The Euclidean Circle is Not Real
Welcome To Aquarius, Volume 2 (February 12, 2006)
Welcome to Aquarius, a new journal of informed dissent
The Rediscovery of Proportion: The Secret of Life in the Universe
Volume 2 (February 12, 2006): The Myth of Exactly Pi
Continuing the content of the Message sent by the Great Pyramid... our infinite Pi (inP) does not exist as the dimension of any real object, but a variable actual value of Pi, or working value of Pi, wvP, can be the concrete physical dimension of a real physical object. I propose that because the polygon shape or polyhedron shape of real objects varies, the value of "Pi" also varies in the real physical universe.
Citation suggested: Manimas, John. The Rediscovery of Proportion, "Welcome to Aquarius," The Myth of Exactly Pi, www.jmanimas.com, Volume 2 (February 12, 2006).
Copyright 2006 John Manimas Medeiros. All rights reserved. Anyone who benefits financially from the work of the author is obligated to compensate the author. Work in progress is not present on my Internet computer. See permissions below as guide to sharing this report.
1) From the Precision of the Message to the Importance of the Message
Link to: (Welcome) or (Geometry Alpha Index) .
The ratio of the area of the square to the area of the circle equals the ratio of the working value of Pi to infinite Pi and equals 0.9999885..., the best ratio constructible, as pointed to by the ancient clues, the Pyramid, the Pentangle, and the Eye of Ra.
For a complete description, see Volume 1: The Precision of the Ancients.
2) The Real Polygon Theory
3) The Euclidean Circle
4) The Mathematical Circle
5) The Psychological Circle
6) The Rorschach Brain Circle
7) The boundary of an object comprised of spherical atoms
8) Conclusion regarding the regular polygon in the real physical universe
9) The implied geometric (polyhedral) atom
10) Mathematical Note
11) Permissions and translations
The Myth of Exactly Pi
1) From the Precision of the Message to the Importance of the Message:
From the information presented in The Precision of the Ancients, we learned how to construct a similar right triangle and by so doing multiply the length of a new side times an established ratio of that right triangle. For example, we constructed the line length that is the best or closest value to Pi constructible: 3.14155655389944902222112102645257... . And we can use this line length as the base B of a right triangle, with the altitude A equal to 1. We then have a tangent that is 1/3.14155655389944902222112102645257... . We then reconstruct a similar right triangle with the base B equal to 1, and the altitude A then is equal to 1 times the tangent, which is: 0.318313543889175753503979966254073... . We then use our compass and straightedge to "take" that line and double it, and make that line the radius of a circle.
That radius will be: 0.636627087778351507007959932508146...
The radius squared will then be: 0.405294048893144874742906303752317... And Pi times that, will be the area of that circle: 1.27326880654636641018324454941636... .
And, we can take our constructed line length of 0.318313543889175753503979966254073..., and multiply that by 4, to get a rectangle. We then use the "squaring of the rectangle" procedure to get a square with an area of: 1.27325417555670301401591986501629... and the ratio of the area of our square to the area of the circle will be the same as the ratio of our working line length for Pi, being 3.141556554... to our infinite Pi: 0.999988509111675262441071001166777... . The inverse ratio is 1.00001149102036676933076020394148..., being the ratio of the circle to the square. That differs from 1 to 1 by 11 millionths of a square unit.
NOTICE that a generic principal is illustrated here: the ratio of our square area to our circle area will be the same as the ratio of our working value for Pi (wvP) to our infinite Pi (inP) when we use 2 * (1/wvP) for the radius of the circle, and 4 * (1/wvP) for the rectangle to be converted to a square by the "squaring the rectangle" procedure.
Computer program output shows (Area of Square) / (Area of Circle) = wvPi/inPi :
wvP= working value for Pi........inP= infinite Pi........
