The Real Treasure Hunt
Welcome to Aquarius, Volume 15 (January 7, 2008) Back to (Welcome)
History as the Real Treasure of the Great Pyramid
The Construction of Pi About 1,000 BCE
This is the Autobiography of my "treasure hunt." In this world, there are many ways to start a journey and stay on that journey until it is over. I can honestly say that I searched for and found Atlantis without ever leaving my home. The treasure I found is the answer to a question about human history, the real treasure of the Great Pyramid. Would your view of human history be changed if you knew for a fact that the ancient Pythagoreans knew the value of Pi to the ninth decimal place around 1,000 BCE and they also knew how to construct a line equal to the length of Pi to the ninth decimal place? A journey of a thousand miles begins with a single step. Here is my journey in seven steps.
1) The first "simple" thought (1957-1971):
There was, and is, a reasonable positive solution to the ancient riddle: "Can we construct a square with the same area as a given circle using only the compass and straightedge?" I first believed this in 1957 when I was in my high school freshman geometry class. The teacher was Leonard Launer. He asked the question. He paused. My brain had an inner voice that spoke to myself and said "There must be a positive response. People as smart as the Pythagoreans would not ask such a question if the answer were simply 'No'." Then Mr. Launer said that the answer was "No," we cannot "square the circle." He was only sharing an accepted mathematical doctrine. Later I would learn that this doctrine was established by the work of Ferdinand Lindemann in 1882. I also would learn that this is possibly the most widely known concept of mathematics (and science) in the world. Everyone, including both children and adults who barely understand the meaning of trigonometry are certain that "we cannot square the circle." They state this scientific dogma just as people who do not know how to wire a light fixture will tell you that E equals MC squared.
Carrying the usual burdens of finding one's way in this world, I had "set aside" my viewpoint that there must be a positive solution. I moved on to the usual: getting a college education, developing a career, learning how to love and be loved, how to be a member of a group of trusted friends, how to manage money in a world that throws credit at you like it is free bread. I have found many friends over time. However, as is true for many people, the friendships of my childhood often seemed to be beyond duplication. Throughout my adult life, I could not allow myself to stop learning. I had to have books and the time to read them. While focusing on human history, politics and psychology, and learning more about carpentry and landscaping and house maintenance and repairs, through my middle years I found my way toward the profession of social work. Got married, had children, got divorced, got remarried. But the Great Pyramid waited for me. My original motivation was never lost. My brain, however occupied with the challenge of a "normal life," wanted to know the positive solution to the ancient riddle.
2) The riddle returns, and generates a more complex thought (1971-1976):
After being held captive by the issues of the 1960's, I got back to something like ordinary life. I read Secrets of the Great Pyramid, by Peter Tompkins. The fire flamed again. There was fresh fuel for that fire. During the 1960's, while struggling with a college education, graduating and getting a job, and with whether I will become a soldier on the battlefield in Vietnam or a peacenik on the streets at home, I also read several books about Unidentified Flying Objects. My UFO reading included the abduction and medical exam story of Betty and Barney Hill in New Hampshire, and Chariots of the Gods, by Erik von Daniken. The UFO literature strengthened another childhood perception of mine. On a clear summer night, I lay down on the ground and looked up at the sky. I saw the great numbers of stars. I had read that probably half of the stars are not stars but galaxies. I was aware that we humans study plants and animals, everything from insects to elephants. We watch them. We try to protect them when they are endangered by loss of habitat and other environmental changes. I have always seen Nature as possessing a logical unity. Life must exist in other galaxies. There must be other beings more intelligent than we are. If they exist, they must be interested in us. The UFO literature represents the human struggle, involving both hope and terror, with the concept that we must share the universe with others who are superior to ourselves. Whether any individual UFO story is factually true is not the real issue. If hierarchy is the way of Nature, there is probably a hierarchy of intelligent beings, and we may not be high on that ladder. We may not even be on it. This thinking about the human identity was fed also by my ongoing studies of history, sociology, the science of learning, world religions and the development of the human mind -- and brain development also if the mind and the brain are not the same thing. Therefore, my old simple thought about the ancient riddle became more complex. The ancients knew more than historians acknowledge. Human history is not a straight arrow. There was a more complex thought taking shape in my brain, slowly, subconsciously. That thought was that the positive solution to the ancient riddle is not trivial, not just a trick of geometry. It has some greater significance. The Secrets of the Great Pyramid describes some of the clues left behind by the ancients. Reviewing the book, which is an extensive study of the history of "pyramidology," the clues include the evidence that the Great Pyramid incorporates the values of Pi (3.141592653...) and Phi (1.618033988...). Phi is the Golden Mean. There is some significance to the value of (4/5) [0.8] and the inverse (5/4) [1.25] and (6/5) [1.2] and (5/6) [0.8333333333...]