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In any case, I've gone further afield than is necessary for the present purpose, and I shall return to the "game" as a "systems analyst" trying to solve the problem by proceeding from what is "known", (the reference points and relations available), then analyzing what operations are available at any given point that could yield other reference points and relations, which in turn, could be analyzed in terms of their likelihood of achieving my goal of finding its "place" in a "well-defined" system, characterized and represented as a "modified bisection".
Given the triple angle result of a "modified bisection", or given a piece of "gold pie", what do I know with certainty? In the case of an angle, I have one reference point, the vertex, L, from which I can view the intersection of two lines, X and Y, which represent an angle less than half a right angle. In the case of the gold, I could assume that there is never enough gold to get a bigger angle. (given a bigger angle, I could bisect it until the resulting pieces are less than 45 degrees and then, divide each of them into three equal shares).
In the general case of an angle alone, the only operations that seem likely
to be of any help is to extend the lines to form vertical angles, or to draw a
circle from L, the only point of reference available. Doing both, I get a
regular X, with four more reference points that are exactly defined. Using the
radius of the gold pie would make one of the two reference points that measures
that radius available from the gold pie coincident with one of the four
references already indicated since the only effect would be the "relative size"
of the construction, not the relationships.
(See Fig. 9). or, (See Fig. 9 - Enlarged)
So that I, as the "systems analyst", can keep track of where I am, I shall
label the results so far obtained in a corresponding manner to the modified
bisection where possible. After all, that is the system in which I am trying to
find the "proper place" for the given angle. Thus I shall now label my "X", or
intersection, oriented in the manner of vertical and horizontal angles, as
follows: The point of intersection is L; the upper right side is X; the lower
left side, being a continuation of X, is X'; likewise the upper left and lower
right sides are Y and Y'.
(See Fig. 9) or, (See Fig. 9 - Enlarged)
The first step was to put the X in a circle, using "any" radius, LG, to get
the intersection of that circle with YL as G, with XL as P, with LY' as Q, and
with LX' as I. What can I do next? I have four reference points, each of which
has exactly the same relationship to the angle L. It doesn't matter which of
them I use for my next step, so I choose the point, I. Notice that the fact that
I chose the point I was implicit in labeling it I to correspond to the modified
bisection.
(See Fig. 8) or, (See Fig. 8 - Enlarged)
The next thing that has to be done is to establish at I, exactly the same relationships that exist at point L. There are many ways to do that. If I stand at the point L and evaluate the basic relationship, all that's there is a relationship of two lines called an angle, with some means to measure it.
I could point with my right hand in the direction of LX toward the point P,
while at the same time I pointed with my left hand in the direction of LY toward
the point G, thus getting an intuitive "feel" for the size of the angle. Many
other ways could be devised to get "exact" measures of that angle, provided of
course that "exact measurement" exists. I believe that it does, but the problem
of measurement, especially of angles, has led to many subtle misapplications of
numbers to geometry.
(See Appendix 2)
This, in turn, has led to "proofs" of impossibility that need to be reexamined.
Pure Euclidean geometry has avoided the pitfalls by dealing with equalities and
inequalities without "number" in a way that the intuitive sense of exactness is
satisfied.
I don't really care how the relationship at L is transferred to point I. One way to do it is to form a triangle LPG at L and then construct a congruent triangle IP'R at I. The easiest way to do that is to draw an equal circle using I as the center, and IL, equal to LG, as the radius. This second step does a lot more than just find the point P' on IX' that corresponds to P on LX. Notice, for example, that the line drawn through the points of intersection of those two circles (which I have not bothered to label) will intersect IL at its midpoint, V.
Two equal, intersecting circles whose line of centers is equal to their radius, is a most fascinating structure in its own right. I shall have to ignore many of the implicit relations it contains in order to emphasize the points of relevance to the present purpose. Suffice it to say that in addition to the point P', the point, R, corresponding to the point G, must be on the same circle as P'.
The third step is simply to find the exact position of that point R on that
second circle. That, too, can be done in several ways. One way is to transfer
the measure of GP to P'R to get congruent triangles. Then having R, the line GR
could be drawn that passes through the midpoint, V, of IL. Since V could have
been obtained as a result of the second step, drawing the line GV extended to
intersect the second circle, the exact same position of R would have been
determined. In either case, it would take four steps to get the line GV, which
is the critical length.
(See Fig. 9) or, (see Fig. - Enlarged)
Now, suppose I proclaim to my partners that it is not necessary to proceed any further. The angle GVL is 1/2 the angle BVL our distant partner started with: it is therefore 2/3 the angle GLP that he sent us. Also, the angle VGL is 1/2 GVL; therefore, it is 1/4 the original angle, and 1/3 the angle GLP he sent us. Would my partners believe me? If not why not?
