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I have done something no one else in the world has ever done before!
That in itself is not such a big deal. There are books of world records just full of such achievements. I have often wondered why some of the record holders did some of the things they did, and I am quite sure that I could do none of them, even if I wanted to.
What I have done is a little different. It has been proven - mathematically - to be impossible, and it was not done to establish a world record. Quite the contrary, it was done so that anyone who is interested can do the same thing. It is a "common sense" approach to solving problems, even "impossible" ones, and it has some importance to everyone because it involves the determination of the truth or falsity of fundamental theories on a fundamental level.
Perhaps I am "pulling your leg". After all, if it has been proven to be impossible, how could I have done it? Perhaps I have made a mistake, but failing to do the impossible is no shame. You can judge my success or failure. In any case, a little mental exercise never hurt anyone.
To solve any problem, or to prove it is impossible to solve, the first step is to analyze the problem. All proofs are based upon - and are limited by - the analysis of the problem. In the case of proofs of impossibility, the requirements are much more stringent. In addition to showing that the analysis is appropriate to the problem, it must also be shown that the analysis used is either the only analysis possible, or that it is the most fundamental analysis that can be applied to the problem. The existence of exceptions is a strong indication that neither of these conditions has been met. Indeed, it can be argued that these conditions are impossible to meet, and paradoxically that it can be proven that it is theoretically "impossible" to prove a nontrivial impossibility.
It should be clear from the title and the preceding remarks that I have some serious questions about the "proofs of impossibility" with regard to the trisection of angles. It might also be noted that it is because of them that I am reasonably assured that no one has yet solved the problem. Furthermore, without them, the trisection of angles would probably fall into the category of a trivial pursuit. Although it would be sufficient to prove the inadequacy of those proofs, I intend to go further with the following claim:
I can trisect nonspecific angles under Platonic conditions.
What does that mean?
Well, the term trisection simply means "to divide something into three equal parts," in the same way that bisection means "to divide something into two equal parts." In this case the "something" is an angle, which I assume needs no definition. I specify "nonspecific" angles because there are some notable exceptions to the "proofs" of impossibility called special angles, e.g., a 90 degree angle and any angle derived from bisections thereof. The "Platonic conditions" are the "rules of reason", which I shall try to explain by example.
I learned about dividing things into equal parts when I was very young. Since I was not an only child, the division of such things as desserts took on some importance at that time, and my father established the following "rule of reason": "You cut, they pick."
It was very effective in yielding the desired result with the least argument, no matter who did the "cutting". If I might boast a little, I thought that I had gotten pretty good at dividing things equally using only my intuitive sense of equality, my unaided eye to "measure" the pieces, and my unguided hand to cut them evenly.
As I grew older, I learned about a device that could be used to measure the equality of pieces called a divider, and another device that could be used to get an even cut called a straightedge. Why, then it wouldn't matter who cut, or who picked. since it could be proven beforehand that they would be equal. Wow! With such magnificent tools, I thought, I could divide anything into any number of equal parts.
Well, it wasn't as easy as I thought it should be.
There might be some objection to my characterization of the "Platonic conditions" in this way, but this characterization is consistent with the "formal rules". The dividers can establish, or "measure" all points that are an equal distance from a reference point, and then those points could be used as reference points to find other relations. Thus the dividers can "measure" at least an equality, and by inference, an inequality. The straightedge, of course, cannot be used to measure, but merely to guide an "even cut" by getting a straight line.
It cannot be overemphasized that it is not the division of an angle into three equal parts that is the important issue here, since there are many ways to do so to any degree of accuracy required. The important issue is the "proof" that it cannot be done exactly using "reason" alone - which is required by the Platonic conditions, and which is insured by the use of straightedge and compass in the prescribed manner.
I am a philosopher, not a mathematician, but I recognize that mathematical proofs are not to be taken lightly. In fact, it would be prudent to use them as a guide to the potentially fruitful areas of inquiry. Having spent some thirty-odd years investigating this and similar problems, I can understand some of the difficulties. Angles, in particular, seem to have a mind all their own with some devilish ways that have helped them keep their secret for so long.
For example: If angles are considered in their trisected form, all transformations and relationships to other angles involve the trisection of other angles. To get a trisection of a given angle, you need a trisection of a related angle. A seemingly "impossible" situation of infinite regression. However, given the fact there is a whole family of trisect able angles that are outside the scope of the "proofs" of impossibility, why couldn't they be used as standards to establish a constructible relationship to a given angle to find its trisection? It would appear that such a process would also lie outside the scope of the "proofs" of impossibility.
On the other hand, if angles are considered in other than their trisected form, they seem to "hide" their "triune" nature in their transformations and relations to other angles by automatically trisecting the differences in "obscure angles. That, of course, is to suppose that angles establish an environment or "frame of reference" wherein everything could be considered in "threes": wherein the only rule known, or the only thing possible is to divide things into three equal parts.
It has long been known that the movement of the hands of a clock automatically trisects angles. If you pick an angle to be trisected, and wait until the minute hand has moved through an arc four times as large, the hour hand will move one third of the original angle during that time. The relation is simple: the hour hand moves at 1/12 the speed of the minute hand. Thus, if the minute hand moves 4 times an angle, the hour hand moves 4 times 1/12, or 4/12, or 1/3 that angle.
It seems that 1/12 cannot be obtained by bisection alone, nor can it be obtained by trisection alone, but it can be obtained by a combination of these operations. Does that imply that there is a relationship between them?
