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I shall now turn to a more traditional area of inquiry by analyzing the way most of us were taught to bisect an angle in order to get at the relationship between biads and triads that seems to have been implied.
To bisect an angle:
1. With the vertex, V, as center, using any radius, draw an arc intersecting
the sides of the angle.
2. From each of those points, using a radius greater than one-half the distance
between them, draw arcs intersecting at P.
3. Draw bisector VP.
This is not the only way to bisect an angle, nor is it the simplest. Notice that it requires 3 circles, all of which can be equal, except in the case of a "straight angle". The closer the angle being bisected is to a straight angle, the closer the point of intersection of the two secondary circles will be to the vertex, making it difficult, on a practical level, to draw the bisector accurately. In such cases, it is practical to use a larger radius for the secondary circles, but theoretically, it is not necessary.
Another thing that could be observed about the "traditional" bisection is that it focuses on only part of the construction. To get the "whole picture", consider that an angle is formed by two intersecting lines and four angles: two pairs of equal angles, all of which have a common vertex. Simply drawing lines through each pair of points forms "bisectors" of all four angles, in the sense that they are equal to one-half the given angles. The additional operations of drawing secondary circles merely repositions those angles to the common vertex, so that the bisection of one of the pairs of angles is more apparent.
Looking at the "whole" picture appears to be much more complicated, yet it is simpler to draw one circle than it is to draw three. Once tha circle is drawn, the bisection is already accomplished implicitly. Moreover, given any "part", the "whole" picture could be recreated by drawing only one circle.
Perhaps I went through that a little too quickly. Let me do that again in a more simpleminded way.
Let "X" mark the spot. The "spot" is the center of the "X", the point of intersection of its two lines, and the point of reference from which I shall examine the properties of "X-ness".
There are 4 angles. They are either all equal (right angles), or there are 2 pairs of equal angles. One pair has an angle directly above, or below, the center. I shall call them the vertical angles. The other pair of angles are to the left, or right, of the center. In geometry, they would also be called vertical angles, I will call them horizontal angles to distinguish them from the pair of vertical angles I have already designated.
Adopting the convention that "horizontal angles always refer to the larger pair permits the consideration of a pair of angles turned on its sides rather than standing on its feet. Note that there will be no conflict if they are also referred to in the usual way as vertical angles. These unequal pairs of angles are similar to those examined in Part 1, except that the equal angles are not adjacent to one another. They, too, can be viewed in terms of transformations from one to the other.
So far I have a point of view, but no frame of reference, except the implied by "x-ness" giving a vertical-horizontal orientation. The two intersecting lines can be any length - extending to infinity. From the point of view of the center alone, I am in the following "spot". I can view the lines rotating in relation to one another, and in some mystical way, get some intuitive insight about the possible relations - equal, larger, smaller, et cetera. By drawing one circle, however, I not only limit ths frame of reference, but I also create 4 other points of reference whose relationship to each other and to the "x" can be defined.
Ah! Simply draw a circle!
From whence could such an idea come in a 2 dimensional frame of reference? That
in itself could be an interesting area of inquiry, and I cannot resist making
some comment about it. If one were to be strict about the "rules of reason", the
use of a compass in a two-dimensional frame of reference should not be allowed,
because its use requires a view from a third dimension. The use of a string to
represent a line would be more appropriate.
However, since the results are more or less the same, there is no serious logical difficulty in using a compass as a convenience, so long as it is understood that the fundamental operation is the rotation of a line of a specific size. This is another indication that the simpleminded approach of viewing angles in terms of the angular motion required to transform one into another in equal steps is a very fundamental approach. That simple parlor game, that can even be played in a drinking establishment, if viewed as an analogy to the relative movement of the hands of a clock could be enlightening. For every step of the hour hand, doesn't the minute hand move exactly 12 equal steps?
Getting back to the "X" in a circle, it can be observed that the analysis of "x-ness" (intersections) has been limited to those "Xs" that are constructed with equal lines that intersect at their midpoints (diameters of a circle). There are four angles whose sides are of equal length. As an example of how language can sometimes get in the way, observe that the sides of each angle form exactly one half of the X. Since there are four angles, does that mean there are four halves?
The problem, of course, is one of shifting reference, something that is easily and frequently done without even being noticed. The X is composed of the sides of the angles; the angles are the spaces between that are merely "represented" by the sides. This may be considered a trivial semantic observation, but the confusion between a concept and its representation is not to be taken too lightly.