. There is something of a dispute as to whether the Great Pyramid incorporates only Pi, or Phi, or both. The word "pharaoh" means beacon, or beacon of light, a pointer that points the way to safety, or to the truth. In ancient times a fire burning at the top of a stone tower functioned as a lighthouse, or "pharaoh." That beacon of fire burned every night in a busy harbor. Although I needed to go on with my ordinary life, I could not let go of this issue. I felt as though I possessed a treasure map and I had a duty to search for the treasure. I had to search alone because no one could be trusted. Regardless of what religion a person might claim, virtually everyone talks as though the authority of Mathematics is beyond question. Anyone who says that the circle can be squared, or even maybe the circle can be squared, is brutally ridiculed and labeled as a fruitcake. Possibly someone does not want us to know that Atlantis really existed, that advanced humans lived on Earth in the past. This truth would change the content of human history and every world religion. We would have to review all knowledge that enters into our concept of who and what we are. The ancient riddle therefore is not only significant rather than trivial; it may be explosive in its effect. It may be the most important question that human beings can answer. This is why my brain, while fixing broken toilets and mowing lawns and repairing old houses, and while planting garden after garden and then moving on, constantly carried my secret treasure map. I would persistently return to the search for the historical treasure, until I found it or died. I looked at the map. I saw that I had to review geometry and mathematics. I began that review in 1977, when twenty years had elapsed in the process of remembering a day in geometry class and elevating it to the mission of my life.
My goal was to show that there is a correct or proper positive answer to the original riddle, and the correct answer does not necessarily result in the construction of "infinite" Pi, because infinite Pi is the result of an infinite summation, and a real physical object cannot have an infinite number of sides. Pi is calculated many ways, I know, but my viewpoint is that if Pi is to exist in the real world, the reality that it can be calculated as the perimeter of a regular polygon with a finite number of sides -- each side having the length of the sine or tangent -- determines the variable and finite and physical Pi that is in control where the size of a two-dimensional or three-dimensional object is involved.
3) The geometry and math is the path to the historical treasure (1977-1990):
Since Hippocrates and others made efforts to "square the circle" around 500 BCE, the original riddle must have been presented around 500 BCE or earlier. It is not a new riddle. It was deemed a real challenge in GEOMETRY (not mathematics) a long time ago, long before the alleged invention of algebra by the Arabs. The original riddle did not intend, necessarily, that the solution would result in infinite Pi (Pi with an endless number of decimal digits) but in a value for Pi that was an excellent working value. They were not looking for (22/7) or (3.142). They (the Pythagoreans) were looking for something more precise, and the purpose of the riddle was to challenge the STUDENT OF GEOMETRY OR PYTHAGOREAN CANDIDATE to engage in advanced constructions and to discover something truly significant about the relationship between Pi and Phi^2 (1.618033988...)^2 or 2.618033988... .
Hidden within my ordinary life, I reviewed mathematics from 1977 to 1990, while continuing my own ongoing studies of world religions, history, and the psychological science around learning, memory and brain development. In 1977 I became an employee in human services for the State of Vermont. In 1989 I became a state social worker. The training provided for professional development included workshops and conferences of 24 to 30 hours each year, and these stimulating sessions kept us up with the latest research in family dynamics, individual personality disorders, addictive behaviors, and child brain development. All of this professional training contributed greatly to my own personal viewpoint as to who and what we are, the possible reality that the "pharaoh" might be pointing to, the reality of the human identity which is of cosmic significance. It might be known to some in positions of leadership. They might keep it a secret on the grounds that the general public cannot handle the truth. In 1938, when Orson Welles produced a radio program that persuaded millions of Americans in the Northeast that Martians had landed on Earth, there was a panic, and people lost their will to be compliant and cooperative. People left their homes and showed signs of regressing to life in the jungle, every man for himself. Law and order fell away. Evidence that superior beings have lived on Earth in the past might even be considered to be a threat to the general welfare. But if it is the truth, it is a truth desired by many. I desired this truth and continued my search from 1977 through the early 1990's by reviewing mathematics and learning how to write simple mathematical computer programs, first in the BASIC language, then in QBasic, then C and C++.
