES that the old method is flawed, unless you want to argue there are two values for pi. The error of this approach is expected to be the catalyst for entire conferences, (according to Jain) and a detailed analysis is beyond the scope and purpose of this article.

Finally the real pi?

I wondered how my late brother Scott might have approached this problem. As I worked on these derivations looking for the answer, (to either prove or disprove claims for the new pi), my brother actually visited me in a dream and I told him, “I think I can solve this thing.” Suddenly I got this image in my head of a square, with sides equal to pi, relating to a circle with diameter 4. This was before I saw Jain’s hint about squaring the circle. This, along with revisiting some of Jain’s information were keys to my derivations. He provided overt hints online as to how a proof might be worked, using a geometric depiction based on squaring of the circle. The following derivations only require a high-school-level understanding of algebra and geometry.

I contend that proof of the real pi is strong evidence of an academic conspiracy on some level.

You can bet NASA knows this, and other elites- especially those involved with theoretical physics work. Since the value pi is now pre-programmed into even the cheapest scientific calculators, one might wonder what value actually comes (stealthily ?) up when technicians involved in high-precision or classified work use its value in calculations.

In the quest for mathematical derivations for the real value of pi, I had many failed attempts. In a particular approach I had devised some equations which I later deemed to be conceptually flawed . I devised the equations based upon geometric constructions which weren’t adequately linked to pi. If I changed one variable, I could get pi to equal just about anything. However, in the context of the ‘squared circle’ depiction, the arguments I had used make a lot more sense, and are presented below. If you don’t like them, I think you may find my final proof more convincing.

So I now offer these derivations of pi, with acknowledgments to Jain for dedicating much of his life to study of the golden ratio and sacred geometry, for pioneering the new value of pi and working to get the message out. I am looking forward to his upcoming world tour. Also thanks to Stephanides for publishing his work, and to vortexspace.org for helping to lead the way, and to all those who continue searching for truth, no matter how that inquiry challenges their fundamental beliefs or elicit ridicule.

Special dedication to my twin brother, Scott Wollum.

He had a knack for reducing complex issues to high-school level mathematics, and a clarity of thought that, on his bad days, rivaled my best days. His problem-solving abilities gave him recognition as among the top students of our state, several years in a row. He showed how complex issues can often be resolved by reducing them to fundamental relationships, and that when things don’t make sense, to always challenge your assumptions. He valued intellectual honesty and moral integrity over popularity and material gain, (putting our institutional academics to shame), always had more cajones than me, and remains a great inspiration. So I offer these proofs in his name. (It is unknown whether these presentations have been discussed elsewhere).

Until we meet again, bro’.

This one’s for you,

Thanks for the visit!

Mathematical Derivations of the Real Value of Pi-

The goal is to present this material in a manner that can be digested by any reasonable high-school math student. It will therefore require a working knowledge of basic algebra and geometry.

Definitions:

concept of proportionality in linear relationships using triangles

Given orthogonal (right) triangles 1 and 2, we can say they are proportional when A2/A1 = B2/B1 = C2/C1. The triangle can be scaled up or down by any factor ‘X’ and this relationship still applies. (Fig. 1)

Also note that when two triangles are proportional, A₁/B₁= A₂/B_{2 ,}B₁/C₁= B₂/C_2,andA₁/C₁ = A₂/C₂

The corresponding interior angles will be the same in the two proportional triangles.

If two proportional triangles each have the same orientation, the corresponding sides will be parallel to each other. (side B will be parallel to BX, A will be parallel to AX, etc.)

The slope of a line is generally defines as the change in its vertical height with respect to the change in its horizontal distance.

When the triangles are oriented with respect to the same cartesian coordinate system, that is, a vertical line would represent (or be parallel to) the ‘y’ axis, and a horizontal line would represent (or be parallel to) the ‘x’ axis, the two axis being perpendicular to each other, then the hypotenuses of two proportional triangles will have the same slope. In these examples it could be defined as B/C.

Also recall that in orthogonal (right) triangles, where one angle is defined as being 90 degrees, knowing the lengths of any 2 sides, there can only be 1 third side which completes the triangle. So knowing the lengths of any 2 sides will be enough to define a unique orthogonal triangle.

More examples of proportional triangles, Fig. 1.1

In fig. 1.2 we have 3 parallel vertical lines, intersected by two horizontal lines and one oblique. Notice the numerous proportional triangles formed and how they relate to each other.