As a systems analyst, I am the foremost expert on the modified bisection system. After all, it was I who developed it. I really don't think that they would accept my appeal to my own authority, nor should they. Perhaps I was too abrupt. I should not be insulted. I should be patient and proceed to complete the picture.
Having obtained the line GR, with its midpoint, V, draw the circle VG, or VR. It intersects the opposite sides of the corresponding congruent triangles at the corresponding points of A instead of P, and A' instead of P'. Since GR is a diameter of a circle, and since the structure os the points VGLA and the points VRIA' are congruent structures, construct their "mirror images" on the opposite side of the diagonal, thus forming a regular X in a circle. Since GR is a line of symmetry, by drawing the one line AR, it can be seen that its counterpart would form a regular V in a circle, and that the four angles constructed congruent to the given angle are the intersections of a regular X and a regular V in the same circle. Those are the defining characteristics of the modified bisection. Not only that, but the line GL is congruent to the dimension of the piece of golden pie. The structures obtained by both procedures are not only similar, but also congruent.
If it so simple, why didn't I start here? After all is said and done, the analysis here is the proof, which must stand on its own in any case. And if it is so simple, how could it have been "overlooked" for so long?
There are many reasons why I did not start here. One of them is that I was pretty sure that I would be "laughed out of town". Think about it.
A trisection as a result of bisection alone!
Worse, a trisection as a side effect of a bisection. Indeed! It seems patently absurd.
The fact that 3 can be derived from 2 by bisecting the 2 and adding the result to itself is easily overlooked, as is the fact that bisecting 1 to get 1/2 is the "flip side" of going from 1 to 2, and bisecting the 1/2 again to get 1/4 results in 1/3, the sum of 1/2 and 1/4: the "flip side" of going from 2 to 3. So what? Did the concept of 4, or 1/4, come before or after the concepts 3, or 1/3? Who cares?
Before I started writing this paper, many of the concepts were not as clear in my own mind as they are now. The process of trying to explain some of the results of my work in the simplest way also forced me to understand them better. Perhaps, had I started here, I would not have been up to the task. In any case, a complete logical system should permit starting at the "beginning", the "end", or any point in between with the ability to move in either direction. If a "generality" is analyzed into its parts, it should be possible to synthesize or derive the generality from its parts. If I put three equal things together to get something else, I should be able to get back to the parts I started with.
As for the question: How could it be overlooked? That's easy. The result looks as "patently absurd" as getting a trisection from bisection alone. Even more unbelievable is that the construction I have presented is not the simplest: the reference point, R, is not really needed. It, like the two reference points, P and P', can be useful in visualizing what's happening. Having found V in the second step, drawing VG creates a triangle, VGL, in which angle VGL is 1/2 angle GVL: if angle VGL is y, then angle GVL is 2y, and angle GLX is 3y, the triple of angle VGL, which must then be 1/3 angle GLX. How simple . . . IF IT CAN BE PROVED that angle VGL is 1/2 angle GVL.
How absurd!
A triangle, VGL, one of whose sides, VL, is 1/2 its side, GL, and the angles "opposite" those sides are in the same 1:2 proportion.
That's obviously not true for a 30:60:90 degree triangle whose side opposite the double angle is not twice the side of the angle opposite the "single" angle. Perhaps it is true for one, or two, "special" angles, but it seems obvious that it cannot be "generally" true.
The triangle, however, was carefully selected so that its single and double angle are together less than 1/2 a right angle. Could this be a result of "periodicity"? I don't think so, yet, if the relationship could be true of one or two "special" angles, could it also be true of a group of "special" angles?
After all, it's not my fault that angles, to fit themselves into triangles,
or circles, might not only adjust their "measure", as measured by the side
"opposite" the angle, but might also "shift their measure" to an "adjacent"
side, or to a different angle, or circle, as the reference of measure.
It appears that they must have some way of doing that. If they did not, how could I believe the geometric principle that, in a triangle, the side opposite the greater angle is always the greater side, the side opposite the smaller angle is always the smaller size? That implies that the side opposite the intermediate angle is always the intermediate side. It also implies that the relative proportions of all three angles must be reflected in the relative sizes of the sides of a triangle. The whole field of trigonometry seems to rest on this "assumption", except that it considers the effect of only two of the three angles by holding the reference angle "fixed" at ninety degrees.
It is also not my fault that "higher mathematics" cannot "measure" some relative lengths as exactly as angles can, but must resort, for practical reasons, to "transcendental numbers" to represent "incommensurable" lengths. Angles, and geometry, do not have that defect.
How can this 1:2 relation of triangles represented by triangle VGL be proven or unproven? How can the points of reference be determined where the shifts in the reference of "measure" must take place? Notice that trigonometric "proofs", or "dis-proofs", could not be adequate, but merely "close approximations", hampered as they must be by "incommensurables". Could a Euclidean trigonometry, using all three angles, be developed to avoid such problems?