1) Selected wvP= 3.140000000000000
wvP/inP = 0.999493042617103
Inverse wvP= 0.318471337579618
* 2= Radius = 0.636942675159236
* 4= Square = [1.273885350318471]
Sqrt = Side = 1.128665295966201
Radius^2 (0.405695971439) * Pi (3.141592654) = circle area
Cir Area = (1.274531483463748)
And ratio of Sq[1.273885350318] / Cir(1.274531483464) is...
Sq/Cir --> 0.999493042617103
Cir/Sq --> 1.000507214519042
2) Selected wvP= 3.141320000000000
wvP/inP = 0.999913211666865
Inverse wvP= 0.318337514166019
* 2= Radius = 0.636675028332039
* 4= Square = [1.273350056664078]
Sqrt = Side = 1.128428135356469
Radius^2 (0.405355091702) * Pi (3.141592654) = circle area
Cir Area = (1.273460578184971)
And ratio of Sq[1.273350056664] / Cir(1.273460578185) is...
Sq/Cir --> 0.999913211666865
Cir/Sq --> 1.000086795866003
3) Selected wvP= 3.141556553899449
wvP/inP = 0.999988509111675
Inverse wvP= 0.318313543889176
* 2= Radius = 0.636627087778352
* 4= Square = [1.273254175556703]
Sqrt = Side = 1.128385650190884
Radius^2 (0.405294048893) * Pi (3.141592654) = circle area
Cir Area = (1.273268806546366)
And ratio of Sq[1.273254175557] / Cir(1.273268806546) is...
Sq/Cir --> 0.999988509111675
Cir/Sq --> 1.000011491020367
4) Selected wvP= 3.140893822011965
wvP/inP = 0.999777555000000
Inverse wvP= 0.318380708380466
* 2= Radius = 0.636761416760933
* 4= Square = [1.273522833521866]
Sqrt = Side = 1.128504689189135
Radius^2 (0.405465101875) * Pi (3.141592654) = circle area
Cir Area = (1.273806185338763)
And ratio of Sq[1.273522833522] / Cir(1.273806185339) is...
Sq/Cir --> 0.999777555000000
Cir/Sq --> 1.000222494492787
5) Selected wvP= 3.141600000000000
wvP/inP = 1.000002338434997
Inverse wvP= 0.318309141838554
* 2= Radius = 0.636618283677107
* 4= Square = [1.273236567354215]
Sqrt = Side = 1.128377847777160
Radius^2 (0.405282839112) * Pi (3.141592654) = circle area
Cir Area = (1.273233589980229)
And ratio of Sq[1.273236567354] / Cir(1.273233589980) is...
Sq/Cir --> 1.000002338434997
Cir/Sq --> 0.999997661570472
6) Selected wvP= 3.142000000000000
wvP/inP = 1.000129662389470
Inverse wvP= 0.318268618714195
* 2= Radius = 0.636537237428390
* 4= Square = [1.273074474856779]
Sqrt = Side = 1.128306020039235
Radius^2 (0.405179654633) * Pi (3.141592654) = circle area
Cir Area = (1.272909426378976)
And ratio of Sq[1.273074474857] / Cir(1.272909426379) is...
Sq/Cir --> 1.000129662389470
Cir/Sq --> 0.999870354420685
So, why does it appear, even after a great deal of study and experimentation, that we cannot construct a circle and square of exactly the same area? Instead of concluding that this is some kind of perversity in the universe, or a fact that indicates the physical universe is obtuse and impossible to understand, let us take a different viewpoint, and say that the reason we cannot construct a circle that has exactly the same area as a square is because circles, as we have defined them, do not exist anywhere in the real physical universe. By this we mean precisely that neither the circle, nor the globe, nor Pi as a dimension of a real object exists in the real physical universe. Nothing has a perimeter or an area that is Pi exactly.