Pythagorean theorem:

for all orthogonal triangle: B^2 + C^2 = A^2. or B² + C² = A²

note: X^2 means: X squared

The ‘Golden Ratio’ (Fig. 2):

The Kepler triangle defined.

A Kepler triangle is an orthogonal triangle with proportions in the square root of the golden ratio with sides Tº, T¹, T², or T^0, T^1, T^2, or any multiple ‘X’, (T⁰X), (T¹X), (T²X).

In a Kepler triangle, we can see the following special relationships (Fig. 3):

Important note: If you have an orthogonal triangle as above with sides A, B, and C, and A/B = B/C, then by definition, this will be a Kepler triangle, and A/C = Φ (Phi), the golden ratio.

Squaring the circle or circling the square (Fig. 3.1)

When a square and circle share the same circumference, the ratio of the diameter of the circle to the side of the square will always be 4/π.

Triangles embedded in the squared circle (Fig. 3.2)

The circle has diameter = 8, the large square has sides = (2)( π).

Note that the horizontal and vertical reference lines pass through the geometric centers of the circle and the square. The triangle on the left has hypotenuse = 4, and vertical leg = π, the triangle on the right has vertical leg = 4 and horizontal base = π.

There are many triangles in this depiction with sides of 4, π, and π²/4. It is easy to identify many triangles with 2 of those 3 sides, but identifying the 3rd side apart from it’s Pythagorean expression is difficult. If all three of those segments were identified in any one triangle, they would define a Kepler triangle which would define a new pi as being related to the golden ratio.

Here are just some of those line segments. (Fig. 3.3)

Other geometric constructions based on the Kepler triangle (Fig. 4.1):

Consider these diagrams, each a Kepler triangle of defined proportions inscribed within a circle, in which each hypotenuse defines the diameter of each circle.

In diagram 2, since C₂ = pi, the circumference of the circle is defined by (A₂) (C₂), which also defines the area of the square, since (A) (C) = (B)^2 (read A times C equals B squared).

In each diagram we can see that the ratio of the square circumference / circle circumference = 4/(T∙π). (Note that this ratio will always be the same for any linear scaling ‘X’ of this depiction).

We will prove that this ratio: 4 divided by T times pi = 1 (or 4 = T∙π), and that these 2 diagrams in fact represent the same triangle.

Derivation of an 8th degree polynomial for pi:

Consider these 2 orthogonal triangles from our squared circle depiction, Fig. 3.2, in which each has 2 defined sides 4 and π, where the third sides (C₁and A₂) are derived using the Pythagorean Theorem. It is important to note that these triangles are taken from the context of the squared circle depiction, in order to adequately relate them to the real pi. (Fig. 5.1):

This next depiction will show each triangle with it’s circumscribed circle (Fig. 6), (The 3 vertices (corners) of every orthogonal (right) triangle define a circle, and the hypotenuse of the triangle will define the diameter of its circumscribed circle. See Thale’s Theorem).

circumference of circle 1 = (A₁) (B₁); <img src="///Users/kevenbarrett/Downloads/apples%20new%205.2%20web_files/apples%2520new%25204.gif" alt="" width="49" height="20" align="BOTTOM" border="0" />circumference of circle 2 = (A₂) (C₂)

We know by definition that these 2 triangles are orthogonal, but we don’t know whether they might be in Kepler proportions. We want to test whether (A) (C) = (B)² (a property of Kepler triangles).

We write this as follows: (insert 6.1)

algebraically equivalent

Note that we end up with the same expression on each side. We cannot call these equations because we are questioning their equality and don’t yet have enough information to solve them. But how could both algebraic expressions be the same? We can say that the 2 expressions are equivalent or equal to each other, because there is a one-to-one correlation among their elements. That is, each algebraic expression from diagram 1, has a corresponding algebraic expression in diagram 2; We obtained these expressions doing the same operations (algebraic manipulations) on corresponding elements of each triangle.

Let us simplify this to more clearly see what is happening, and represent each triangle as follows.

We will represent the 2nd triangle as being in unknown proportions to the first triangle using the scaling factors X_1,X₂, and X₃. We need more investigation to see what relationships they might have to each other. (Fig. 7)

If these expressions are equivalent (the same)

then X₁∙ X₃must = (X₂)²to enable the X factors to cancel out.

The only way these 2 expression can be the same is when the X factors on the right all factor out of the equation, that is, when (X1) (X3) = (X2)².

note; This may also happen when X₁ = X₂ = X₃, but we don’t yet know whether they are equal to each other.

