The Regular Polygon Becomes a Circle
Welcome to Aquarius, Volume 8 (August 16, 2007), Squaring the Circle Exactly.
Essay: The Missing Sensory Test, and the denial of science (4 pages)
Modern psychological science, widely respected today, was born during the nineteenth century and began with a heavy burden of association with hypnotism and philosophy. The practitioners of "hard" science, such as physics and chemistry, felt that one could not conduct real scientific experiments involving the individual human mind. To physical scientists, psychological viewpoints were based on speculation and interpretation of subjective experience. It took a long time for the early believers in psychology to demonstrate that we could study the mind and conduct experiments that produced objective evidence pertinent to how the mind works. Even the name of this science, "psychology," refers back to the Greek root word "psyche," which means "mind" and not brain. It is a word that is close in meaning to our modern word "personality," and therefore does not mean the same thing as the strictly physical organ that we call the "brain." The difference between "brain" and "mind" is still subtle, because the word "mind" connotes something ephemeral or intangible about a person that is not necessarily restricted to the brain alone. The word "mind" or "psyche" might be described as something that bridges the physical brain with the non-physical "spirit." Back in the beginning of psychology, the first advocates of this new science needed desperately to produce hard physical evidence, through experiments meeting the same standards as the experiments designed to observe the physical laws of the universe, in order to have "psychology" gain admission to the club of men (mostly men) who called themselves "scientists."
There are surely several good books about the history of psychology. An excellent example is The Story of Psychology by Morton Hunt (Doubleday, 1993). With 653 pages of text, and abundant notes, Hunt does a good job of describing the steps or phases of development as psychology grew from suspect to acceptable science. Hunt's third chapter is entitled "The Protopsychologists" and includes brief intellectual biographies of such "science minded" philosophers as Francis Bacon, Rene Descartes, Spinoza, Thomas Hobbes, John Locke, David Hume, Gottfried Leibniz and Immanuel Kant. What this diverse group of men, who did not all live at the same time, had in common was their conviction that the human "mind" could be understood in a scientific manner. This might be stated in other words as the conviction that human behavior was understandable as to its causes, and was scientifically predictable. What we do is a product of what we have experienced in the past and what we have done in the past. But these scientifically minded men did not conduct psychological experiments.
Hunt's fourth chapter is entitled "The Physicalists." This means physicians and physicists who pursued an interest in how humans physically sensed the world around them, as this was the most mundane and readily observable functioning of the nervous system, or "neural" system that society was beginning to accept as the physical source of how humans experienced the external world. One of the "physicalists" was Ernst Weber, who measured the "just noticeable differences" that the sensory organs and neurons of human beings can detect and record. He used knitting needles and a drafting compass to lightly touch human skin on different parts of the body in order to observe how well the human subject could identify exactly where they were touched. He learned, and we learned, that this ability varied with different parts of the body surface, and this information has been confirmed whenever such experiments are repeated today, because of the fact that we have different quantities of touch-sensitive nerve endings at different parts of the skin surface. This was an entry into physical science. This was called "psychology" because the psychologists claimed the study of how humans sense the world as part of their new science. Today, we are more likely to call this experiment an experiment in "neurology," but we would also acknowledge that neurology and psychology are closely related and mutually supportive sciences.
The fifth chapter describes the grand entrance of psychology into the world of real science in the late 1800's, with the experiments of Professor Wilhelm Wundt at the University of Leipzig. Wundt's earliest experiments tested the speed of human perception. For example, the subject responded to a sound by pressing a telegraph key. The measurement of the interval of time from sound to response was the measurement of a sensory function. Wundt's later experiments focused on "mental chronometry" or intervals of perception. The central concept that was being developed with these sensory function experiments was the concept of what interval of duration is required in order for a physical event to be detected. The experimental method was being used to answer such questions as: How long does a tone need to be sounded in order to be heard by a human being? How long does a light need to shine, or object appear, in order to be seen by the human eye? This kind of experiment led indirectly to the invention of the moving picture, because observing how short an interval is necessary for a picture to be seen resulted in the observation that if a second picture was only slightly different from the first, the human eye -- and brain -- would perceive a smooth movement rather than two separate pictures, but only if the interval of time between the viewing of the first and second pictures was very short, approximately one-fiftieth of a second. Another form of experiment that involved precise physical measurements was how different must the tone of a sound be, such as vibrations per second, or pitch, in order for the human ear to distinguish one sound from another. These experiments showed that individuals varied in the sensitivity of their eyes and ears, especially in the area of sound. But, there was a range into which all humans fit. There were sounds that humans could not hear, both at the low frequency end and the high frequency end of sound waves. And, there was a "spectrum" of visible light that is only a very small piece of the full spectrum of electromagnetic radiation. This information was vitally important to the modern understanding that there can be a lot happening in the universe that is not detected by our rather restricted physical senses. We do need our instrumentation in order to gather more precise and more extensive information about what is going on around us.