How could this be true? This is blasphemy, an outrageous attack on the religion of Western civilization (which is, you should know, Mathematics). Mathematics is the religion, and Pi is the God, the incomprehensible, invisible, magical God of Mysterious Science. But, I will take all the risks entailed in challenging this god and this religion. I am accepting, instead of the doctrines of Western Civilization, the guidance sent to me (to us) by the builders of the Great Pyramid. I place my faith in them. I entrust myself to them. They are far more trustworthy. They are the teachers of Jesus. I do not "mix" religion with science; I reconcile human religion and human science. Each must be disciplined to move toward the truth that comes from all sources. The danger in the world is not only that "religionists" are fanatical and authoritarian, but rather the greatest danger is that scientists are lying. Scientists are mixing politics with science, using false science to defend government policies or business interests that are contrary to the interests of the many, the interests of the patient majority.
I will propose four primary reasons why inP does not exist as the physical dimension of any real physical object, and then elaborate:
A) The original Euclidean definition of Pi is an abstraction that is made up of nothing.
B) We cannot identify the beginning and ending of a line length of Pi, as we can in fact do for the line length of the square root of 2.
C) The curve of all things, including the circumference of a circle, is an optical illusion based upon the ratio of the straight-line length of the side of a polygon to the radius of the polygon.
D) The Rorschach Brain. The curve of all things is based upon the functional effect of the human brain that compels "filling in" of missing parts of any image, or any perception for that matter, including both simple and complex perceptions that provide incomplete data. Our brains always select data from our memory inventory to fill in the missing data so that we can "recognize" the new perception.
The purpose of the arguments presented here is to distinguish between the theoretical infinite Pi and the variable working value of Pi in the real, physical world. This is extremely important, because in order to understand the real, physical world, we must describe it accurately. If our description is wrong, our understanding is wrong and our learning is obstructed. This argument may not be original with the author, John Manimas, but if similar ideas have been published, they have not been published well, because most people believe that circles are real. Some people believe that the circle is sacred. Some mathematicians believe that Pi is sacred.
2) The Real Polygon Theory
Lets call this a "theory," the Real Polygon Theory, because it is an argument that only polygons and polyhedrons are real. No one has my permission to name any theory or idea after me.
The original Euclidean definition of Pi is an abstraction that is made up of nothing.
3) The Euclidean [definition of a] circle:
According to theoretical geometry, also known as Euclidean geometry, named after the Greek whatever named "Euclid," who was deemed by the Europeans to have invented geometry about 2,000 years, or more, after the North Africans built the great Pyramid, the circle is:
A line that is composed of an infinite number of points, each point having no dimensions, all being equally distant from a single point, that central point also having no dimensions.
A Thousand Points of Something:
Notice first that the line should be rather difficult to see, since it is made entirely out of points that have no dimensions. We have to admit that when we draw or construct a circle, we do not use points without dimensions, because if we did we would not see any of them. We use a pencil or a scribe of some type, to make a mark of some type, that does have dimensions. To begin to understand the real regular polygon that exists in nature, we first must give our points dimensions. This may bring us into the realm of a field of knowledge called "physical geometry." If you want to know whether this is physical geometry, ask a mathematician. In any case, we now have a line that is composed of points that have dimensions, or size. We can think of them as little cubes with a side length. It is not good to think of them as tiny little spheres, because spheres will roll out of place. We can say that the number of cubic points that are all equidistant from a center point (We do have a problem with that center point, don't we. Is that a cube also?) is 1,000, or better, 1,024, which is a power of 2. We want to use the base 2, or binary, number system for our points--or sides of the regular polygon--because it is very useful for readily halving or doubling the number of sides, and vertices (angles). If you are having difficulty adjusting to cubical points with dimensions, that's okay.
Question authority. The most important thing to keep in mind here is that since our points have a dimension, there has to be a finite [limited] number of them. We cannot have an infinite number of points with a dimension to make up the perimeter or our polygon because that would make our polygon "infinite" in size.
