A Child, a Particle and pi

Copyright 2009, John Manimas Medeiros

A Child Contemplates the Universe:

As I lay in bed as a young child, I contemplated the universe. The universe is made of something. We call the something “matter.” This is the real, physical universe, the universe that we detect with our sensors. I knew that the universe was very large, made up of hundreds of millions of galaxies, and a great quantity of space. I knew that we seem to see more of it as our telescopes improved, and this suggested that there was still more to see if our telescopes improved again in the future.

I tried to imagine the end or outer boundary of the physical universe meeting up with “nothing.” Then I wondered what “nothing” could be, and how could nothing exist outside of the something of the universe. The nothing would then have to extend infinitely outward and this would mean that the real universe had no size. It would mean that while being the largest of all things that contained all other things composed of matter, it would also be infinitely small, because it would be contained in a container of “nothing” that was infinitely large, with no end, no outer boundary. This “nothing” outside of the physical universe would not be “space” or “void” or “null.” It would be far less than the absence of matter and would not simply be a place where matter was thin or scarce. It would have to be absolutely nothing, an absence of real existence. It would not even be darkness alone because darkness could still be something because we detect darkness. If total darkness is only the total absence of light, then “nothing” would be even less than the total absence of light. Nothing would be the total absence of anything.

I could not imagine this state of being called “nothing.” Nothing could not have a state of being because it is not. It seemed that even giving it a name made it something. For me this absence or emptiness was impossible to imagine, or manipulate or process as a thought or concept. It did not seem possible or certainly it seemed impossible to do anything with it in my imagination. This made sense of course, and still makes sense, because one cannot do something with nothing.

This problem perplexed me. This question was imprinted on my mind as a question I had not resolved, although I came to resolve many others to my satisfaction. This question suggested that the physical universe is infinite, that there is no outer boundary. Then when I learned in school that physicists were searching for the smallest possible particle of matter, this cosmic question had to be modified. Because, if there is a smallest possible particle of matter, then that means the universe is finite, limited, in at least one direction, the “downward” direction from larger to smaller. And, if the universe was infinitely large, with no outward boundary, then it was infinite, not limited or enclosed or bound, in the “upward” direction from smaller to larger. This still was not a proof or entirely satisfactory picture of the universe for me. It seemed odd, incomplete in some way, this theory that the universe is finite in one direction, rather than two, or neither. However, it did occur to me that these two other alternatives are rather odd also. A universe that was finite in size would be like my original mind-straining effort to imagine an outer boundary of reality -- all the stars and galaxies and gas-filled space -- that was enclosed in an infinite shell of “nothing.” The other alternative, a universe that was infinite in both directions, led to an even wilder conclusion. If there is no smallest possible particle, then we would just find smaller particles as our instruments of detection improved, and just as we find the universe getting bigger and bigger as we are able to see further upward, the fundamental substance of the material universe would get smaller and smaller as we are able to see further downward. Except, we know before we arrive at infinitely small, by means of common logic, that if there is no smallest possible particle, then the smallest unit of matter is infinitely small. And if the smallest unit of matter is infinitely small, then the smallest unit of matter is “nothing.” Therefore, a universe where there is no “smallest possible particle” seems impossible. How could all of everything be made out of “nothing.”

While these thoughts may seem fantastic, or even childish, questioning things that go nowhere or do not appear to produce any meaningful result, I have never lost my concern that this particular set of questions is in fact extremely important in our understanding of the universe in which we live. This is our home, and we must understand it accurately in order to live here. I felt satisfied that there was probably a smallest possible particle, and probably an infinite universe. But not ultimately satisfied. Satisfied for today, but not for life. Then I noticed that physicists and astronomers talked about the universe as though it were a stick of dynamite that exploded long ago, and made a “big bang.” I have always held this concept in the greatest contempt. I believe that those who developed this theory are little boys who like to play with their chemistry sets and make mixtures of explosive chemicals, and they make the universe what they like it to be, an explosion. These physicists also talk about the shape and size of the universe, and some say it is shaped like a giant elliptical egg. When they present the world with a “picture” of the universe, they imply that the universe is finite in size. Then some of them go so far as to say that there may be other universes. This is the most contemptible of all concepts because it discredits common logic by changing the definition of “universe” from that which encompasses all of reality into a kind of town within a nation, or buffalo in a herd, or star in a galaxy, or galaxy in a universe. I hold this teaching to be dramatically irresponsible. There is no scientific basis for physicists to change the definition of the word “universe,” which means the event or action or word that encompasses everything. The universe is the “big setting” if we think of all events as occurring in a setting, or a place. There is no setting larger than the universe, and all other settings exist within it. If the physicists continue to designate a “universe” as something that occurs in a larger setting, then they need to invent a new word that encompasses everything. But I don’t recommend that any such new word be invented. I recommend that we stick with the word “universe” and its traditional meaning: that which encloses all objects and events and that which is the largest of all things that exist.

