Part 3

There are many constructions that can be viewed as systems that automatically create an angle and its triple angle, or conversely, an angle and its third, or trisector. The problem is to find one that is reversible, capable of being visualized, explained, and proved.

Anyone who is familiar with trigonometry, the study that focuses on angles and their measure, should also be familiar with the property of "periodicity" that the "measure" of an angle appears to exhibit. Translated into the current context, that property is reflected in the fact that different sizes of angles may require different operations to find their trisectors; i.e., the trisection of an angle less than a right angle may be different than the trisection of an angle greater than a right angle; so too, for angles greater or less than an equilateral angle, which can be considered as the "primary standard", the "natural" angle that represents the "1/3 relationship".

Another way to view this is from the point of view of measurement. Measurement is a comparative process: a refinement of the intuitive sense of equality, larger, and smaller. It requires some "standard" of comparison. The choice of a standard may make the comparison easy or difficult, perhaps even possible or impossible.

Developing a system, or a structure to represent that system, which contains angles in a one-to three relationship is only a first step since "three" is difficult to visualize or explain without "one" and "two". A system, or structure, that contains both a 1:2 and a 1:3 relationship would be more helpful, and one such system has already been presented. It has the "defect" that larger angles are found in one place, smaller angles in another: but so long as one can get from any one point in the system or structure to any other point, the "defect" is not significant. Perhaps that's the best that can be done. If the "defect" is the result of "periodicity", which itself is a result of a comparative process, how could it be removed?

The most well-known, and perhaps the best, example of a system and structure that automatically contains these relationships is the movement of the hands of a clock, as was already mentioned. But no one yet has found a way to represent that in a geometric construction. Attempts to do so seem to get very complicated very quickly. Even given such a construction, the problem of periodicity, tied as it appears to be to the fundamental concept of Time, would probably not "go away".

Taking the "modified bisection" one more step, and analyzing it in terms of what has been presented so far, a relatively simple structure is obtained that can be visualized, explained, and useful in the proof of a true trisection that can be obtained by starting at one of the triple angles of that structure.

"Completing the picture" of the modified bisection by drawing the full primary circle (See Fig.8), or, (See Fig.8 - Enlarged) combines the implicit bisection (See Fig. 6), with that of the modified bisection (See Fig. 7) of the preceding section producing the "X in a circle", which can be used as a "theorem" to prove its known properties and relations.

(See introduction).

Hereafter, when I refer to the modified bisection, I shall be referring to this extended version of it.

To maintain consistency with the original structure, the intersections of the extensions of lines AV and BV with the circle are labeled A' and B', i.e., the X is composed of intersecting lines AA' and BB'. Using a small "x" to represent the intersection of the two lines, that can be represented as AA'xBB'. In addition to those 4 points on the primary circle, the bisector FV intersects it at G and its extension intersects the circle at R.
Draw AR and BR, intersecting A'V at I and B'V at J.
Draw GA' intersecting BV at K and BR at M.
Draw GB' intersecting AV at L and AR at N>
Define all the angles less than a right angle in terms of angle A'GR equal to y.
(See Fig. 8) or, (See Fig. 8 - Enlarged)

That looks more complicated than it is. Notice that A'A and B'B both form X's in a circle with bisector GR. Thus, all of the angles with their vertex on the circumference of the circle are equal to y. The 4 central angles are equal to twice that, or 2y, which are also equal to the angle x designated in the original modified bisection, i.e., 2y = x. The angles of interest are the 8 vertical angles at I, J, K, and L, which are formed by the intersections of the lines of the large X (AA'x BB') and the base lines of horizontal triangles of the thinner X's (GRxAA' and GRxBB').
(See Fig. 8) or, (See Fig. 8 - Enlarged)

It should be clear, by symmetry alone, that those angles are equal to each other. Taking one of them, e.g., angle GKB and viewing it as an external angle of triangle KA'B, it should also be clear that they equal 3y.
(See Fig. 8) (See Fig. 8 - Enlarged)

This structure has many interesting properties and relationships. With a careful study of it, you might come up with a simpler reconstruction of it than I can. A word of caution: The proofs will not be easy, though the constructions appear "self-evident". In fact, if you do a careful study of that structure, you will find some interesting "paradoxes", which, "taken out of context", make this structure itself appear to be "impossible".