A Whole That Has Dimensions:
A whole that has dimensions must be comprised of parts that have dimensions. After geometry class, when you sit down to eat your lunch, will you be eating a sandwich that is made up of a thousand slices of pastrami that have no dimensions? Not very likely. The real world always has dimensions. That is why we are always measuring things, incessantly counting and measuring. The most powerful people in our society do not know how to evaluate the quality of anything except by using numbers, by referring to the size, the cost, or the age, or maybe just a number on a scale. Another very sophisticated measuring device that humans have invented for evaluating things, is to substitute "A, B, C, D, E" for "1, 2, 3, 4, 5." This is how we give a "quality" rating to how much you have learned today, or this year. These five letter grades determine who will be a leader, who will be rich, and who will be an employee to be used, and who will be poor.
We cannot identify the beginning and ending of a line length of Pi, as we can in fact do for the line length of the square root of 2.
4) The Mathematical Circle:
It must be either rather clear to a person, or at the very least suspected, that in order to address the "squaring the circle" problem, and in order to talk about Pi and in order to see it as a real, concrete value and not a mystical or "transcendental number," we must identify Pi as a straight line. To be a straight line with a designated length, that line has to have a beginning point and an end point. We do not have this problem with other "irrational" numbers, such as the square root of two, for example. We construct the diagonal of a square, and that line, although it is also an endless non-repeating decimal just as Pi is, has a starting point at one corner and an end point at the opposite corner. With Pi, however, which we identify as the circumference of the circle, we have a line that blends into itself. Notice that if we use any instrument or tool, including our own brain, to identify a point on the circle as the "beginning" of the line that has a length of Pi, we know that the end point of that line of Pi length must be immediately adjacent to the beginning point. But they are not the same point. Then, since the line has concrete length, with a beginning point and an end point, the substance of the line in between these two identified points must have finite, concrete dimension. All of the "length" has a dimension, and therefore if we divide that length up into parts, or into points, each point must have a dimension. Now, being comprised of points that have dimensions, it is no longer the circumference of a Euclidean circle. It no longer has an "infinite" quality, but is finite and has relationships to other lines that are mandated by the laws of physics, and of mathematics, and of proportion. It can no longer "escape" us and be out of reach. It is just as limited and fixed in length as a nail or a stick or the width of a cell of living tissue.
There is another problem. If we designate the length of the diameter of the circle, another straight line with a beginning point and an end point, as having a length of 1, then the circumference of the circle has a length of Pi. But then we can take our diameter of 1, and with our compass and straightedge, make it a little longer. Then we can construct a new circle with the radius a little longer than the old circle. Now we know that the circumference of the new circle is Pi times the new diameter. However, it is not Pi times the old and original diameter that we designated to be our 1. Therefore, the new circle has a circumference that is more than Pi -- in proportion to our original 1. We could also construct a third circle that has a shorter diameter, and of course the circumference of that third circle would be less than Pi, in relation to our original 1. Thus, with our three circles, we have begun the construction of a universe in which there is really only one circle with a circumference that is supposed to have the length of exactly Pi, and all other circles have a circumference that is either greater than exactly Pi or less than exactly Pi, in relation to our original 1. And that is in fact the way that our real, physical universe has to be, because if and when we designate the line length of 1, it must be the only 1 and cannot be changed thereafter. Once there is a 1, there is only one circle that actually has a circumference that is supposed to be equal to exactly Pi. All other circles have a circumference of a different length. This tells us, screamingly, that Pi is a proportion and not a line length, and it implies, of course, that it is only a proportion, and not a line length. Does this mean that our infinite Pi is, in reality, never a line length, that is, in the real physical world?
The curve of all things, including the circumference of a circle, is an optical illusion based upon the ratio of the straight-line length of the side of a polygon to the radius of the polygon.