I read Hunt's book and I suggest such a book be required reading for any mathematician or physicist. Mathematicians and physicists need to study psychology. They need to understand that their way of knowing is no different from the way all humans know whatever they know.
This brings me to my purpose. There is one sensory test that was not studied by scientific experiment, either by the early psychologists or any modern scientist -- as far as I know. That missing sensory test is an experimental design that answers the question: When does a human observer view a regular polygon and see a circle? A regular polygon can be precisely described in terms similar to that of a circle. The polygon (always referring to regular polygon here) has straight sides and vertices, where the straight sides intersect. A straight line from the center of the polygon to the mid-point of one of the sides is the "internal" radius, and a straight line from the center to a vertex of two sides is the "external" radius. In construction of a polygon, it is a common practice to place the polygon "inside" a circle, such that each of the vertices rests on the circumference of a circle. In this way, we can construct a circle and then construct a polygon with four sides, then eight, then sixteen, and so fourth, until we can hardly draw the straight sides when the polygon has sixty-four sides, or 128 sides, or 256 sides, or 512 sides or 1024 sides. When do our eyes no longer see the "straightness" of the sides and the "sharpness" of the vertices? If this occurs when the polygon has 1024 sides, we could say that this is the event that changes our perception from polygon to circle, when the polygon has 1024 sides. But it might be a more generalized scientific statement to consider the proportions of the polygon that is "becoming" a circle. That would mean using the proportion of the internal or external radius to the length of the sides or the number of sides as the point where the polygon "becomes" a circle. This is more generalized in the sense that we are considering the proportions of the structure because proportion removes the effect of size. By this I mean that if the polygon becoming a circle had a radius of one-hundred feet, or a thousand meters, we might be able to see the separate sides and vertices quite clearly, even though there were 1024 sides. However, if we use proportion to define the point at which the polygon appears to be a circle, our statement will be true no matter what the size of the polygon. This missing experiment may be the most important missing piece of information in human history, because it is pertinent to whether we believe that a "circle" can exist in the real physical universe. There is a very important difference between a physical circle, or globe or sphere, that exists in the real physical universe, and the "circle" that is described in Euclidean geometry. Euclidean geometry is what we all studied in high school, and it does not define a physical circle. It describes an abstract or theoretical circle, a circle that cannot exist in the real, physical world. And that is the issue that needs to be fully resolved. And to resolve this issue fully, all physical scientists have to agree on a common concept as to what forms do exist in the real physical world, only polygons or some form of "circles" and "spheres." This is not a casual matter, or a concern restricted to geometry. This is as fundamental as physical science can be. If we conclude that there are no abstract circles in the real physical universe, then all objects that appear to be circular or globular in form are actually polygons and polyhedrons. And that means we have to seriously question the orbital or planetary model of the atom. If we conclude that the planetary or orbital model of the atom needs to be replaced with a polyhedral atom, all of twentieth-century physics has to be revised, as well as chemistry and biochemistry and electromagnetism. This proposed change may be vigorously resisted, but the very survival of the human species is at stake, because in order to survive, we must be good stewards of the Natural World. And in order to be good stewards of the Natural World, we have to understand accurately how Nature works. And in order to understand accurately how Nature works, we must know whether atoms are spheres or polyhedrons.
To take my argument a little further, I present here a very brief statement of my view that Relativity Theory is wrong. I am not alone in this conviction. Relativity Theory tells us that if a human being is in a vehicle, such as a train, that is traveling a little faster than the speed of light, the train will physically stretch and become longer than it is when it is traveling at a velocity much slower than light. Those who defend Relativity Theory insist that this is not just a perception of stretching. The train actually gets longer. But the theory also tells us that if we are on the train and pass by a deer in a meadow, we will still see the deer, and the deer will see us, but only for a very fleeting fraction of a second. Also, the theory tells us that the velocity of the train, relative to the deer, is actually the same as the velocity of the deer relative to the train. In other words, this means that our situation with the train traveling a little faster than light is exactly the same as if the train were still and the deer is traveling a little faster than the velocity of light. That means, according to the theory, that as the train passes by, the deer, which is traveling faster than light relative to the train, will stretch physically. Since we know that the deer is a living organic being, we know that if it is actually physically stretched, it will die. We have no scientific knowledge to tell us if the effect of passing by at greater than the speed of light changes with the distance between the deer and the train. We just have this theory, that traveling faster than the speed of light is in some way impossible, because it would cause physical objects to become longer, or bigger. We need to know whether this is physically correct, or is Relativity Theory really a type of observation that belongs to the field of psychology rather than physics. I believe that it belongs to psychology, and that it is about perception and detection, and not about what happens physically when an object is traveling faster than the speed of light.