A Return to One’s Instinctive Knowing:

So, my question was still not resolved to my complete satisfaction, until I re-discovered, in 1970, my instinctive response to the question about squaring the circle. Years earlier, when I was a freshman in Geometry class in high school, I first heard the ancient question: “Can we construct a square exactly equal in area to a given circle using only the compass and straightedge?” My Geometry teacher said the correct answer is “No.” The voice of my own mind said “Yes.” For reasons I cannot explain -- to your satisfaction or mine -- I thought that the ancients were too smart to ask that question unless they knew there was a positive answer. But, I could not dwell on this geometrical problem then, or even for many years later. I had to grow up, go to college, fumble and stumble through the process of learning how to love others without destroying myself, or them, and how to be a parent, and how to live in the very small universe of a lifetime that is lived within a very small physical territory. Still, I have always been aware that I had some advantages, living in America, where even though I was relatively poor in terms of money and property, I could survive, live and learn, and have free time, in the early mornings, to search for my truth. I did search, and what I found is that the squaring of the circle question is probably the best simple proof, that requires only a logical mind and no telescopes or microscopes, that there is a smallest possible particle. It also can be accepted as a proof that the universe is indeed infinite in one direction -- outward -- and finite in the downward direction. What this means is as poetic as science can get. It means that the universe is made of something, and that is all there is. There is no “nothing” to be the opposite of “something.” There is no “nothing” in the universe or outside of the universe or anywhere. This conclusion, if accepted, carries us further than one might have anticipated. If the physical universe is all that there is, then all that exists is contained within this “big setting.” God, heaven or any form of “other world,” any spiritual reality or event or state of being has to exist here, at home within the one existence that encompasses all things. There is another way to describe my search for my truth.

One can ask a question about the question. That is: Why did the ancients ask this question about squaring the circle?

“Can we construct a square exactly equal in area to a given circle using only the compass and straightedge?” [and vice versa]

Is it a foolish question? Is it a question that cannot be answered? Does it have a negative answer, simply “No.”? Or, if it has a positive answer, “Yes,” then how can it be done? Is there some trick or secret construction with the straightedge and compass? Or is there some other explanation?

Does everyone know that great scientific minds of ancient times, and for much of human history, attempted to find a way to “square the circle”? Does everyone know that when mathematicians accepted a “proof” that the circle could not be squared, that the proof was not necessarily a proof that it could not be done but rather that the number pi is a “transcendental” number and trying to construct pi as a straight line is a waste of time? (Ferdinand Lindemann, 1884). But none of this tells us why the ancients asked the question. It seems to have been deemed to be a very important question for centuries. Why did we turn away? Why did we say “Oh the hell with it. It’s not an important question anyway.”?

My search led me to the conclusion that it is a true riddle in that there is a hidden assumption or “misleader” that determines the true nature of the question. The hidden assumption is that the “theoretical circle” or “infinite circle” exists in the real, physical universe. This theoretical circle can properly be called the “Euclidean” circle because the geometrical definition of a circle comes to us from the ancient geometer Euclid. His geometry was the best we had for many centuries. Regardless of how we have improved on Euclid’s work, and if we have really, the circle we learn about in geometry class does not exist in the real world. This is where my efforts to “square the circle” led me, not over a few months or over a few years, but over fifty years. For some reason, I never let go of the question and my “irrational” conviction that the ancients must have known there was a positive answer or they would not have asked the question.

My search for the solution to “squaring the circle” led me to a handful of conclusions, which I feel I prove here with my geometric constructions and the verbal descriptions that accompanies the constructions. I use only accepted mathematics and geometry. I use no tricks. I do not challenge, not here, any other doctrines, just the doctrine of the infinite circle and infinite pi.

The dissection of the problem begins with the traditional circle. What is a circle?

A circle is a space enclosed by an infinite number of points, which have no dimensions, each of which is equidistant from a given point (which also has no dimensions).

When we do geometry, we accept this. Or, maybe we accept it only to conform and go along with the teacher. After all, we do see apples and oranges, and coins and disks. It does appear that there are real circles in the real universe. But, there is a real problem here. If the circumference of a circle is made up of an infinite number of points, but each point has no dimensions, then each point is that “thing” that drove me crazy as a child. Each point with no dimensions must be “nothing.” And we are taught, by the mathematicians themselves, that if you multiply “nothing” or zero times a million, or even times infinity, you still have nothing. This results in the circumference of the circle being non-existent. This circle, interesting as it is, is theoretical. It is a line that is composed of an infinite number of points of nothing. I call it the “infinite” circle, and if the ratio of the circumference to the diameter is “pi” then this version of pi based on the infinite circle should be called infinite pi. That is what I call it.