5) The Psychological Circle:
One of the most important functions of the early psychologists was to study human senses and the acuity, or precision, of our perceptions. Some of the early experiments are described in books on the history of psychology. These stories are fascinating, because they taught us that what appeared to be smooth or continuous events to us are actually perceived (or sensed or received) as a series of smaller, distinctly separate events. The most useful example is the moving picture, or film. One of the first types of "moving pictures" was the flip pad. Each page of the flip pad had one picture on it. The next page of the flip pad had a slightly different picture. The operation of the flip pad by rapidly "flipping" one picture after another before the eyes, created the "illusion" of a "moving picture." What this proved psychologically, and perceptually, however, was that our eyes and the visual cortex of our brains perceived distinctly different pictures, each picture for a very short and limited interval of time, approximately 1/50th of a second, but each image lingered briefly on the retina and in the visual cortex, so that before one perceived image faded the next came into focus, and the brain perceived both images together as one smooth motion without perceiving the interval of "nothing" in between those two images. This "toy" was then shown to be a powerful experiment in human perception, leading to a profoundly important understanding about how we perceive reality and therefore a new scientific basis for thinking about how we "know" reality. New questions were asked, and the essence of a certain set of questions was "for how long of a specified interval of time does a stimulus have to exist in order to be perceived." This question has been applied to all of the five common senses: sight, hearing, smell, touch, taste. How long does an image have to exist in order for us to "see" it, whether that image be a black point on a white background, or a white square on a black background? Are we more likely to see an image if it is bright? If it flashes on and off? If it is of a particular color? For how long does the tone of a bell need to sound in order for us to hear it? For how long does a pressure need to be exerted against the skin in order for us to feel it? What is the density of chemical molecules, or number of molecules, that must come into contact with our olfactory lobes and for what interval of time in order for us detect a smell? All of these forms of perceptual experiments have established that each type of stimulus must endure for a minimum interval of time in order to be perceived. Also, with all of our senses, a series of stimuli, such as a series of musical tones, or percussive sounds, or touches, or images, will "blend" in a manner that enables us, or compels us, to interpret the ongoing perception as an event with meaning. What this means in terms of physics is that if physical objects had the capacity to exist for short moments in time, such as 1/50th of a second, then disappear for approximately that same short interval of time, we would still see them as being permanent objects before us. We would not perceive the "disappearances" that occurred in between the appearances. What does this have to do with the circle?
All perceptions occur only when a certain threshold is reached in terms of the duration of the stimulus. According to the perceptual experiments, it is possible for a stimulus to occur, such as an image or a sound, for an interval of time that is too brief for us to perceive. This actually means that an event can occur without our perceiving it. This applies to the circle in a very special way. The circle is a visual illusion that occurs because the visual capacity of our eyes, and the perceptual capacity of our visual cortex in the brain, "smoothes" the straight sides of a regular polygon when a certain proportion of the polygon "size" or radius to the length of the sides is reached. In most cases we view a circle that far surpasses the proportion required to make our perception of a polygon into a circle because the "side length" is usually no more than the width of the point of the "scribe," being a pencil or a pen. With the "side length" being the width of the scribe, the number of "sides" of the regular polygon that we see as a circle is in the thousands at least, and possibly in the tens or hundreds of thousands. However, we can appreciate the way that straight sides become a "smooth curve" by describing a couple of familiar phenomena.
Flight in a rising helicopter:
Let's go out to a large dry lake bed in Nevada. Let's make a very large "construction" of a regular polygon. We will secure a pole in the ground with concrete and that pole will have a precision swivel on the top so that a metal ring can swivel smoothly around the pole 360 degrees. We will attach a lightweight but strong cord to that ring on the swivel. This will be the center of our regular polygon. The cord, our radius, will be 477.5 meters in length. The outer end will have a "scribe" attached, a large steel "pen" for us to scratch a groove into the ground in the shape of a large circle. However, we are constructing a polygon. So, for the perimeter of our polygon, we will use three thousand black rods, each five centimeters in diameter and one meter in length. We will place these meter-long rods on our scribed circle, so that our construction will be a polygon with 3000 straight sides, each one meter in length. You will see that 477.5 * 2 * Pi equals 3000.22, close enough. The diameter of our polygon is 955 meters, nearly 1 kilometer.