Phi was the center of the ancient geometer's universe, the number that when squared equals itself plus 1, like biological reproduction. It is reasonable to conclude there is a connection among several ancient clues. There is a trail of clues. First, there is the riddle itself. Then, there is the construction of the pentagon and pentangle. There is the tangent=2 right triangle (the first step in construction of the pentagon), and the laying down of the hypotenuse, which demonstrate a productive construction process and the reality that the tangent=2 right triangle enables multiplication of any line length (or ratio) by 5. I concluded that the Great Pyramid is a Phi pyramid. The height on a face of the Great Pyramid is equal to Phi/2 or the cosine of 36 degrees. I saw the clues now familiar to many, the clues of the pentangle being incorporated into a common symbol used by the American military, and the eye at the top of the Great Pyramid on the dollar bill, the historical documentation that the Freemasons played a major role in the creation of the United States of America, the evidence that there is probably some connection between the Freemasons and the Knights Templar, hinted by such researchers as Michael Baigent, Richard Leigh and Henry Lincoln, who wrote Holy Blood, Holy Grail, published in 1983. My search through my treasure map was never discouraging enough to cause me to quit, always encouraging enough to keep me going.
4) The search keeps me motivated (1990-2000):
My records indicate dates for the development of different strategies. I labeled the folders in my computer as numbered strategies and there were thirty-six AStrat folders at the end of my work. Here I am summarizing from memory. In 1990 I read Serpent in the Sky by John Anthony West (published in 1987). West was timely for me. By this time I had read several of the "classic" works in "pyramidology" including Schwaller de Lubicz. West pushed me over the edge with his narrative argument that the ancient Egyptians employed a binary system, in their brains, to multiply numbers. The binary system is deemed by most scientists today to be the "number system" of Nature. Binary phenomena are found in genetics and in electronics. This reality in itself has persuaded most scientists that this is IT. This is how Nature works. In 1990 a theory formed in my mind that the binary system is very important, but it is not the fundamental "number system" of Nature. Because of another book I read long ago, The Science of Measurement: a historical survey, by Herbert A. Klein (1974), my thinking is guided by a concept of cosmic importance, the concept that Nature has no absolute values, the concept that all measurements are comparisons. Therefore, the fundamental "number system" of Nature is not a number system as conceived by the human brain, but is Proportion only, the comparison of one measurement to another.
In 1990, all of my studies and thinking met up with the ancient statement that "Proportion is everything," also known as "All is number." I concluded at that time that I had to follow my treasure map to the end. I must never quit because the stakes were so high. The issue was not just whether we can "square the little circle." The matter at issue here is whether or not Proportion is the number system of Nature, and the binary number system is only one manifestation of Proportion. With this conceptual issue overwhelming me, I could not give up, no matter how lonely, or how ridiculed, I might be. I was willing to fail 10,000 times in order to be right once and find the treasure.
Proportion, and "Proportion is everything," is ancient, but also new. The statement "Proportion is everything," is just another way of summing up all of the concepts that support the modern studies of "Chaos Theory" and fractal geometry. In other words, if the ancients understood that "Proportion is everything," then they understood that proportion, or fractal geometry, is the origin of all of the shapes and forms produced by Nature. This may mean what I believe it means. This may mean that there are no Euclidean circles or spheres in the real, physical world, only polygons and polyhedrons. This may mean, and I personally do believe it means, that the atom is a polyhedron. This is the issue my treasure map is dealing with, not only "can we square the little circle," but is Proportion the fundamental number system of Nature, and is the atom a polyhedron, and do Chaos Theory, fractal geometry, and "Proportion is everything" all point the way to a clear understanding of how physical matter is self-organizing -- the Secret of Life in the Universe. This is what is at stake here. This is why I persisted for 30 years, or 50 years, against a doctrine of mathematics that was believed by everyone in the world. If Proportion is the beacon of light that points to the self-organization of matter, then it is the pharaoh that shows us the way to understanding who and what we are, how Nature really works, and how we can be good stewards of the natural environment that supports life on Earth. So, although I know that these thoughts will cause others to accuse me of being grandiose and foolish, I don't care. This is what I have seen on my treasure map. A treasure map that I had to follow to the end.
