History is Changed

The Completed YES Solution to the Pythagorean STC Riddle

Work posted from 2000 through 2021 is superseded here and

offered as evidence of the author's long search and credibility

The Two Proposed Changes to History

The two changes that I propose to the history of geometry and mathematics, and to the history of the world, are as follows:

ONE: The Pythagoreans had to know that the correct answer to the riddle is YES, simply because of what a riddle is. A serious riddle is a question or problem that reveals surprising or interesting and important information when properly solved. If the true answer to the STC riddle were "NO," then the riddle would be a dud, not worth the ink used to write it out. Either the Pythagoreans knew the positive solution all along or the original proposition was a declarative sentence rather than an interrogative. This was my initial conclusion in 1957 when I first heard the riddle from my math teacher, Leonard Launer, at Andrew Warde High School in Fairfield, Connecticut.

TWO: The YES solution to the ancient Pythagorean riddle is a new addition to the archeological evidence that ancient or pre-historic people possessed advanced knowledge.

I wish to add my own personal interpretation of the complete solution, as follows:

The complete answer to the question (Can we (or one) construct a circle exactly equal in area to a given square using only the compass and straightedge? ) includes a revelation regarding the "misleader" or distracting suggestion that is built into a good riddle. The misleader in any good riddle is either missing information or words in the riddle that redirect mental attention to a misguided assumption or misguided understanding of the question.

In the STC question the challenge appears to be can we construct squares and circles exactly equal in area using only the compass and straightedge?" The part that states

"using only the compass and straightedge" is the riddle's "misleader" because the implication is that we are able to construct (or express precisely) the value of pi exactly by some other means, such as by algebraic expression or digital number notation. Therefore, both amateur and professional mathematicians have "gloried" in the fact that we cannot write "pi exactly" in digital format because pi is an endless, non-repeating decimal (an irrational number). However, THIS IS EXACTLY the point embodied in the riddle, the surprising solution that we cannot write pi exactly but we can construct pi exactly ONLY when we do so with the compass and straightedge.

The solution teaches us therefore, we cannot represent the value of pi exactly (or sqrt[2] or sqrt[5] et cetera), with a digital number system, but we can represent pi exactly with the compass and straightedge and ONLY WITH THE COMPASS AND STRAIGHTEDGE. My conclusion, a proposal to all concerned, is the person or persons who asked this question (or made this statement) about 2,500 years ago knew all the details of the correct positive solution. The reason I made this discovery is because I trusted the Pythagoreans. And because I wanted a clear and convincing description of the human condition that I call "institutional conservatism." We humans create institutions, at least partly for good reasons, such as to protect information, to protect all of us from con artists who pretend to know, and to promote research that produces new knowledge – the revision of old knowledge. But the members of institutions organize and promote themselves, their subject matter and business or service to society, and their status in society as the authority to be trusted and respected. I WANTED TO SHOW THAT EVEN THE INSTITUTION OF MATHEMATICS CAN MAINTAIN AND DEFEND A DOCTRINE THAT IS WRONG. For centuries mathematicians searched for the solution and other mathematicians scoffed and berated them for not accepting that the correct answer to the riddle was "No." In 1882 a German mathematician (Ferdinand Lindemann) produced a proof that pi exactly could not be constructed and in his thesis he berated all mathematicians for wasting time on this problem. That proof has been accepted and propagated as sacred doctrine and led to the fixed belief throughout the world that it is impossible to "square the circle." There are awards offered for the solution of "famous" mathematical problems, but no award is offered for "squaring the circle" because the institution of mathematics has concluded, beyond their doubt, that there is no solution. But the solution is here. See and remember how the defense of institutional doctrines, the closing of doors, leads us away from the truth.

Today, June 2024, there are numerous articles posted online celebrating mathematicians of many nations and their various proofs that pi is either transcendental, or incommensurable, or irrational or all of the above. But here is the construction of pi exactly, squares and circles exactly equal in area, and a straight line with the numerical length of pi exactly. All of these doctrines have to be revised. Revision is the essence of science.

I Propose That Geometry and Trigonometry Comprise the

Universal Number System for All Intelligent Species

I believe that what we know as geometry and trigonometry is the universal number system, being what I would call "Natural Proportion" because it is not a symbolic or iconic human language but rather the description of the actual, real physical universe. To clarify: we use language to talk about mathematics and geometry, but a circle is a circle, a square a square, and a right triangle is a right triangle. None of these structures represents something other than itself. The same is true for all line length values and ratio values. This is obviously different from the word symbols of written and spoken languages. The word "chair" represents something that one sits upon, but we do not try to sit on the word "chair" written on a piece of paper. For all intelligent species, geometry and trigonometry – natural proportion – does not need to be translated. By the way, I do not believe the human species is an intelligent species. We are technological animals. We will reach the status of intelligent species when our technology causes no unintended consequences. AND …

the past limitations on the search for the solution to constructing squares and circles exactly equal in area rests squarely on the use of the digital decimal number system. Neither the decimal system nor the binary system guides a human mind toward the solution. In order to find the solution, one must use the Natural Proportion number system. This I did, and that is what is presented here.

THE TRUTH IS THE SAME NO MATTER WHENCE IT CAME.

Therefore, here is the inescapable situation for historians and mathematicians and scientists. You must choose A or B. You cannot have both.

A) As John Manimas Medeiros believes, the Pythagoreans of 500 BCE knew the complete positive solution to the riddle, and they therefore had to know the numerical values involved to a level of precision denoted at least to the tenth decimal place in the digital decimal number notation system.