Now we get into our helicopter and hover over the center of our giant polygon at a height of about 200 meters. We can hardly see our polygon. Then we rise slowly, and we see it more clearly as we rise upward. We see the straight sides, until we get to a certain height and we no longer see the straight sides any more. We see a circle. The reason we see a circle is because when the proportion of the radius of a polygon to the length of the straight side of that polygon exceeds a certain number, I believe it is around one hundred sides, possibly higher, our brain begins to switch from seeing a polygon to seeing a circle. This is explained further under "The Rorschach Brain Circle."
The curved edge of a corner:
We see a curved edge or form on the same basis that we see a circle, when the "straightness" of a dimension is too small for us to distinguish it from the next straight side of an object. This has been demonstrated by viewing blade edges or "corners" under a microscope. This phenomenon proves that not only a circle, but also any curve is a function of the proportion of the dimensions of the object being viewed. Even the edge of a razor blade looks like a rough, curved surface when viewed under our best microscope.
6) The Rorschach Brain Circle:
The Rorschach brain circle is simply the circle that we see because our brains fill in or add data from our memory banks whenever we view an object. As we develop as children, we develop the concept of a circle, and whenever we view an object and are cued by the image that it is either a circle or similar to a circle, our brain adds any missing information that is compatible with a circle. Our brain "wants" to recognize what we observe, and it pushes us in the direction of seeing an object as similar to what is already in the brain's memory. This mechanism is limited because if the object really differs from anything in memory, our brain then switches from the process of recognizing the object to "discovering" a new object.
The [neurologically] Invented Circle:
The essential process of making a polygon into our concept of a circle is simply that we add or insert "implied" points on the perimeter of a polygon in order to make it a circle. As shown in the drawings below, a curved line cannot exist between two points only. In order to be a "curve" made up of points with dimensions, there has to be many proportionally small points. Otherwise, we will see the straight sides of the polygon. The truth is that the curved line actually becomes impossible when we demand that it be comprised of concrete points joined by "lines" that represent the length, or dimension of the object in the space between these points. Another way of saying this is that we have no real basis to draw or construct a curved line between two points unless we are asserting that there are other points that are part of the object and that are outside of the "straight line" space between the two points.
7) The boundary of an object comprised of spherical atoms:
There is one more "final assault" on the value of infinite exactly Pi in the real world. If a more or less round object is comprised of spherical (orbital model) atoms, then the edge or boundary of that object would appear, under a microscope of appropriate strength, to be "scalloped." Rather than being a smooth, curved boundary line, the line would look like the drawing below, like a row or succession of half-circles. That means that the real perimeter of the object would not be comprised of "points" exactly the same distance from a point in the center of the object. That means the section of the object is not a circle according to our original definition. Neither does the object meet the definition of a "sphere," which is an abstraction the same as the infinite Pi definition of the circle. A real physical object has sides made up of a finite number of concrete points or sides. The actual "length" of the perimeter of such an object could be described or defined as a sum or a series, especially if it is a regular polygon, but in all cases the perimeter is a finite measurement. Pi is a numerical value, an endless non-repeating decimal, but it is not and cannot be a measurement or a dimension of a real, physical object. If we cannot construct the exact value of Pi as a line or as an area, this is the best explanation of why. The reason why is that Pi is an abstraction that does not exist in the real physical world and a construction is not an abstraction but a real physical object in the real physical world. This is at the core of the message sent to us from the ancients.