My work led to my discovery of "sequential halving" and how we can have a "super-binary" number system based on the square-root-of-two rather than on two itself. I wrote computer programs that illustrate how if we double and halve numbers that are powers of the square root of two, in the right precise sequence, we can "construct" any numerical value by summation, which is essentially only the same method we use in our decimal system or binary system. The special quality of "sequential halving," however, which also naturally involves "sequential doubling," is that it can be accomplished with a compass and straightedge. In other words, we can actually construct any numerical value, including any fractional value, using only the square-root-of-two (the diagonal of a square) as the number base [instead of 10 or 2] and the compass and straightedge, and the process of "sequential halving." This is described in The Pyramid Mark (Two) on my website, which I originally posted in December, 2000, when I first established my website (jmanimas.com). It will be more or less obvious to the reader that sequential halving appears to illuminate a biological and genetic process. That is, sequential halving looks a lot like a fertilized cell dividing (haploid) and doubling (diploid) upon conception. This conceptualization was very powerful in persuading me that I should stick with my treasure map and the "beacon" of the Great Pyramid and never give up.
5) The treasure is found (2000-2007): (Phi = 1.618033988...) (phi = 0.618033988)
From 2000 through 2006 I was stuck on the value of (Pi*Ho) 3.144605511... . Pi times "Ho" is Pi (3.141592653...) times (1.000959022...) and is based on the fact that the square root of phi (0.618033988... or twice the sine of 18 degrees) equals 0.786151377..., and 4 times that equals 3.144605511... (Pi*Ho). I spent a few years trying to find a way to get from (Pi*Ho) to construction of Pi, or from Pi to construction of 1, but did not find such a destination. The riddle is solved if we can construct both the value of Pi and the value 1. I call the inverse of the "Ho" value "Gh" (0.999041896...), meaning "Ghost." This value, 0.999041896... is equal to Pi times the square root of Phi, divided by 4. My "possession" by Gh and Ho was based on a rather simple reality that also shows why there is confusion over what are the values incorporated in the Great Pyramid. If we say that the height of the Great Pyramid is 1, it looks like the length of the side of the base is Pi/2 or 1.570796327... . Obviously, if these measurements were precisely correct, the proportion of the base to the height is Pi/2. This would mean that the builders of the Great Pyramid (around 2,000 BCE, and some say 8,000 BCE) knew the value of Pi to a very precise level. However, there are other possibilities. In terms of proportion it could be that the base has a length of 1 and the height is 2/Pi or 0.636619772..., which yields the same result in the proportion of the base to the height (Pi/2). Further, the square root of Phi (1.618033988...) is 1.27201965..., and that divided by 2 equals 0.636009824... which anyone can see is very close in value to 0.636619772... . This means that it could be that the base of the Pyramid is 1, and the height on the face of the Pyramid is Phi/2 (0.809016994...) or the cosine of 36 degrees. (The right triangle with angles of 36 degrees and 54 degrees is found when we construct a pentagon.) And with these two dimensions, the vertical height of the Great Pyramid would be the square root of Phi/2 or 0.636009824... . One can see this by constructing a right triangle and applying the Pythagorean Theorem. You have to imagine the right triangle that has the height on the face as its hypotenuse, being 0.809016994... . The base of that right triangle would be one-half of the base of the Pyramid, or 0.5. Therefore, the hypotenuse squared equals 0.654508497..., and that value minus (0.5) squared, or minus 0.25, equals 0.404508497..., and that value equals the height squared. Therefore, the height equals the sqrt[0.404508497...] which equals 0.636009824..., the square root of Phi, divided by 2. For this reason, and because of a measurement made by Charles Piazzi Smyth, described by Peter Tompkins in Secrets of the Great Pyramid, I believe that the Great Pyramid is a Phi Pyramid. And I suspect that the unique qualities of a truly Phi proportioned pyramid may cause such a pyramid to have interesting qualities with regard to the capture and focusing and radiation of energy. This I believe is the possibly scientifically valid cause behind the "mystical" belief that a pyramid shape can cause dead animals to mummify (rather than decay) or keep a razor blade sharp. There may be some truth to these ideas, but possibly only if the pyramid shape is precisely proportioned as a Phi pyramid. This is one example of the many convergent clues that kept me going.
In 2006 I knew I had to shake myself lose from Ho and Gh and try to find a way that we can begin with a line length of 1 and then construct a value that was equal to Pi exactly, or very close. I also had developed further my conviction that infinite Pi does not exist as the entire dimension of any object in the real physical universe. That means that Pi, to me, is variable in the real world, depending on the number of sides of any object, idealized as a regular polygon. In 2006 and 2007 I reviewed my work and looked for new paths. I re-discovered the value that I labeled as "MQ" 1.000015321..., and the fact that Phi^2 = Pi * (5/6) * (MQ). This reality includes the clues of Phi and (5/6). Therefore, I got very interested and explored the meaning of MQ, and asked myself whether this value of MQ could be constructed.