What this means is that in a real circle, in the real physical universe, the circumference of the physical circle has to be composed of “points” that are real and that do have dimensions, and there must be a finite number of these real points that are equidistant from a center, whether that center has no dimensions or does have dimensions. The ratio of this physical circumference to the physical diameter would be physcial pi. My search led me to the discovery that there is a difference between infinite pi and physical pi. And my proof, given here below, describes and illustrates what that difference is. The main difference is that physical pi can be constructed as a straight line, and when we work with physical pi, it is possible to construct a square that has exactly the same area as a given circle, and construct a circle that has exactly the same area as a given square.

The real reason the ancients asked the original question changed for me. The search for the solution to the “squaring the circle” question leads to a simple or “low technology” proof that there is a smallest possible particle, and the universe is finite in one direction, downward, and infinite in the other direction, upward. These are no small potatoes.

A Low Technology Proof of the Basic Reality of the Physical Universe:

So here is what I believe is proven by my work with a compass and straightedge:

1) In the real, physical universe, only polygonal and polyhedral objects exist, or “physical circles” that are regular polygons, because it is not possible for a theoretical or infinite circle to exist.

2) In the real, physical universe, the ratio of the “circumference” or perimeter of a regular polygon is physical pi, and not infinite pi, and physical pi is an irrational number that can be constructed using only a compass and straightedge.

3) When we accept physical pi in the physical universe, we can construct squares and circles that are exactly equal in area because the circles are regular polygons. These circles cannot be infinite circles as defined by Euclid in traditional geometry.

4) This process of “squaring-the-circle” does not lead to any trick or secret geometrical construction. It leads to the simple, low-technology proof that there is a smallest possible particle of real, physical matter.

5) The proof that there is a smallest possible particle of physical matter serves equally well as proof that there cannot be “nothing” outside of a finite physical universe. This means that the real, physical universe must be infinite and without any upward or outward boundary.

6) Most important of all with regard to our understanding of physics, since only physical pi and a physical circle -- meaning a regular polygon or polyhedron -- can exist in the real, physical universe, this means that atoms must be polyhedral in shape, not “spherical” in accordance with the Euclidean theoretical circle. The sides of the atomic polyhedron may be rather small, but they are much larger than the almost infinitely small points of contact between two solid spheres. Therefore, if it is true that atoms are polyhedrons, their polyhedral shape would almost certainly change our understanding of how atoms bond to one another.

Therefore, my crazy obsession with the ancients and their question, with squares and circles and geometry, led me to find the answers to my childhood questions, to my satisfaction. All that exists, including God, exists right here in the one physical universe. There is nothing that is equal to “nothing” and no “other worlds” that we need to be concerned with. If there is a heaven or a hell, they are here, not in “another realm.” Mother Nature covers all in her nest. There is no “other realm.” All this is derived from playing with squares and circles. We do not need to travel in one direction infinitely in order to know that we will never finish our journey. We only need to travel to different points of interest, and every point of interest will have dimensions. Even death will be an experience or event that occurs in the same universe where we live. I do not know that experience yet, but I am convinced I will not have to leave the universe, my home, in order to get through that process. I do not understand death, but it is comforting to me to know that I can understand things each day in a new way for as long as I live. I have no obligation to stop searching. I can walk in the garden of thought and smell the flowers. I can stop and enjoy, then move on.

The Construction That Reveals the Essential Nature of the Physical Universe:

Construction of a physical circle, being a regular polygon, using the “smallest possible particle” as the length of the polygon side:

Later, everyone who objects will get a chance to prove this thesis wrong, simply by constructing a “circle.” So, if you want to be sure you know what my alleged error is, you just have to continue your study of the steps I describe for the construction, which could have been completed by the ancients. Since they built the Great Pyramid and could obviously count to a million or more, constructing a large regular polygon by dividing angles in half would certainly be easy for them. They could have constructed their giant regular polygon on the flat, plane base that they prepared for the Great Pyramid, which has a square base of thirteen acres. That square base would therefore be 752.5 feet on a side, or 9,030 inches. One half of the side, or the length of a radius from the center of the square perpendicular to a side, would be 4,515 inches (376.25 feet). This plane sheet of stone would then serve as the drawing board upon which they could construct a regular polygon, with sides having a length of two units (approximately inches) and with a vertex to vertex diameter of up to 4,500 units, or centimeters, or goat horns, or Pharaoh‘s toes, or whatever practical unit of measure they chose to use.