8) Conclusion regarding the regular polygon in the real, physical universe:
The appropriate conclusion to these arguments is that Pi exactly is a myth. There is no dimension of Pi exactly in the real physical universe. We need to consider that this may be true. And, if it may be true, we must use the scientific method, as best we can, to determine with certainty if it is true. Because, if it is true, we may in fact need to change our entire description of the universe. We may need to discard our "mathematical" universe and replace it with a "proportional" or geometric universe. The two are different, and incompatible. One is correct, the other is wrong. The proposition presented here, in three volumes, that we live in a proportional universe should not be dismissed without thorough investigation. To do so is irresponsible and unscientific. It must be either proven true or ruled out. Otherwise, we risk rushing forward with a false perception of how the universe works.
9) The implied geometric (polyhedral) atom:
The evidence presented here thus far, and presented as my interpretation of what the Great Pyramid is intended to convey to us about Phi and proportion and how the universe works, implies that the atom is a polygon, or more accurately, being three-dimensional, a polyhedron. The implication is that the atom has straight sides, and is not shaped like a tiny little solar system with electrons revolving around a nucleus in circles like planets revolving around a sun. The implication is that the planetary or "orbital" model of the atom is wrong. We need a model that depicts atoms as polyhedrons. This is what comes next, from my interpretation of the message, in Volume 3 of Welcome to Aquarius: Medicine and the Polyhedron Atom.
Mathematical Note: The Universality of Phi and Proportion and the Pythagorean Theorem and the quadratic equation and certain revisions of mathematical rules or doctrines which are required in order to understand the original purpose of the Pythagorean Theorem and the quadratic equation as tools for learning the realities of Proportion. All this so that we can re-discover Proportion and how and why Proportion is Everything.
If you studied Part C) The Universality of Phi in The Precision of the Ancients, you should have had your mathematical tree shaken at least a little, and you may have noticed that with the output of AnyNrt01 something was not as expected. The output is interesting, but it looks "contrived." How the output was obtained is not fully explained. An initial explanation is presented here, as follows: To get the output of AnyNrt01, which means finding the X value which, when squared, yields a value that is X + N for our pre-selected N, we "adjust" our traditional application of the quadratic equation:
The computer (C language) code for the program reads as follows:
"Adjusted" computer code:
N= N + 0.5;
C=N; BS=B*B; FAC=4*A*C; DE=2*A; TR=fabs(BS-FAC);
TM=sqrt(TR); NS=fabs((-B-TM)/DE); PS=(-B+TM)/DE;
Translated into English, this means that in THIS application of the Pythagorean Theorem,
A*X^2 + B*X + C = 0, I have made certain changes or "adjustments" in the formula for the two traditional solutions to a quadratic equation:
Formula: -B (+ or -) square root of (B^2 - [4*A*C] )
all divided by 2 * A
First, I have selected a target value for N, such as 2, and then added 0.5, so that the "working value" of N that is actually used in the equation is 2.5, and this N value is the C value in the quadratic equation and in the formula for the solutions.
One will recall that when we use the quadratic equation to find the solution to a problem, we get two solutions, and one of them may be negative in value, such as -5, and if we are looking for a real solution, such as the weight of something or a length of time or any one of many other measurable things, a negative solution does not make any sense. Therefore, that negative solution is "discarded." I always thought that was a strange convention. In any case, I do not discard the negative solution, because there is reason to believe that it is in fact an error in mathematics that the solution is deemed negative. I make it a positive solution, and it turns out to be the correct solution to the problem I am addressing. Let's look again at the computer code and how I made changes in the application of the quadratic formula:
Formula: -B (+ or -) square root of (B^2 - [4*A*C] )
all divided by 2 * A
"Adjusted" computer code:
N= N + 0.5;
C=N; BS=B*B; FAC=4*A*C; DE=2*A; TR=fabs(BS-FAC);
TM=sqrt(TR); NS=fabs((-B-TM)/DE); PS=(-B+TM)/DE;
First, for the value of C, as previously stated, I added 0.5 to get the "working value" of C actually entered in the equation. If 2 was the target value, 2.5 was the working value for C.