Before making any attempt, I shall therefore define some rules to insure a "fair game". The first is that I be given triple angles that are a result of the above procedure. The second is that I will be declared a winner if I give back either the original angle or one of those whose vertex is on the circumference of the circle. Is that a "fair game"?

To do a "modified bisection" in the manner shown requires that the original angle be less than that of an equilateral triangle, to insure that the primary length used will be long enough to intersect the opposite side from the points of reference found with the primary circle. Starting with an angle less than sixty degrees, which is defined as equal to 4y, then bisecting it twice, it should be clear that y is less than fifteen degrees. Thus, the angle given to be trisected, 3y, must be less than forty five degrees.

The only effect of this first rule is to limit the size of the given angle. Not only does it limit the maximum size of the given angle, but also the smallest, since it must be equal to at least three times the "smallest conceivable angle" that can be obtained by bisection.

I should explain how the given angle, 4y, was bisected twice, and I shall do so by showing the relationship to an X in a circle. Taking any X in a circle and moving the two lower legs toward each other an equal amount ("steps") until they coincide produces a Y in a circle. Being derived from an X in a circle, it retains some of its properties. In particular, the lines connecting its external points form isosceles triangles, which in this case implicitly bisect the half angles of the remaining vertical angle.

The X and Y relations can be useful. Learning the "x y z's" of angles could be likened to learning the "a b c's" of language. They could provide a consolidation of many geometric relations in a simple, "intuitive", and abbreviated manner. The analogy to language is not so far fetched. Who can say that the letters of various languages were not derived from "pictograms" that represented, among other things, various geometric relations as they were observed in the "real" world?

Of course, given an angle greater than an "infinitesimal" but smaller than half a right angle, there would be no way to be sure where it came from, but it could validly be assumed to be one of the angles, say the angle GLA, in the "modified bisection" construction, or the structure, or the system of angles thereby represented. For example: Given just a single angle, its vertical and its horizontal angles are automatically known and can be considered as given. That, of course, depends on how it is viewed, which implies an interesting generalization. Viewing something in terms of itself may yield some limited knowledge. Viewing it in terms of its "place" in a system of "fixed" relations implies that given a small "clue" the "whole picture" can be reconstructed.

The second "rule of fairness" simply acknowledges the fact that, given an angle equal to 3y and returning one equal to y, a trisection has taken place. Giving back an angle equal to 4y implies trisecting the given angle, 3y, and adding the result, y, to that given angle. Of course, if the modified bisection could be reconstructed from a triple angle, and if the two structures obtained from each of these methods could be shown to be congruent structures, that would constitute a "proof by analogy".

Such a "proof" might not be acceptable to some, but it should al least require the concession that the trisection "could be" valid, since the rejection of one structure would require the rejection of the other. This approach is important at this time because the modified bisection has some peculiar properties that may have gone unnoticed. Indeed, many constructions of triple angles have some peculiar properties that appear to be contradictory and which point to some serious problems in mathematical thought. Those problems cannot be addressed until they are recognized. In any case, the existence of such problems should not be taken as "proofs" of impossibility.

I will retain my sense of perspective by presenting an equivalent "game" from a practical, everyday, common sense type of problem. Hopefully, some of the conceptual problems will be elucidated by such a presentation that could also answer the following question: If I demanded the size of the primary circle, would that make it an unfair game?

Assume that I am one of four equal partners in a distant goldmine. One of the partners stays at the mine. When there is a distribution to be made, he simply takes his share and sends the rest to the three of us to divide. In the early days, he sent cash and we never had a problem dividing it into three equal parts. The curious thing is that the smallest denomination of coin never had to be divided into thirds; only into halves and quarters. That seemed curious, until we realized that for him to send us an amount such as $1.00, he would have had to start with an amount equal to $1.33 1/3.