Early in 2007 I became convinced that MQ was the value that brought me to the door of the treasure. I searched for ways that this value could be constructed, because I saw that if we could construct MQ, we could construct Pi. Early in 2007 I discovered that the clue regarding (1.2) was a misunderstanding for all the long centuries. The true clue and value used in the solution is not 1.2 but rather 1.222222222... or (11/9). This value of (11/9) and the inverse (9/11) play a major role in the solution, which includes the central role of the secant of 18 degrees raised to the 4th power, 1.222291236... (my "SF" solution).
My SF solution complies with the original limitation of using only the compass and straightedge, and results in construction of Pi to the 9th decimal place (or greater, subject to evaluation by a high precision computer) which is of course 3.141592653... . The summation method for calculating or constructing Pi is to construct a regular polygon of 8 sides and then imagine what happens if we could multiply the sides by 2 indefinitely, to 16, 32, 64, and so on. The perimeter of this regular polygon then becomes the "circumference" of the potential circle that we approach as we continue to double the sides. The perimeter is the sine, or tangent, of the division of the circle (360 degrees, also known to scientists as two radians or 2 times Pi). For example, for a regular polygon of 128 sides, the angle of each "pie slice" of 1/128th of 360 degrees is 2.8125 degrees. The sine of 2.8125 degrees is 0.049067674..., and that times 64 equals 3.140331157..., and this value "approaches" Pi but is still some distance away. Each time we double the sides of our regular polygon, we "improve" our perimeter value in its approach to Pi. When we use this summation method to calculate Pi, we do not get to Pi to the 9th decimal place until we have "constructed" a regular polygon of 262,144 sides. Therefore, because this construction is possible, using only the compass and straightedge, my position is that this is the correct positive solution to the riddle, and it is correct from the historical perspective in that it is a very precise and practical value for Pi, it enables construction of squares and circles of the same area to a very high level of precision, and it is accomplished without using any of the mathematics that was developed AFTER 500 BCE ! Pi to the 9th decimal place is Pi to the billionths of a unit. This level of precision was accepted for most if not all technological purposes throughout the twentieth century. It appears that we are expected to apply even greater precision as we begin the high technological work of the twenty-first century.
How could those who asked this riddle at 500 BCE or earlier have known these numbers to this level of precision? This is the significance of what I have done. My position is that construction of a line length of infinite Pi is probably not a meaningful goal in the first place, because Pi is not like the diagonal of a square or the trigonometric functions. It is not just a line length. It is the entire dimension of a two-dimensional shape. It has to be finite. Since that dimension is a perimeter (or circumference) and it has to be finite, then the calculation of infinite Pi, though very interesting, CANNOT BE THE DIMENSION OF A REAL OBJECT REGARDLESS OF WHETHER WE DEEM IT TO BE A CIRCLE OR A REGULAR POLYGON WITH 262,144 SIDES. If a small regular polygon has 262,144 sides, and its "diameter" (vertex to vertex) is five units, what power of microscope do we need to actually see a vertex? And, if we were to construct a large square equal exactly in area to the large "circle" or regular polygon of 262,144 sides, it looks to me like the difference or deviation from perfect equality would be close to the inverse of 262,144 which is equal to 0.0000038147... . My hand-held calculator registers a deviation that is far smaller than that. My work is intended to change history, and the history of mathematics, but not necessarily to change mathematics, which cannot be changed. Infinite Pi cannot be constructed as the entire dimension of a real object, and it may be that it cannot be constructed by any means, but that is not the issue I have addressed.
The issue I have addressed is: Was there a solution that could be reasonably deemed to be THE CORRECT POSITIVE SOLUTION SOUGHT BY PYTHAGOREAN TEACHERS WHO KNEW THE SOLUTION THEMSELVES AND WANTED THEIR STUDENTS TO DISCOVER IT BY THEIR OWN EFFORTS? My proposition is: YES, THE SF SOLUTION WAS, AND STILL IS, THE CORRECT SOLUTION OF THE RIDDLE, BECAUSE IT USES GEOMETRY ONLY, AND THE LEVEL OF PRECISION TO THE 9TH DECIMAL PLACE IS SATISFACTORY FOR ALL PRACTICAL PURPOSES. The construction steps are described in detail, with drawings, in Values and Constructions Required, which is section two of Squaring the Circle Exactly. A summary of the mathematical facts is presented here. Mathematical facts can be neither patented nor copyrighted. Once you understand this, the treasure is yours. The treasure of history belongs to everyone.