Perhaps, I should try to show the "paradox" that even a geometric "proof" that seems to require the conclusion that such triangles are "impossible", must itself be "impossible", since I have already demonstrated the existence of such a triangle by its appearance in two geometric structures that are congruent?
If such a "proof" were to be presented, perhaps I could then show its error, or the resolution of the "paradox". That is not necessary, nor should it be required at this time. Philosophy begins with wonder, and it is not such a terrible thing to leave something to wonder about. Sometimes there's not much choice: but, - lest I be accused of trying to "sweep a serious problem under the rug" - I shall point out a paradox that some might wish to interpret as a "fatal flaw".
It doesn't matter whether the modified bisection, or the trisection, construction is examined since they are congruent. Examining either of them, it could be noticed that, in the same structure, there are triangles side by side that have the same angular relations but different linear ones.
For example: (See Fig. 8) (see Fig. 8 - Enlarged) or (See Fig. 9) or, (See Fig. 9 - Enlarged)
Triangle GVL contains angles y and 2y: its third angle is therefore equal to
the supplement of their sum. Triangle GA'L also contains the same angles.
In triangle GVL, VL is 1/2 GL. Thus the side opposite the double angle is
twice the length of one of its adjacent sides, which is the side opposite the
single angle. (GL is the radius of a circle: IL is a radius of the same circle:
and VL is the midpoint of IL).
However, in triangle GA'L, the side , A'L, which is the side opposite the
double angle is clearly less than twice the length GL, which is the side
opposite the single angle?
How is that possible?
There would be no point in trying to answer that question if those relationships did not exist. Notice that in triangle GVL, the double angle is a central angle of the circle VG, while in triangle GA'L both angles are inscribed angles. Notice, too, that there is a shift in the relationship of angular to linear measure from the radius of the smaller circle whose radius is GL, to a larger circle whose radius is GV, and that the relative measure of GL and GV is determined by the given angle.
Since these observations imply a simpler reconstruction, I shall present it
here with no further explanation than that it merely creates another congruent
structure.
(See Fig. 10) or, (See Fig. 10 - Enlarged)
Given a triple angle and retaining for the time being the restriction to
angles less than 1/2 a right angle:
1. From L, using radius GL, draw a circle intersecting the vertical angles at G,
I, Q, and P.
2. From I, and Q, using the same radius (GL), draw two circles intersecting at
R.
3. Draw GR. Then: angle LGR = 1/3 angle GLP.
To try to answer some of the questions raised here would require a careful reexamination of the concepts of measure and relative measure that would require a careful reexamination of the concepts of "counting", and "number", which, in turn, would require a reexamination of the concept of unity, which is an integral part thereof. Quite an Herculean task! Why should I be the only one responsible for doing that?
The fact that I am willing to make some attempt to resolve the paradoxes inherent in the present understanding of these concepts should not be misconstrued as an indication that I have any doubts about the results I have obtained so far. In fact, it would be fruitless to pursue the matter any further if the analysis I have presented so far was not accurate. Why shouldn't mathematicians have some responsibility for resolving the "triple angle paradox"?
The "proof by analogy" presented here is less than satisfactory only because of that "triple angle paradox", which it highlights, and which, up until now, has apparently gone unnoticed. I would be willing in a follow-up work entitled The Triple Angle Paradox", to analyze the causes, and some possible resolutions of that paradox.
The constructions presented here, i.e., the modified bisection
(Fig. 8) or. (Fig. 8 - Enlarged) and
its reconstruction (Fig.
9), or, (fig. 9 -
Enlarged) are either congruent or they are not.
The triple angle and its third are either correctly defined, or they are not.
The geometric theorems on which those definitions are based are either valid, or
they are not.
Just what are those geometric theorems on which those definitions are based?
In linear terms:
1. The base angles of an isosceles triangle are equal.
2. The sum of the angles in a triangle equal a straight angle.
3. Alternate interior angles of parallel lines cut by a transversal are equal.
In curvilinear terms:
1. A central angle is measured by its intercepted arc.
2. An inscribed angle is measured by 1/2 its intercepted arc.
3. An angle formed by two lines intersecting within a circle is measured by 1/2
the sum of the intercepted arcs.
Since the constructions can represent either of those sets of theorems, perhaps the paradox is a result of the inherent differences in linear and curvilinear measures. Be that as it may, there are many interesting questions that would still remain unanswered, but they are for another time and another place. I have done what I set out to do. The fact that the results conflict with some preconceived notions is not my fault. I lived up to my end of the bargain. I have tried to play a fair game.
As to my partners in the gold mine, if they still don't believe it, I shall simply take my share, and dissolve our partnership. They could easily divide the remainder in half, and could check that my share has exactly the same weight. I would be willing to bet double or nothing. Would you take that bet? I could then keep that magnificent piece of golden pie as a memento. It will, at least, give me personal satisfaction every time I look at it.
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