The method I describe here is different from any I have described before, and it is rather unusual because the fixed unit of measurement is the length of the side, which represents the width of the smallest possible particle. Because Proportion is Everything, it does not matter whether we know what the width of the smallest possible particle is. It only matters that the smallest possible particle is real, concrete matter. It is not made up of energy or “plasma” or ether or anything else. It is material, physical, not probable but certain. We know that the smallest possible particle is not two inches wide. We can agree for purposes of this demonstration that the smallest possible particle is one billionth of an inch in width. So, we are going to use a piece of material, a large molecule, or a unit of stone or wood or feather or string -- and it doesn’t matter what -- so long as it real matter. We could say, to make ourselves feel better, that the “particle” or piece of material we are using is two billion times longer, or wider, than the smallest possible particle. It doesn’t matter, because the length of two inches is a geometrical and mathematical metaphor. It is standing in for the smallest possible particle, like when we say X = 5 trillion, the X is standing in for the 5 trillion. In our construction, we begin with a hexagon, which has sides of 2 units length. The vertex radius of the hexagon is 2. Since the perimeter (or circumference) of the hexagon is 12 (6x2), the ratio of the perimeter (12) to the vertex diameter (4) is 3. Simple, and we are off. For our first regular polygon, the ratio of the perimeter to the vertex-to-vertex diameter of our regular polygon is 3. This actually means that “physical pi” for a simple hexagon is 3. Just 3. The whole integer 3. The mathematicians are now tearing off their clothes and shouting. Ignore them. Pi, in the real physical universe, is variable. The reason pi is variable is because there are only regular polygons (as well as many other shapes of course) in the real, physical universe, and no Euclidean “circles,” and the construction described here proves this fact beyond any reasonable doubt. Of course we can improve on 3. We can double the sides of our polygon to 12. With 12 sides, the ratio of the perimeter to the vertex diameter is 3.105828..., and with 24 sides we already reach 3.132628... . All that really needs to be done is to demonstrate the method for constructing larger and larger regular polygons, by doubling the sides, and keeping the length of the side always the same, two units, representing the width of the smallest possible particle.

The Expanding Polygon Construction: initial preparation on the giant plane:

We, or the ancient geometers, use a device that is a flat cross, shaped exactly like the simplest form of cross that is commonly presented as the shape of the cross that was used to “crucify” a person in ancient times. The tool looks much like a letter “T” with the longer vertical leg extending above the arms and the upper extension being about the same length as an arm. The arms of this cross-shaped tool represent the fixed length of the polygon side and is used to locate the outer side of the expanding polygon’s shrinking central angles. Only the lower outer corners of the arms are used as the points marking two vertices of the polygon. On the cross tool, an inscribed vertical center line, perpendicular to the center line of the arms, is used to locate the cross tool under the line cord that divides a central angle in half. That center line is of course perpendicular to the center line of the arms. This “cross” tool is similar to a T-square and is essentially a straightedge. The traditional “straightedge” used in geometric constructions was not a straight bar but a square, so that the right angle could readily be constructed.

------ > Small version of cross-shaped tool: U

Three taut ropes or cords are the straight lines stretched from the center of the polygon outward to locate the two sides of the central angle and the center line that divides a central angle in half and that is perpendicular to the polygon side. The center pivot would be a short vertical metal pin, with a single long cord placed around it and stretched out into the two straight lines that are the sides of the shrinking central angels. One side would be fixed in place, the other moved each time the central angle is divided in half. The middle cord, being used as the middle line each time the central angle is divided in half, would be pivoted on the center pin by connection to a ring or “S” hook. Each of these three cords, functioning as straight lines in this construction, would be held taut and straight by a heavy but movable weight at the outer end. The distance across the length of the two arms is the fixed length of the polygon side. After an initial start on a smaller plane of paper, one line would be longer than the other two, to serve as the center line each time the central angle of the polygon is divided in half. This way the weight used to hold the center line in place does not interfere with the weights stabilizing the sides of the angle being divided in half. After using the Expanding Polygon Construction to produce a working value for physical pi, the geometers could then incorporate that pi value in the proportions of the Great Pyramid, as suggested in Secrets of the Great Pyramid (Tompkins) and many other works.

The ratio of the height of the Great Pyramid to the base side is described as being a height of 1 to pi/2, the same as the ratio of the radius of a circle to the circumference of the circle. If this is correct, then the right triangle that is the vertical section of a face of the Pyramid has an Altitude A (height) of 1 and a Base B of (pi/4). This ratio or tangent of that right triangle could obviously be constructed using the values established by the Expanding Polygon Construction, because the tangent of 1 over (pi/4) is the same as the tangent of 4 over pi. We obviously can construct 4 over pi simply by multiplying A (1) and B (pi/4) by 4, a simple construction. Then, 4 times pi produces a rectangle with area 4*pi. Square the rectangle and the side is then 2*sqrt(pi). Divide the side in half and we then have a line length of sqrt(pi). Construct a right triangle with A = sqrt(pi) and B = pi and the tangent ratio is obviously 1/ sqrt(pi). Therefore we are able to construct both a line length and a ratio equal to 1/sqrt(pi).