B) No one in ancient times knew the positive solution to the riddle, and therefore, John M. Medeiros, an American who was born and raised in Connecticut, graduated from Brandeis University, lived most of his adult life (46 years) in Vermont, whose major employment was as a state social worker (25 years), and who worked on the riddle for 54 years and was the first person in all of human history to solve the riddle, at age 81. Thank you.

One last thank you. Thanks to all the "circle squarers" over the centuries. My work does not diminish yours. My success is not due to exceptional intelligence. My success in solving the riddle is due to the fact that I am a stubborn ass and I had an electronic calculator. If I had to perform my calculations by hand, as you did, the task would not have taken 54 years, it would have required 540 years.

Another Final Note

Construction of pi exactly and (Pi*MQ): 3.141640786 ( pi * 1.00015321)

The label or designation as "Pi*MQ" – again I use capitals out of respect, is that MQ is a multiplication factor, not an addition. The numerical value we will construct here is Pi times the value MQ (1.000015321). This factorial difference makes the line length of Pi*MQ very slightly longer than pi exactly. Pi*MQ is far closer to pi exactly than 22/7, which equals 3.142857143. Consider, for example, if we used a giant rope compass and constructed a circle (on a flat salt plain or hard desert plain) with a radius of 317.5 feet, a diameter of 635 feet or 7,620 inches, then the circumference using pi exactly would be 23,938.9 inches, and the circumference using Pi*MQ would be 23,939.3 inches, a difference of 0.4 inch or less than one-half inch. That is a high level of precision to measure, in the past, or today. A book I discovered in a used bookstore in Northampton, Massachusetts in 1971 (Tompkins, Peter. Secrets of the Great Pyramid. New York: Harper and Row, 1971) describes Tompkins theory that the ancient Egyptians understood the value of 3.141640786 as a very close approximation of pi exactly. I believe that the ancient Egyptians and Pythagoreans, and King Solomon's architect Hiram Abiff, all knew pi exactly, for which theory I present the evidence here. The evidence being that we can square the circle and the Pythagoreans would not have posed the question as a means for their students to learn unless they knew it could be done, and it is very important to the meaning of number and geometry and science that it can be done. It is evidence in support of the proposition that "Proportion is Everything." I believe the declaration that proportion is everything is as important as Hawking's theory of a Big Bang.

Definition of MQ:

MQ = 1.000015321, and MQ * 2 = MQ^2 + 1

Note that is different from Phi + 1 = Phi^2 (actually kind of the opposite)

Further: (MQ + 1) / 2 = sqrt(MQ)

and MQ + 1/MQ = 2 (1.000015321 + 0.999984679)

There is no other number that fits these definitions of MQ.

Let us add:

MQ = [Phi^2 * (6/5)] /pi

There is no other number that fits this definition of MQ.

… further note on history – for Ancient Alien Theorists:

Here's the thing: Why this riddle? [Can we construct a circle exactly equal in area to a given square using only the compass and straightedge?] What is the purpose of this riddle? Is it just a challenge to geometry students? Is it a test for advanced geometers?

We could respond "Yes" to these questions, but I believe the true purpose of the riddle is to serve as a pointer to an extremely important message about human history and how the universe works. In particular, what number is and what "proportion" is. The profound scientific truth that the riddle points to – IF AND WHEN ONE SOLVES IT (one has) is a secret hidden in plain sight: Proportion is Everything. What this statement means is that proportion is what exists in Nature, and is – or appears to be – the primal force that gives unorganized neutral particles their character, and thereafter all of the material realities that we discover in the real, physical universe. In Nature there is proportion and only proportion, and what we know as "number" is the phenomenon of an intelligent brain measuring proportions. We see math everywhere, and we see number everywhere, but only because proportion is everywhere. We see proportions as ratios, or fractions, as in a ratio-nal number. But all numbers are ratios, including all whole numbers. Part two of what the riddle is pointing to is that this was known on Earth in the past. AND, the precise measurement of pi exactly (to at least the ninth decimal place) was known in the past on Earth. The pointer here points to "Atlantis " or an advanced civilization on Earth in ancient times at least 12,000 years ago, and probably more. BUT WHAT IS MORE INTERESTING STILL is the nature of the way in which the riddle points to high precision being known to ancient civilization: this is an ancient artifact that cannot be refuted by verbal argument. Old devices, old maps, bones and pottery found buried, ancient artwork made of exotic medals, fused sand, amazing monumental structures, can all be beaten down with words of doubt and contrived explanations. But knowing the solution to the Pythagorean riddle is an "artifact" that cannot be beaten down with words. The riddle points to the "arti" fact that ancient peoples knew that we can construct pi exactly as a straight line and there is no argument that offers a meaningful alternative explanation. They knew, and now we know, that proportion is everything, and we now know that even advanced civilizations can be set back thousands of years by mistakes, by bad behavior, or by natural forces. If history is like an arrow, it does not simply fly forward. It can be pushed backward, or broken. And that is not only the message of the Pythagorean riddle. It is also the message in the parables attributed to Jesus Christ: The Primacy of Stewardship, by John Manimas, 2009 If we do not take proper care of the people and property entrusted to us (life on Earth), we could lose everything. The essence of this message sent to us from the ancients is NOT simply be nice, or be kind, or be good or be gentle. It is all of these, but it is above all BE RESPONSIBLE. When civilized people are not responsible, they risk all. All could be lost. This is the way the universe works. It is not about space travel or portals or endless wars and light against darkness. The essence of life in the universe is WHO ACCEPTS RESPONSIBILITY. These are pointers sent to us by responsible men and women from the past, like our grandparents. Be responsible. Do not underestimate the power of Nature to punish without mercy. Nature has no regrets, only survivors.