-- A=1, and B= 1 always, for this "project."
-- BS= B squared, FAC = 4*A*C to calculate the value "under the radical" the amount we are taking the "square root of." FAC is a positive value, therefore we will always have minus FAC under the radical to subtract from BS or B squared. But...
-- For the value under the radical (BS-FAC) I am not allowing any negative total, I take the absolute value which means any negative or minus value for (BS-FAC) is changed to a positive value. If this value is negative, the program and calculation cannot be done because we cannot have the square root of a negative number. This is different in "higher" mathematics where we, or they, use the imaginary value "i" which means the square root of (-1) and is used to manipulate what are called "complex numbers." For our purposes, in the real world of physical geometry, we can say that "complex numbers" are bullshit numbers, and we are going to say that the value under the radical is always converted to a positive value. There is an excellent reason for this which I will explain later. Now, I just want you to begin to see how this works, and that it does work for our purposes.
-- For any negative solutions where -B - [ square root of + (B^2 - [4*A*C] ) ] results in a negative value, we are considering that to be an error also, and we convert the negative solution into a positive value, which I have designated as "+ NS" meaning the absolute value or positive value of the traditional negative solution.
Why do I do this? Why should we do this? There are a few reasons and the first is that it works, it gives us the solutions I/we are looking for, those values for X where X^2 equals X plus the selected value N. I have coded the program so that one first enters the desired value for N, and then the program calculates, using the "adjusted" quadratic formula, the exact value for X that yields the target results: X^2 = X + N. So, you might ask, who cares? What does this mean? What it means is that this is on the path to understanding the meaning of proportion and why proportion is everything. You might also ask, what was the business about adding 0.5 to the value of N and to the value of C in the quadratic equation and quadratic formula. The reason is easy, as follows:
When we make the target value 1, and then add the 0. 5 to change the C to 1.5, we get the following results:
Formula: -B (+ or -) square root of (B^2 - [4*A*C] )
all divided by 2 * A
The 4*A*C value (or FAC in computer code) is 4 * 1 * (1.5) which equals 6. Then, the value under the radical = (1 - 6) = -5, but our "adjustment" to the total value under the radical converts that value to +5. And then our numerator for the traditional positive solution becomes -1 + (square root of 5) or ( -1 + 2.236067978... ) or 1.236067978... and that is divided by 2*1 to give us 0.618033988... and the traditional negative solution becomes + [ -1 -(square root of 5) ] or
+ ( -1 - 2.236067978... ) or 3.236067978... and that is divided by 2*1 to give us 1.618033988... . As a result of our adjustments, the positive or absolute value of the negative solution is the correct solution, so long as we add 0.5 to N (and C in the quadratic formula) and keep A=1 and B=1, for every (positive) value that we select for N. The adjusted formula works for this purpose.
Understanding fully why this works comes later, with experimentation with the formula and with study. We are beginners. Proportion is a new field for us, but it must be studied or else we risk completely misunderstanding how the universe works. That is why I said in the beginning that "the stakes are very high." What could be more at risk than our very survival, which depends upon our understanding of how the universe works. The ancient clues have now been correctly interpreted and they point us to proportion. Proportion is everything and the clue to you is to begin studying the Universality of Phi and Proportion so that you can participate in the reformation of mathematics and the rediscovery of proportion and the next phase of human evolution through which we will depart from our destructive emotional traits and become an intelligent species and good stewards eligible for membership in the kingdom of heaven.
In Volume 3 of The Rediscovery of Proportion, I will explain further why we need to "adjust" the quadratic formula in order to properly study proportion (geometry). The traditional application of the quadratic formula is out of touch with reality. The adjustments I recommend put it back in tune with physical reality.
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Citation suggested: Manimas, John. The Rediscovery of Proportion, "Welcome to Aquarius," The myth of Exactly Pi, www.jmanimas.com, Volume 2 (February 12, 2006).
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