Drawings and Procedure for the Expanding Polygon Construction:

The construction is started, as a practical matter, on a large sheet of paper on a flat floor or table. We begin by constructing the hexagon, using any commonly known procedure with compass and straightedge, so that the side of this regular polygon has a length of two inches or any two workable units. We are not measuring any line, just identifying the length of two units. The most familiar example is to construct a circle with radius of two units and then keeping the compass open to the length of two units, mark off the six sides of the hexagon, which are of course this same length of two units. Here below is a computer-calculated progression of the ratio of the perimeter (or circumference) to the diameter, as the regular polygon is expanded by doubling the number of sides which is the same as halving the central angle of the previous regular polygon. We, or the ancient geometers, do not need to actually construct the entire polygon in each step. We only need to construct the central angles and then divide each central angle in half while maintaining the length of the polygon side as the fixed two units of the arms of the cross tool.

Figure 1: The cross shaped construction tool

Figure 2: The cross tool used to construct polygon expansion by halving angles

Figure 3: Basic illustration of the results of polygon expansion

First: The procedure for dividing an angle in half, using the cross tool, and maintaining the fixed polygon side length of two units. All we are doing is using the commonly known procedure for dividing an angle in half. Then we slide the cross tool beneath the center dividing line and align the center line of the cross so that the arms are perpendicular to that center dividing line. Then we align the corner points of the arms with the sides of the angle, resulting in the fixed side length for the new regular polygon that has twice the number of sides as the previous regular polygon (6, 12, 24, 48, 96, 192 etc). This procedure is repeated the number of times made possible by the size of the plane surface. Keep in mind that the Bonneville salt flats in the United States provide a plane surface that covers several square miles.

TG means one half of the polygon central angle;

Vrad means radius (from vertex to vertex), equal to the cosecant of angle TG;

Vdia means diameter from vertex to vertex;

Per means perimeter or circumference;

Physpi means physical pi or Perimeter divided by Vertex Diameter.

The last entry following each expansion of the polygon is the ratio of the constructed physical pi to the mathematical conventional value of “infinite pi.”