The trisection of an angle is as much a conceptual problem as it is a geometric one, and mathematicians are not the only ones who have difficulty with the concept of "trinity". Theologians, for example, have also considered the nature of the Trinity to be so profound that it was inaccessible by reason, and could only be known through Divine Revelation. I ran into this equivalent theological "proof of impossibility" while I was still at college. At that time I had received a formal letter of heresy for having suggested that everything, including the Trinity, must be accessible by reason. Philosophy - indeed, any theory - based on reason would be a useless art if all relevant knowledge to that theory was not accessible by reason, which in the final analysis is the measure of its validity. Even the sciences based on observation and experimentation require a logically consistent interpretation, that is, reason, to establish the validity of their theories. What about psychology? Could "irrational" behavior, i.e., behavior "inaccessible by reason", be properly treated by a rational theory that puts such behavior outside the scope of reason?
My insistence on that point while taking a course in psychology, brought the response: If you think you could do better than the recognized "experts" being studied, go write your own theory! Believe it, or not, that actually happened; and at that point, I realized that I was not a student of psychology, nor a student of philosophy, but I was a philosopher!
I walked out of the class. I quit school . . . and I did what I consider to be the most important work of my life. No one else thought it was important. Even now, more than thirty years later, I know that it was the most important work of my life. That's why I took the trisection of angles so personally. It is an affront to a philosopher's integrity to say that some knowledge is outside the scope of reason. Some things are impossible. It is impossible that knowledge that can be obtained by any means could at the same time be inaccessible by reason. The trisection is an excellent test case.
Part of the conceptual problem is reflected in the fact that the concept of "three ness" is rarely encountered in a useful form. Even the basic term of a "third" is used in the sense of "that which follows the second". The concept of "two ness", however, is abundantly reflected in all sorts of "pairing" relationships.
Getting back to geometry, there are many pairs of angles, e.g., complements, supplements, conjugate, reflex, et cetera. Other than the equilateral, or equiangular triangle, what other triad standards are there? The 30, 60, 90 relation, of course, can be considered as a triad. Interestingly enough, it can also be considered as one of the pair of triangles derived by bisecting an equilateral triangle. Doesn't that imply a relationship between bisections and trisections?
I shall now present a simple problem in geometric construction, which, for lack of a better term, I shall call ambiguous. Given two unequal pairs of angles, each of which are composed of two equal and adjacent angles, determine the angular motion required to transform one into the other. Let me restate that as follows: given one pair of adjacent angles each of which are "x" degrees, and given another pair of angles each of which are "x+y" degrees, determine the angular motion required to transform one into the other.
Two equal and adjacent angles are defined by 3 lines: the common center line and two sides. Probably the simplest solution would be to start from the center line and construct an angle on either side equal to the angles of the other pair of angles. Thus each side of one angle would have to move through an angle of "y" degrees away from the center line, or toward it, depending on whether the smaller, or the larger, angle was used as a starting point. The answer to the question posed would be 2y degrees.
Notice that this construction requires motion in two directions. If one side is moved clockwise, the other must be moved counterclockwise. What happens if there is movement in only one direction?
For ths sake of simplicity, I shall start from the smaller pair of angles, and use one of the sides, instead of the center, as a reference. Construct an angle equal to one of the larger angles that would go past the center line of the smaller pair of angles by "y" degrees. From that line, construct another equal angle that would go past the other side of of the smaller pair of angles by 2y degrees. Thus, in this construction, the angular motion required to transform one pair of angles to the other is 3y degrees.
Is the answer to the original question 2y, or 3y degrees? Is the relation between these two pairs of angles a "biad", or a triad relation? Neither? Both? Either, depending on the analysis?
As simple, or simpleminded, as this may appear to be, some people get "turned off" as soon as they see "x" or "x+y" and by such terms as angular transformations. The problem is so simple that it can be turned into a parlor game, and that "third" angle is so obscure that it could easily have been overlooked for centuries.
I have actually played the following "game" in a drinking establishment.
Taking three straws of one color and three of another, I set up two pairs of equal angles. The colors are not relevant, but they make it easier to distinguish the starting pair of angles from the target pair of angles. Then I would ask: How many equal steps would it take to change one into the other? To show what I meant by an "equal step", I would move one of the outer straws of the smaller angles to equal one of the larger angles, pointing out that this was one step: then I would move the other outer straw of the small angle an equal amount to point out that I did it in two equal steps.
Then I would offer to buy a drink to whoever can do it in three equal steps. Enough people are able to do that, so it would appear that I could lose on the deal. In that case, to get even, I show that I can do it in four steps by using "half-steps", and try to get a "taker" to try it in five equal steps. Would it be worth buying me a drink if I could do it? How about doing it in six or seven equal steps?
The two transformations shown are not the only ones possible either. Constructions in which all three lines are moved to transform one pair of angles into the other are also possible, and such a construction is required for a five step transformation. In thus studying the relations of angles to each other, angles appear to behave as chameleons, always hiding their true colors. If viewed as "biads", they behave as biads. If viewed as triads, they behave as triads. If viewed in some other way, they would probably behave in some other way, which could raise an interesting question. If a legitimate construction to trisect an angle were found, would it be provable?
That, of course, is a self-defeating question, because it must be provable to be any good: but it leads to the suspicion that the proof could be more difficult than the construction to insure that it did not involve circular reasoning. On the other hand, to elude so many people for so long, the construction could be so simple that it is easily overlooked, and the proof could be self-evident.
I make these comments because, over the years, I have done many constructions that I could neither prove nor disprove, and because I wish to emphasize the need for very careful analysis. The simple minded approach taken so far might be the best way to proceed to find the fundamental relationship being sought, but the study of transformations of angles gets very complicated very quickly. There are such a myriad of possible relations that all seem to vie for attention at the same time that they create "mental illusions" very much like the familiar "optical illusions". They boggle my mind.
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