By definition we know the values of X₁, X₂ and X₃. We can substitute these values in the above equation as follows, giving us an 8th degree polynomial for the expression of pi.

(Note that expressing pi as a polynomial is something the experts say can’t be done.)

We will not go into detail how to solve this equation. (won’t need to).

An 8th degree polynomial will have multiple solutions, but only 1 of them will make sense in the context of our problem. We suspect pi is somewhere between 3.1 and 3.2;

If we substitute the old pi = 3.14159…into the equation, we get:

65296 = 65536.

The old pi is not a good fit for this equation. Using the proposed new pi = 3.14460, we get:

65536.1 = 65536, calculations were carried out only with 7 significant digits. Using more digits will improve accuracy. This new pi satisfies the equation, the the old pi does not.

What does this mean? See Fig. 8.1

circumference 1 = A₁∙ B₁; circumference 2 = A₂∙ C₂

Consider these diagrams of circles with inscribed triangles (the 3 vertices (corners) of every orthogonal triangle defines a circle, see Thale’s Theorem). In our depictions the circumference of each circle is defined by the hypotenuse of each triangle multiplied by the corresponding leg defined as pi.

The circumference of circle 1 is represented by (A) (B), or the diameter multiplied by pi, the circumference of circle 2 = (A) (C). We have discovered a curious relationship between these triangles, derived a mathematical equation that satisfies pi in our depictions that contradicts the old pi value.

We have demonstrated that the old pi does not survive algebraic derivations based on these figures, . But the purported new pi, 3.14460 does to at least 6 places. The old pi does not satisfy the definition of pi in the squared circle or in the above diagram, and would give an erroneous result for the circumferences of these circles.

In our diagrams, the ratio of circumferences of circle 2/circle 1 =

The value of pi in our diagrams must satisfy the 8th degree polynomial we derived. If you use the old pi to describe this ratio, you of course will get a different number. And that number would be wrong,

The same 8th degree polynomial may also be derived as follows:

Since we have determined that (X₁) (X₃) = (X₂)^2,

we may say that:

Making substitutions and reducing this expression will yield the same polynomial:

π⁸+ 16π⁶ + 16³π²= 16⁴

By now you may be wondering whether X_1,X₂and X₃are all the same factor, Indeed they are, but we haven’t demonstrated that yet.

You will see that demonstrated in the next section.

Now we will derive a 4th degree polynomial for pi

We will do this after establishing proportionality between 2 specific triangles.

Consider these 2 triangles again from our squared circle depiction, and we will write the Pythagorean expression for each (Fig. 9).

Pythagorean expressions:

Again we find a situation where we perform the same algebraic manipulation on 2 different triangles and get the same algebraic expressions. Let’s again simplify this, by representing the triangles more generically.

We represent each side of the 2nd triangle as some unknown factor of the 1st one with the factors X1, X2, and X3 (Fig. 10).

same expression, therefore X₂ = X₃ = X₁to allow all X’s to factor out triangles must be proportional

The only way in which these 2 equations can be the same is when X₁= X₂ = X₃ and they all factor out of the second equation. This means that the 2 triangle are proportional to each other, and we can solve for another expression to define the 3 sides. (Fig. 11)

This allows for an easy derivation to express pi as a 4th degree polynomial.

This time we won’t bother to plug in any numbers for pi, as having derived other ways in which to express these triangles results in a very simple solution for pi.

The simple solution:

consider triangle 2 (Fig.12)

We will re-scale the triangle so that the base C = 1. and solve for the other sides maintaining the same proportions.

The new triangle has sides 1, 4/π and 16/π²

Now we write the Pythagorean expression for this as:

1² + (4/π)² = (16/π²)² or 1 + 16/π² = (16/π²)²

P.S.

Dear Krassimir Ivandjiiski I thank for reference to my name [ Stephanides ( actually Stefanides)] and to my "proof" in this article, under "The Real Value of Pi With Mathematical Proof, by Mark and Scott Wollum", Posted by Kevin Barrett on February 5, 2015 I attach relevant references to my original work for those being interested: 1) http://www.stefanides.gr/pdf/BOOK_1997.pdf 2) http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf 3) https://www.youtube.com/watch?v=ZXKQFXlWvE0 4)http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf 5) http://www.stefanides.gr/pdf/2012_Oct/PHOTO_11.pdf 6) http://www.stefanides.S.