Polygon #1: has 6 sides, angles (60.000000000000) degrees

TG=30.000000000000 deg, sine= 0.500000000000, Vrad= 2.000000000000

Vdia= 4.000000000000, per= 12.000000000000, physpi= 3.000000000000

Phys pi / inf pi = 0.954929658551371910

Polygon #2: has 12 sides, angles (30.000000000000) degrees

TG=15.000000000000 deg, sine= 0.258819045103, Vrad= 3.863703305156

Vdia= 7.727406610313, per= 24.000000000000, physpi= 3.105828541230

Phys pi / inf pi = 0.988615929465369140

Polygon #3: has 24 sides, angles (15.000000000000) degrees

TG=7.500000000000 deg, sine= 0.130526192220, Vrad= 7.661297575540

Vdia= 15.322595151081, per= 48.000000000000, physpi= 3.132628613281

Phys pi / inf pi = 0.997146657349636810

Polygon #4: has 48 sides, angles (7.500000000000) degrees

TG=3.750000000000 deg, sine= 0.065403129230, Vrad= 15.289788298679

Vdia= 30.579576597357, per= 96.000000000000, physpi= 3.139350203047

Phys pi / inf pi = 0.999286205822908500

Polygon #5: has 96 sides, angles (3.750000000000) degrees

TG=1.875000000000 deg, sine= 0.032719082822, Vrad= 30.563203909079

Vdia= 61.126407818159, per= 192.000000000000, physpi= 3.141031950891

Phys pi / inf pi = 0.999821522787608050

Polygon #6: has 192 sides, angles (1.875000000000) degrees

TG=0.937500000000 deg, sine= 0.016361731626, Vrad= 61.118225309427

Vdia= 122.236450618854, per= 384.000000000000, physpi= 3.141452472285

Phys pi / inf pi = 0.999955378904973210

Polygon #7: has 384 sides, angles (0.937500000000) degrees

TG=0.468750000000 deg, sine= 0.008181139604, Vrad= 122.232359843701

Vdia= 244.464719687403, per= 768.000000000000, physpi= 3.141557607912

Phys pi / inf pi = 0.999988844614245090

Polygon #8: has 768 sides, angles (0.468750000000) degrees

TG=0.234375000000 deg, sine= 0.004090604026, Vrad= 244.462674359721

Vdia= 488.925348719443, per= 1536.000000000000, physpi= 3.141583892148

Phys pi / inf pi = 0.999997211146561150

Polygon #9: has 1536 sides, angles (0.234375000000) degrees

TG=0.117187500000 deg, sine= 0.002045306291, Vrad= 488.924326063088

Vdia= 977.848652126177, per= 3072.000000000000, physpi= 3.141590463228

Phys pi / inf pi = 0.999999302786202750

Polygon #10: has 3072 sides, angles (0.117187500000) degrees

TG=0.058593750000 deg, sine= 0.001022653680, Vrad= 977.848140798936

Vdia= 1955.696281597871, per= 6144.000000000000, physpi= 3.141592105999

Phys pi / inf pi = 0.999999825696523350

Polygon #11: has 6144 sides, angles (0.058593750000) degrees

TG=0.029296875000 deg, sine= 0.000511326907, Vrad= 1955.696025934368

Vdia= 3911.392051868736, per= 12288.000000000000, physpi= 3.141592516692

Phys pi / inf pi = 0.999999956424128980

TG=0.014648437500 deg, sine= 0.000255663462, Vrad= 3911.391924036998

Vdia= 7822.783848073996, per= 24576.000000000000, physpi= 3.141592619365

Phys pi / inf pi = 0.999999989106032050

Polygon #13: has 24576 sides, angles (0.014648437500) degrees

TG=0.007324218750 deg, sine= 0.000127831732, Vrad= 7822.783784158130

Vdia= 15645.567568316259, per= 49152.000000000000, physpi= 3.141592645034

Phys pi / inf pi = 0.999999997276508010

Polygon #14: has 49152 sides, angles (0.007324218750) degrees

TG=0.003662109375 deg, sine= 0.000063915866, Vrad= 15645.567536358325

Vdia= 31291.135072716650, per= 98304.000000000000, physpi= 3.141592651451

Phys pi / inf pi = 0.999999999319126980

This print-out shows the results for fourteen divisions, but we could complete only twelve divisions on our thirteen-acre square plane, because even the diagonal radius of 6,385... Units (inches) is less than the radius of polygon thirteen (7,822...) inches. As is true for any valid geometric construction, we do not “measure” our working unit. I use “inches” here only to illustrate the practical size and dimensions of such a construction.

We, or the ancient geometers, get our ratio and then line length simply by comparing the known perimeter to the concrete, physical constructed length of the vertex diameter of the expanded polygon, taken at the greatest expanded size of the regular polygon that we can construct. We do not need to complete any polygon construction, or use the unwieldy lengths of any actual perimeter. We know the length of each perimeter is the number of sides times 2 units. We have the length of the vertex diameter physically constructed on our plane. We can obviously divide that vertex diameter in half several times, as many times as is needed in order to obtain a practical working length. For example, in the twelfth construction, the calculated length -- and actual constructed length -- of the vertex diameter is 7,822.783... units and the perimeter 24,576 units. All we have to do is divide the vertex diameter line in half repeatedly, until we are down to a manageable length, as follows:

Rounded approximations for this illustration only:

1) 3911 units, the vertex radius, is the vertex diameter divided in half

2) 3911/2 = 1955.5

3) 1955.5/2 = 977.75

4) 977.75/2 = 488.875

5) 488.875/2= 244.4375

6) 244.4375/2 = 122.21875

If the units were “inches,” then 122.21875 would be a bit more than 10 feet. There is no intent or need for high precision in this part of the discussion, because I am only demonstrating how the geometer can arrive at manageable dimensions.

Having constructed a line length that is approximately 10 feet in length, we now have a proportional length that is equal to the actual vertex diameter divided in half 6 times. We then divide the value of the perimeter, 24,576 in half 6 times, and we can see immediately that we do not need to construct any line lengths other than our final result. We simply calculate:

1) 24,576 units/2 = 12,288 units

2) 12,288/2 = 6,144

3) 6,144/2 = 3,072

4) 3,072/2 = 1,536

5) 1,536/2 = 768

6) 768/2 = 384

And we can see that if our units were inches, 384 inches = 32 feet.

Returning to our ACTUAL UNIT VALUES. We have a proportional perimeter of 384 units, and our actual proportional diameter from polygon #12, divided in half six times, is:

7822.783848073996 units / (2)^6 = 122.2309976261561875...

And 384 / 122.2309976261561875... = 3.1415926>193653835841659112496938...

Infinite pi is more like 3.1415926>535897932384626433832795...

In any case, we have constructed a ratio that is physical pi for a regular polygon with 12,288 sides. If we were able to construct a regular polygon with 12,288 sides on an ordinary sheet of paper, we would of course not see a polygon. We would see a circle.

But, if we designate the proportional length of the side of 2 units as being the width of the smallest possible particle, we CANNOT sub-divide the units of length of the perimeter, or circumference, any further. Therefore, even though we could expand our regular polygon, at least theoretically, to a much higher number of sides -- possibly constructing our regular polygon on the Bonneville salt flats, and even if we achieved construction of a regular polygon of 6 sides doubled 20 times, or 6,291,456 sides, we still would be a very long way -- every mathematician must agree, from a regular polygon (circle !) with an infinite number of sides, which is of course not even possible in the real, physical universe. This is proof that real physical pi is variable, and is dependent upon the actual physical perimeter with the smallest possible particle being the unit of length.