Things got a little more complicated when it became difficult for him to convert all the gold into cash. What he did then was to melt down the refined gold and pour it into a mold that was like a pie pan with one radius being a fixed wall and another radius a moving wall. Then, since the size of the pan, including its depth, did not change, he would end up with a piece of "golden pie" of some unknown angle.

For him, there was no problem to get his share. He simply represented that angle on a piece of paper and bisected it twice to get a template. He then made a single cut to take his share. That was certainly easier for him than having to make two cuts in the somewhat harder alloy of the smallest "coin of the realm" which he did from time to time when he got an odd amount. The fact that a half, or quarter, of a coin was useless didn't bother him. He took pride in being exact. He was also a little peculiar.

He liked this procedure of taking his slice of the pie. Not only did he solve the problem of being sure that he got exactly his proper share, but he also solved the potential problem of bickering about whether or not he got the proper rate of exchange.

What did he do to us, his three partners?

He sent us an unknown angle that we had to divide into three parts!

We knew that it was exactly divisible since we knew that he had exactly his proper share. If he would only send us the "measure" of his share . . . but he refused. Angles seemed to be as capricious as our partner, so as a temporary measure, we resorted to melting down the gold and putting it into a rectangular mold since we knew how to divide a rectangle.

Thus it appears that given just one dimension is not sufficient. We knew the size of the circle our capricious partner used. The fact that there are other "implicit dimensions may or may not help at this time. For example: If the "curvature" of a circle is "everywhere the same", then the curvature of my partner's piece is identical to the curvature of the piece he sent for us to share. Our view of that curvature is a "wider angle view", but if we could find a "true measure of curvature", our measure of the curvature would have to be three times his. In fact, an "angular measure of curvature" can be defined to show this. It is not helpful, since it just creates another smaller angle to be trisected.

It might be worthwhile simply to note that a "circular length" implies at least two dimensions that, in terms of linear dimensions, can be interpreted as the "ratio" between two forces (linear and centripetal), or two types of motion (linear and circular) in much the same way as trigonometry measures angles as the ratio between two lines. Does the fundamental concept of an angle necessarily imply more than one dimension as does the fundamental notion of "circular length"? Can the concept of "equal steps" be used as a "one dimensional" measure of angles? Or does it depend on how it is viewed?

In fact, given one of the dimensions might actually make the job harder, because then a congruent structure would have to be reconstructed, not just a similar one. But let me get back to the original "game", and the "modified bisection".

Now I have the problem of proof squarely before me. It is more formidable in some ways than understanding how angles behave, because I must somehow get at fundamental misconceptions that are not only at the very foundation of the philosophy of mathematics, but have also been incorporated into the way most of us were taught to think. How can I avoid the problem of "conclusion jumping", which is such a universal tendency? I must admit that I am not immune from that tendency; at times, I have jumped to conclusions on insufficient evidence. Then, too, there are so many ways to proceed that it is difficult to decide on the "best".

I am quite sure that the construction of the modified bisection will not be given as careful scrutiny, as will any construction that reverses the process by starting at the triple angle to reconstruct the structure. I realize that rigorous proof is required, and I could show you one or more "formal" constructions, but their proofs could be very tedious, depending upon what geometric theorems are used. Before I attempt to do that, I will show the "easiest", "intuitive" way of viewing the structure, so that it might be "self-evident" that given any angle in that structure, all angles are determined, and, given any linear measure in that structure, all the linear dimensions are determined.

Viewed in terms of the "intuitive" X, Y, and V relations, the modified bisection structure can be seen to be the consolidation of the definitions of the X in a circle, the Y in a circle, and a V in a circle, that are all in the same circle. Since any of these can be derived from the other, the processes of transformations are implicit in their defined relationships. Given any one of them, the others can be constructed in the same circle.

Viewed in this manner, the triple angle that is the present focus of attention (there are others) can be defined in various ways.
It is the intersection of the leg of a regular X with the leg of its associated regular V in the same circle.
It is also the intersection of the leg of a regular X with the baseline of the horizontal triangle of the associated regular Y.
The angle that is 1/3 the triple angle is 1/2 the angle of the associated regular V, or, it is 1/4 of the vertical angle of the associated regular X
It could also be described as the external angle to the triangles formed by connecting the endpoints of a regular Z that can be considered as two equal and parallel line segments connected by a third egual line segment to form a "bent" line. Though not previously defined, it is implied. It is "intuitive" enough, and it could be defined in terms of the others, since their relationships are exactly defined in both angular and linear terms.