6) The geometry and the core concepts behind the constructions:
Important values and constructions to know (using degrees as angular measure),
(ratios can be converted to line lengths and line lengths can be converted to ratios):
Cosine 36 = 0.809016994374947424102293417182819... times 2 = Phi
Phi = 1.61803398874989484820458683436564... and inverse is
phi = 0.618033988749894848204586834365638... or "little phi" or 1/Phi
and = (Sine 18) * 2, Sine 18 = 0.309016994374947424102293417182819...
also, (Sine 18) = (Cosine 36) minus 0.5
(phi)^2 = 0.381966011250105151795413165634362...
phi + (phi)^2 = 1 (known to all mathematicians and geometers)
(Phi)^2 = 2.61803398874989484820458683436564... = (1.618033988...)^2
and (Phi)^2 / 2 = (sine 18) + 1 = 1.30901699437494742410229341718282...
and note that (sine 18) + 0.5 = 0.809016994374947424102293417182819...
Also, Phi is the limit of the Fibonacci series: 1+1, 2+1, 3+2, 5+3, 8+5, 13+8, and so on, where the last sum divided by the previous sum = 1.618033988... as the sums grow larger. Previous sum divided by last is inverse (0.618033988...).
Begin with phi = 0.618033988749894848204586834365638...
(Pi*Ho) = 3.14460551102969314427823434337184... = 4 times sqrt(phi)
sqd = 9.88854381999831757127338934985021... = 16 * phi
(Pi*Ho)/ 4 = sqrt(phi) = 0.786151377757423286069558585842959...
FX = (5/6)=0.833333333333333333333333333333333...
SX = (6/5)=1.2
Phi^2=2.61803398874989484820458683436564... (also = Phi +1)
divided by Pi =0.833346100984274536545032628319445...
which = (5/6) * MQ, where MQ = 1.00001532118112944385403915398333...
and sqrt(MQ) = M= 1.00000766056122262280424348031297...
or M = 1.000007661...
QM = inverse of MQ = 0.999984679053605550929814794430672...
which = Pi * VP, or Pi * 0.318305009375087626496177638028635...
and VP = (5/6) * (phi)^2 or (5/6) * 0.381966011250105151795413165634362...,
therefore VP = 0.318305009375087626496177638028635...
and (phi)^2 = (0.618033988749894848204586834365638...)^2
= 1/ 2.61803398874989484820458683436564... = (phi)^2 because I use upper case and
lower case to distinguish phi [0.618033988...] from Phi [1.618033988...]
Note that a hand-held scientific calculator may show Phi as 1.618033989...
Variations in the decimal value of MQ (after the ninth digit):
Later, we will see that our construction of MQ will yield a slightly different value that has
different decimal digits following the 9th decimal digit, due to the limitations of the precision of the math processor in a desktop computer. My desktop computer writes the constructed line MQ as 1.00001532123408073944164930824006... . The first version of MQ, above: 1.00001532118112944385403915398333..., is the result of using the computer calculator to obtain Phi^2, and then dividing by Pi, and by (5/6). The difference in decimal digits following the ninth decimal digit is due to the limited precision of the calculator. Limited, but still good to the ninth place (billionths).
PF=0.55 (label for this value derived from "Point Five")
(label for this value derived from "Eighteen Series")
*1.2 = 2.18181818181818181818181818181818...
*1.2 = 2.61818181818181818181818181818181...
*1.2 = 3.14181818181818181818181818181818...
*1.2 = 3.77018181818181818181818181818181...
*1.2 = 4.52421818181818181818181818181818...
*1.2 = 5.42906181818181818181818181818181...
*1.2 = 6.51487418181818181818181818181818...
And, 2.618181818... = 432/165, and 3.618181818... = 864/275
Notice that 55 and 89 are the 9th and 10th numbers or "terms" in the Fibonacci series.
And 89/55 = 1.618181818... . Experiment with 55, 5.5, and 0.55 and the number
values in the "Eighteen Series."
LN= (11/9) = 1.22222222222222222222222222222222... (eLeven Nines)
NL = 0.818181818181818181818181818181818... (Nine eLevenths)
* 1.2 = 0.981818181818181818181818181818182...
SF = (secant 18)^4 = 1.22229123600033648574532213002996...
(label for this value derived from Secant 18 to Fourth power)
inverse of SF = FS = 0.818135621484342140063933385739262...
sqrt of that = (cosine 18)^2 = 0.90450849718747371205114670859141...
and that = (Phi/4) + 0.5 [0.5 + 0.40450849718747371205114670859141...]