Notes for construction of square exactly equal to regular polygon circle:

Computations and proportional constructions:

Perimeter: 12,288 sides x 2 inches = 24,576 inches, or 2,048 feet

Divided by 2^6 (64) = 32 feet

Diameter: 360/12,288 = 0.029296875 degrees, divided in half

= 0.0146484375 degrees, sine = 2.5566346186241731406164952083865e-4

= 0.00025566346186241731406164952083865 (1/radius)

Radius = inverse = 3911.3919240369975380626843218332 inches

3911.3919240369975380626843218332 inches

Diameter = 7822.7838480739950761253686436665 inches

Or = 651.89865400616625634378072030554 feet

Diameter / by 64 = 10.185916468846347755371573754774 feet

Perimeter of 12,288 * 2 inches = 2,048 feet, / by 64 = 32 feet (or 384 inches)

With proportional radius of 5.092958234423173877685786877387 feet

R^2 = 25.938223577578812525726150414671

* (infinite) pi = 81.487332638491161843073947184636 (proportional circle) *

*4,096 (64^2) proportional factor = 333,772.11448725979890923088766827

Area of circles in square feet

Square: radius = 3911.3919240369975380626843218332 inches

Or 325.94932700308312817189036015277 feet

/ by 64 = 5.092958234423173877685786877387 feet

* (1/12) = 0.42441318620193115647381557311559 foot

* 12,288 = 5215.1892320493300507502457624443

* 64 (proportional factor) = 333,772.11085115712324801572879644

Area of square in square feet, subject to desktop calculator precision.

And 333,772.11085115712324801572879644

DIVIDED BY 0.999999989106032050 (proportion of physical pi to infinite pi)

= 333772.11448725984107595512079999... (from program above)

/by 64^2 = 81.487332638491172137684355664038 (proportional square) *

Compare 81.487332638491161843073947184636 (proportional circle) *

and the --- 81.487331750770782042972590038193 square constructed below,

subject to precision of the desktop calculator is in fact the actual area of the regular polygon “circle” and the square exactly equal in area.

Therefore, when we use a calculated/constructed value for physical pi the area of the square is in fact exactly the same area as the area of the regular polygon circle, the only form of a “circle” that is possible in the real, physical universe where the sides of a regular polygon cannot have a length less than the width of the smallest possible particle. This is true unless physicists name and describe a new third form of existence that is not matter and not energy.

Constructing the “square” that is equal exactly to the “circle” is in a sense already done,

as follows:

The radius of 5.092958234423173877685786877387 feet,

squared, = 25.938223577578812525726150414671

* (physical pi, from program above - 3.141592619365)

= 81.487331750760822927422288329901 our constructed polygon “circle”

compared to constructed square below:

Square: 5.092958234423173877685786877387 feet

* 16 feet, = 81.487331750770782042972590038193 our constructed square

Compared 81.487331750760822927422288329901 our constructed “circle”

subject to precision of desktop calculator.

But we would of course get a slightly different calculated/constructed answer if we used infinite pi to construct the “circle,” which we SHOULD NOT DO, because we cannot further subdivide the concrete sides of the real, physical “circle” which must in reality be a regular polygon, because each side of the regular polygon circle is already the smallest possible particle. This fact of the real, physical universe determines the physical reality of (and laws governing) the atom, molecules, and all matter larger than a molecule. As precision in the formulas and equations of physics continue to improve, mathematicians and physicists will plan and then execute a transition from infinite pi to physical pi, because precision is the essence of technology and precision is the essence of truth.

The regular polygon of the real, physical circle, with the smallest possible particle being its sides, cannot be modified by “curved space” between the particles. The real circle can only be comprised of real matter and not any form of “space” or anything that is not matter. Therefore, this is how the “squaring the circle” riddle can be accepted as scientific evidence, or a proof, that the universe has a smallest possible particle of matter, and there is nothing that can be defined as being “nothing.” This result supports the conclusion that the material universe is not enclosed in a container of “nothing” or any other “thing” that is “not matter.” The physical universe therefore must be infinite with no outer boundary. The child has found the answer to his question.