Although this is "self-evident" to me, I realize that it is probably "obscure" to others. It is sometimes very difficult to set aside "usual modes of thought" to take a "fresh look". It takes time, and serious thought, to "digest" and incorporate even the simplest of ideas into a "new mode of thought". It is worth the effort because other, perhaps more important, relationships might then be seen. So far I have barely "scratched the surface".

Since no "perversions" of the usual geometric theorems were incorporated in th X, Y, V relations herein defined, it should be evident that a more "formal", and more elaborate, "proof" can be presented using those "subsidiary" geometric theorems in a more usual way. After all, an X in a circle is nothing more than the pairs of central angles formed by two diameters of a circle; the V is nothing more than an inscribed angle whose sides are equal; and the Y nothing more than the relation between them obtained by connecting the vertex of the regular V to that of the central angle whose chord they have in common.

Having the X,Y,V relations that I have referred to as "regular" because their linear parts were equal, I could now define a similar set of "lopsided" relations by putting X,Y,V's in semicircles. A regular relation simply won't fit in a semicircle unless one of its sides is lengthened or shortened, i.e., unless it is lopsided. Such relations could not be obtained by tilting or rotating their "regular" versions, since these operations would have no effect at all. They could, however, be obtained as transformations of their regular versions by shifting the center of its reference circle to the midpoint of the line connecting the "extreme" points on one side. In a V, that line is already there. For an X or a Y, it would be the baseline of one of the "horizontal" isosceles triangles. Of course in the X or Y transformations, two or more different circles could be used as the reference circle of linear measure. Notice that the lopsided X retains almost all the properties of a regular X in relation to its horizontal angles, except that the "bases" of the vertical angles are not parallel. They are the legs of an isosceles trapezoid, that when extended, meet at an angle of 1/2 the vertical angle.

The fact that an exact relation of the relative linear dimensions of the same angles in different size circles could be represented in this way raises a very intrigueing possibility. Given the same angle inscribed in two different size circles, and considering their chords as their "measure", the difference in their "measure" in the two circles could be viewed as a result of the difference in "curvature" of the two circles, which could be defined in terms of pi. Since that same "difference" in "measure" could be obtained independently through a lopsided X (viewed independently of circles as the diagonals of an isosceles trapezoid) or as a function of square and cube roots (by application of the Pythagorean theorem, for example), does that imply a relationship of pi with square or cube roots?

I don't know whether or not such a possible relation was ever postulated. It might be just another example of the universal tendency of jumping to conclusions. Someday, I might get back to it.

I thought that the concept of a lopsided X might help to see the relationship between the intersecting lines ACxBD in the original modified bisection. (See Fig. 7) But, since that structure will not fit in a semicircle, it is not a "regular" lopsided X as so far defined. If the definition of a lopsided X was extended to those in which the baselines of the horizontal angles were parallel, it would be necessary to determine the center of the reference circle that contains all four points. The restricted definition, designated as "regular", may be a special case of a more general one, but it at least permits the exact relationship of related circles to be defined, e.g., relating the diameter of the circle that would contain the regular X to the diameter of the circles represented by the horizontal baselines of the lopsided X.

The restricted definition of a regular lopsided X as an Xin a semicircle might be useful in analyzing the modified bisection model (See Fig. 8), or (See Fig.8 - Enlarged) since it contains two such lopsided X's side by side that are congruent to each other. The diameter of the circle is a common baseline of the horizontal isosceles triangles of the larger sides. In fact, that structure is fully defined as a pair of congruent lopsided X's whose common horizontal baseline is the diameter of a circle that bisects the associated regular X. It might also be noted in passing that the lines drawn parallel to the regular V's from the endpoints of the regular X form two "thinner" X's (BB'xGR and AA'xGR), which are also congruent to each other and have a diameter of the circle in common.

Continue Part 3