TN= SF/LN = 1.00005646581845712470071810638815...
NT, inverse of TN = 0.999943537369751504522585249236876...
(labels "TN" and "NT" from mind of author)
The value 0.2222 (stopped there at 4th decimal place)
The value 0.7778, = 1 - 0.2222
Also, 1.222 and 8.778 and 0.1222 and 0.8778
John Manimas, December 27, 2007
Historians and mathematicians are invited to reconsider the ancient riddle.
Five minute statement of the MQ and squaring-the-circle exactly proposition:
1) Squares: | [_] --> from line to square, originally geometry, not mathematics:
1*1 = 1 --->--> 2*2 = 4 --->--> 3*3 = 9 --->--> 4*4 = 16 --->--> 5^2 = 25
Phi = 1.618033988... = Phi (big Phi) --->phi (little phi) = 0.618033988...
Phi^2 = Phi + 1 = 2.618033988... ---> phi^2 = 0.381966011... = 1/(Phi^2)
and note also that [phi + phi^2] = 1, or 0.618033988... + 0.381966011... = 1
2) Fibonacci series: 1, --->--> 1 + 1 = (2) --->--> 2 + 1 = (3) --->--> 3 + 2 = (5)
5 + 3 = (8) --->--> 8 + 5 = (13) --->--> 13 + 8 = (21) --->--> 21 + 13 = (34)
--->--> 34 + 21 = (55) --->--> 55 + 34 = (89) --->--> 89 + 55 = (144)
--->--> 144 + 89 = (233) = ---> 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... --->
and 55/34 = 1.617647059... ---> 89/55 = 1.618181818... --->144/89 = 1.617977528...
233/144 = 1.618055556..., et cetera ---> Phi = 1.618033988...
and 144/233 = 0.618025751..., et cetera ---> phi = 0.618033988...
3) 0.618033988... + 0.381966011... = 1 and Phi^2 + (1/Phi^2) = 3 and the square root of [sqrt(3), plus 2] = 1.931851563... [= X] and (1.931851563...)^2 = 3.732050808..., and 1/(1.931851563...)^2 = 0.267949192... and this X^2 + (1/X^2) = 4 [because 1/X = 0.51763809...]. However, there is no X such that X^2 +(1/X^2) = 1, and the X that complies with X^2 + (1/X^2) = 2, is 1. The next step is based on Phi^2 = [Pi * (5/6) * MQ], and MQ or a value very close to MQ can be constructed. [Pi * (5/6) * (phi)^2] = 1/MQ, which = 0.999984679..., and MQ = 1.000015321..., therefore, M = sqrt(MQ) = 1.000007661...
4) Can we calculate and construct the value of MQ? Yes. There is Pi-A and Pi-B.
Pi-A is Pi calculated with electronic binary technology (calculator, computer)
Pi-B is constructed as follows: (A) SF = (secant 18)^4 or secant of 18 degrees to fourth power, = 1.222291236... [The secant is H/B, inverse of the cosine B/H] (B) 11/9 = 1.222222222... and 9/11 = 0.818181818..., through a procedure using SF and 9/11, we get value 1.222215321...
and then 1.222215321... - 0.2222 = 1.000015321..., we have our MQ value and then Phi^2 = [Pi * (5/6) * MQ] and therefore Phi^2 / MQ = Pi * (5/6) or 2.618033988.../ MQ = Pi * (5/6), and [Pi * (5/6)] * 1.2 --->--> or [Pi * (5/6)] * (6/5) = Pi-B, and Pi-B is the solution to the original riddle because Pi-B is a proper value, to the 9th decimal place, for physical Pi.
The scientific experiment:
Think of my work as a scientific experiment that can be repeated by others to see if they get the same results. Here is the experiment. On my calculator I have saved six number values: A, B, C, D, E, F, as follows:
A = 3.141640787... B = 1.222215321... C = 1.222291236... (secant 18)^4
-------------------- --> D = 1.000015321... E = 0.999937891... F = 0.999943537...
These values are obtained by the following calculations, which I claim can all be performed by construction. By "performed by construction" I mean that all of the values I present here can be constructed as line lengths and ratios using only the compass and straightedge, and those constructions are described in detail in Values and Constructions Required (62 pages):
A = 3.141640787... is Pi * MQ. Enter the square root of 0.8 (4/5) on your calculator. Find the arctan of that value, that is, arctan of 0.894427191... . The result is, that when the tangent of a right triangle is 0.894427191..., the angle of interest is 41.81031489... . The cosine of that same right triangle is 0.745355992... . The secant of that same right triangle is the inverse of the cosine, which is 1.341640787... . Add the secant squared value (1.8) and the result is the value of A = 3.141640787... (Pi * MQ). If you divide 3.141640787... by Pi, the result is MQ, the value which is saved as the value of D = 1.000015321... .