Here I present an experiment to test whether Relativity Theory should be re-named "Media of Perception Theory." Using a sonar device, assure that the design of the instrument measures the velocity of an object using only the sound reflected off the moving object and the calculation of the time intervals required for the sound to travel from the instrument to the moving object and back from the moving object to the instrument. First test the moving object when it is traveling less than 500 miles per hour. The most suitable object is a jet plane that can exceed the velocity of sound at a level that is very close to the ground. The most suitable place for such and experiment is on vast salt flats in the American desert. Use this sonar device to measure the velocity for the plane at 400 miles per hour, then 500 miles per hour, then 600 miles per hour, then 700 miles per hour, then 800 or 900 miles per hour. I predict that the result will be that the instrument will show only a velocity at or close to the velocity of sound, even when the plane clearly exceeds the velocity of sound. This is evidence that the inability to measure a velocity that is greater than the velocity of the "media of perception" or the "media of detection," which is in this case the media of sound, applies to all media of detection and not only to electromagnetic radiation. This may not be accepted as final proof, but it certainly would be persuasive evidence.
And so, I question a theory accepted by scientists. And when science is questioned, many scientists respond in a manner similar to a medieval priest or bishop. The traditional Roman Church said that Mary is the mother of God. When someone questioned this statement and asked, "How can the creator of the universe have a human mother?" the church's answer was "It's a mystery." The scientist does not respond in exactly those words, but with words that have essentially the same meaning. The scientist says, "Nature's laws do not need to conform to a human being's logic." Scientists can be very smug when they give this answer to questions they don't like, but they need to be advised that when they make this statement they are denying the fundamental principle of evolution. The fundamental principle of evolution is that when an animal evolves a survival trait, that is, a trait than enhances survival, that trait is perpetuated in the offspring of that animal. Therefore, the scientist's statement that "Nature's laws do not need to conform to a human being's logic," means we humans have evolved brains that do not and cannot understand how Nature works. This viewpoint actually refutes the fundamental principle of science itself, which is that we can understand how Nature works. If we cannot understand how Nature works, then all of science is a waste of time. It will only lead to more efficient methods of self-destruction rather than to enlightenment. If the theory of evolution is true, and I believe it is fully supported by the evidence, then our brains have evolved, and continue to evolve, so that our logic does conform to the logic of Nature's laws accurately. To understand Nature accurately is obviously a survival trait. Perhaps we cannot survive with knowledge alone, but it certainly appears rational to me to assert that accurate understanding is required in order to avoid the catastrophe of trying to oppose Natural Laws that cannot be opposed. Therefore, I assert the opposite of the scientist's explanation for "science" that appears illogical. I say, "Human logic does conform to Nature's Laws, and likewise Nature's Laws must conform to human logic." This is so because if our logic does not and cannot comprehend Nature's Laws, then evolution led to the development of a suicidal trait rather than to a survival trait. The scientist who discredits human logic discredits all of science.
The Euclidean circle and a real circle:
Euclid, and geometry, defines a circle as a circumferential line composed of an infinite number of points, which possess no dimensions, all equidistant from a central point which possesses no dimensions. A real circle must have a physical circumference, and therefore that circumference must be composed of a finite number of points, or sides, with dimensions, all equidistant from a central point, with or without dimensions. It cannot have an infinite number of sides, because it would then be bigger than the universe, or at least as big. The sides of a real "circle" must have real, physical dimensions, because if they have no dimensions, an infinite number of nothing still equals nothing. If a physical circle must have a finite number of sides, with dimensions, then it must be a polygon, and likewise, a physical "sphere" must be a polyhedron. If called upon to explain the crystalline molecular structures of the metals and minerals, which would be easier to explain: 1) How spherical atoms create polyhedral molecules? Or, 2) How polyhedral atoms create polyhedral molecules?
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