April 2009 (from September 1996): Now the child, grown through a lifetime, believes he knows why the original question was asked. It is because there is an intimate, direct relationship between the ancient riddle and the ancient statement that Proportion is Everything. The riddle was not originally stated as a question or a riddle; it was stated as a declarative sentence: “It is possible to construct squares and circles exactly equal in area using only the compass and straightedge.” This declaration so confounded the ancients -- and geometers and mathematicians for 2,500 years -- because they all could not let go of the Euclidean theoretical circle, the circle that has an infinite number of sides with no dimensions. Although this theoretical or “infinite” circle is interesting, and serves a useful purpose, it is the misleader in the ancient riddle or declaration. When we let go of the theoretical, infinite circle, and replace it with a real, physical circle, and impose on our physical circle the concept that there is a smallest possible particle, we then see that the true and profound original reason for the declaration and then the riddle is to show that IF THERE IS A SMALLEST POSSIBLE PARTICLE THEN IT IS TRUE THAT WE CAN CONSTRUCT SQUARES AND CIRCLES EXACTLY EQUAL IN AREA USING ONLY THE COMPASS AND STRAIGHTEDGE. This is true because if there is a smallest possible particle, and proportion is everything, we actually can simply construct a regular polygon with 128 sides on a large sheet of paper on a flat table the size of a dining table or billiard table. The reason this is true is because proportion is everything and we can use any units, units that are approximately an inch or a centimeter or the width of a thumb, and construct our regular polygon with 128 sides (as the Eye of Ra is the division into 64 parts, the next division being 128), each side having a length of two units. Then the central angle will be 360/128 or 2.8125 degrees. And one half of the central angle is 1.40625 degrees. The sine is then 0.024541228522... and the inverse, the cosecant is 40.747756334462... . Therefore the vertex diameter of our regular polygon physical circle is 81.495512668925736151949489206831... And the perimeter of our regular polygon circle is 128 sides times 2 = 256. And we see that the ratio of the circumference (perimeter) to the vertex diameter (the larger diameter) is a value for pi that is finite (and actually variable) for our physically constructed physical circle: 3.1412772509327728680620197707882... . It is a finite ratio, a rational number, although it is still an endless non-repeating decimal. The difference between this rational pi and the so-called “transcendental” pi that is based on the theoretical, infinite circle IS PRECISELY THE ISSUE ADDRESSED BY THE RIDDLE: WE CAN CONSTRUCT PHYSICAL CIRCLES AND SQUARES EXACTLY EQUAL IN AREA USING ONLY THE COMPASS AND STRAIGHTEDGE BECAUSE THERE IS A SMALLEST POSSIBLE PARTICLE IN THE KNOWN PHYSICAL UNIVERSE AND THAT MEANS THERE IS NO THEORETICAL INFINITE CIRCLE. Because there is a smallest possible particle, there is the smallest possible triangle, square, pentagon, hexagon, and the smallest possible regular polygon with any number of sides. Because the sides are of the smallest possible width, they cannot be divided further. We cannot have a “circle” or regular polygon with sides the width of less than the smallest possible particle. Because proportion is everything, our regular polygon with 128 sides is proportional to a real physical circle that is very small, but it is still precisely accurate and valid. If the smallest possible particle is actually one billionth the length of our unit of construction, then our “circle” that is physically a regular polygon is a billion times larger, in proportion to the real physical circle that is constructed by God or Nature with sides of the smallest possible particle. This, I say, is the correct solution to the riddle. Like all riddles, one must identify the “misleader” before the riddle can be solved. The Euclidean circle is not the circle that needs to be constructed. The circle to be constructed has to be constructed physically in the real physical universe, and therefore it cannot be constructed with parts that are smaller than the smallest possible particle. This true solution may be a disappointment to the mathematicians, but it is fantastic in its importance. It tells us that the ancients knew there is a smallest possible particle and that the universe is infinite with no outer boundary. They knew what we “modern” humans still agonize over: All that exists is contained in the one physical universe. Things that can be imagined in the mind do not necessarily exist in the real physical universe. Our brain is a picture maker. We can imagine things that are not real. And out unending challenge is to separate the two.

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How to prove that this proof is in error:

The mathematicians of the world have made it clear that they believe I am wrong, and possibly worse they wrong, such as an insulting waste of time. But, many mathematicians feel obligated to employ their allegedly superior intelligence to correct math errors that sprout up like weeds and confuse the world of simpler folk, so here is your chance. You, being a brilliant mathematician and geometer, can prove me wrong in the simplest way. No elaborate “higher mathematics” is required, and no arcane construction with Newtonian compass and a Riemannian straightedge. All you have to do, in accordance with a few simple rules, is construct a circle. You actually have three choices.

1) Construct a Euclidean circle, showing the points that have no dimensions, an infinite number of them, comprising the circumference of your circle.

2) Use points that are real, concrete, physical matter, with dimensions, and construct the circumference of a circle with an infinite number of these points.

3) Use points that are real, concrete, physical matter, with dimensions, and construct the circumference of a circle comprised of a finite number of these points (as though they were the smallest possible particle), and then show why and how your “circle” is not a regular polygon.

A fourth choice would be to ignore my proposition, continue to accept the “no squaring the circle” doctrine and refuse to question the authority of a mathematician.

You still should be concerned with this issue. It is not trivial, and it illustrates the means to produce an understanding of the real universe using only a compass and straightedge.