There are other ways to obtain the value of Pi*MQ as the result of a trigonometric function. For example, Phi^2 (2.618033988...) * square root of 5 (2.236067978...) = 5.854101966..., add 2 and the result is 7.854101966..., divide by 10 and multiply by 4, and the result is 3.141640787... .
B = 1.222215321... is equal to MQ (or D) plus 0.2222. Therefore, when we subtract 0.2222 from B, the result is MQ (D). How do we get B? B is the product of C * E. How do we get E?
C = 1.222291236... is the secant of 18 degrees raised to the fourth power, (secant 18)^4. There are many ways to construct this line length and ratio value.
D = 1.000015321... is MQ, equal to M^2. M = 1.000007661... . These two values comply with a specific precise definition. M^2 plus (1/M^2) = 2 exactly. This is the same as MQ plus (1/MQ) = 2 exactly. MQ is the value that is the end result of our experimental constructions, because if we can in fact construct MQ exactly, then we can construct Pi exactly and then 1/sqrt(Pi) exactly.
The precisely defined value of MQ [equals M^2] is M such that M is not equal to 1 and M^2 plus (1/M^2) equals 2 exactly. M therefore is kin to Phi, because Phi^2 plus (1/Phi2) = 3 exactly.
E = 0.999937891... and F = 0.999943537... are values that we construct employing the value of C = 1.222291236... (secant 18)^4.
F = 0.999943537... is calculated (or constructed) as follows: C * (9/11) = 1.000056466... and the inverse of this value is F = 0.999943537... . Study of this calculation will show that there are other ways to get the same result 0.999943537... .
E = 0.999937891... . There is more than one way to obtain this value. However, the most "enlightening" or "visible" method is as follows: to 0.999943537... (F) add 9. The result is 9.999943537... . Divide by 10. The result is 0.9999943537... . (In this new value we have 5 nines following the decimal point, not 4.) Next, multiply 0.9999945373... * 0.999943537... (F) and the result is E = 999937891... .
That is the experimental procedure to construct the value MQ. It can be done with an ordinary hand held scientific calculator by saving the number values A-F shown here. It can be done on a high precision computer in the same steps, of course.
7) Is "squaring-the-circle" proof of Atlantis (2008)?
The problem for historians is more or less obvious. Who originally discovered how to construct Pi to the 9th decimal place using only a compass and straightedge? Was in John Manimas in July of 2007, or was it the Pythagoreans before 500 BCE? If it was understood by the Pythagoreans, how could they know values to this high level of precision, a level of precision that could be applied by our "modern" civilization only after the rise of science and the development of precise technologies in the nineteenth and twentieth centuries?
There are many who have argued that the ancients knew far more than we have previously believed. There is a magazine called Atlantis Rising, a book called Forbidden History, and there are programs that have been aired on the Discovery Channel that describe truly impressive technological accomplishments in ancient times, including the construction of weapons, bridges, machines and even forms of industrial production. Why does the human species appear to have developed with dramatic unevenness? Why have there always been humans running around the jungle nearly naked shooting monkeys out of trees with a blow gun while other humans are building rockets of one kind or another, and castles, and machines, and printing books and communicating with radio waves? What happened that makes it look like we have some kind of problem, a problem with sexual drives, with violence, with religion, with our place on Earth and in the universe? Why are we called so clearly to be the caretakers of life but continue to be the destroyers of life? What is the truth about who and what we are? If the ancients possessed knowledge of Pi to the 9th decimal place, and the value of MQ, then I have not just squared the little circle. I have knocked on the door of the cosmos, and it appears to be inclined to open.
The fact that we can square the circle by constructing a straight line length of Pi to the 9th decimal place, using only the compass and straightedge, is a new piece of evidence in the controversy as to whether the story of Atlantis is true, and a civilization with advanced knowledge existed on Earth in the distant past before the first page of recorded history. The constructions that I claim to have RE-discovered, and not discovered for the first time on Earth, are evidence that the Pythagoreans presented the original riddle because they knew there was a positive response, just like a correct answer on a modern quiz or test. The correct response to this riddle never was and is not now a simple "No, it cannot be done."
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