Drawing lines connecting the points of intersection with the circle yields four isosceles triangles each of whose base angles are one half the vertex of the adjacent triangle. Notice that the base lines of the vertical triangles are equal, parallel, and horizontal, while those of the horizontal triangles are equal, parallel, and vertical. Thus the vertical lines form angles with the sides of the vertical angles equal to one half the vertical angles. The same is true for the horizontal lines and angles.
The only problem is that these lines do not pass through the center of the verticle or horizontal angles, so that they are not easily seen as "bisectors". Of course, drawing the secondary circles as is done in the "traditional" approach, will have the desired effect of constructing a line parallel to either pair of "base" lines and midway between them, but that requires in most cases going outside the limited frame of reference so far defined. Can it be done within the limited frame of reference of a single circle. Better yet, could it be done without a circle to avoid the problem of "nonlinear lengths"?
It may have been noticed that the process of analysis has taken the form of simple questions that require ever more fundamental, more primitive concepts for their answer. I have made many assumptions. I could list and try to explain them, but that would be an unnecessary diversion to the task at hand. Simply relying, as I have done, on your "intuitive sense" rather than on your knowledge alone should be sufficient. Trying to get to the "ultimate primitive concepts" is a process that can go on indefinitely, since there is always a certain amount of ambiguity and "circularity" in the simplest of concepts and definitions.
Consider a point. There can be no simpler structure in geometry. It is defined as having position, but no dimensions. It is circular in that the concept of "position" implies a "point" of reference to establish its position. Thus it is "impossible" to consider a point so defined without also considering implicitly its "point of reference". Going one step further, it is "impossible" to consider two points without also considering implicitly the relation between them. One of the three without the others is "meaningless".
The closest approximation to considering a single point is to consider a "conscious point of reference" automatically creating the "point being considered" that remains in existence in a "meaningful relation" only so long as it is being considered. That's a pretty close analogy to the concepts of "trinity" that I have run across. As if that were not enough, the concept of "non-dimensionality" is as close an analogy to the concept of "nothingness" that I can imagine!
The saving grace, if you'll pardon the expression, is that most of us are sure we know what we are talking about when we talk about a point. The fact that any particular area of knowledge can be considered in many different ways is not particularly remarkable. What is profound to some may be ridiculous to others, but the mystical quality is ever present, even though it is ignored. What has that got to do with geometry?
Imagine a point. It is difficult to do without representing it in some way, usually by a dot. The dot is not the point because it has dimensionality. Some people "dot" an "i" with a circle. The representation of the circle might have as much similarity to a dot, or a point, as it has differences. There could also be square, or other shaped "dots". It depends on how one looks at it.
It seems reasonable to assume that some things could be agreed upon. Taking a point as the simplest geometric structure, no matter how it is represented, the next most simple, or "primitive", structure would be a straight line. It seems to flow naturally from the concept of a point as the simplest relation between a point and its reference.
A straight line, considered as the shortest distance between two points, introduces some other basic concepts, such as: motion to or from a "fixed" reference point; a concept of size of a specific measure; a concept of time, or of a particular kind of motion that would represent the shortest distance; a concept of number, at least the numbers 1 and 2 for the two points, with the number 3 lurking below the surface as the third element - the relationship between them; and so on.
The circle might be considered the next of the simple geometric structures, since the only thing required is a "slight modification" of linear motion. Once a linear distance is established, motion that maintains that linear relationship to a fixed reference point would generate a circle, with another "nonlinear" distance that reflects the modification of the concept of linear motion. If the movement of the point a fixed distance from another point, which is a normal way to define a circle, is the focus of attention then "curvilinear" motion and the circumference of the circle are the result. If the movement of the line around the fixed point becomes the focus of attention, a circle is still generated in the form of a circumference. The modification of the concept of motion is then referred to as "angular" [quite different from linear motion], with a view of a circle as a surface, or a "space" rather than a curved line.
Another contender for the position of "simplest" geometric structures after the line could be an equilateral triangle, since it could be constructed using linear motion only with different constraints. It might be interesting to demonstrate that, but I want to move a little more quickly to more important things.
I shall conclude these comments by noting that even the simplest of geometric structures can be viewed as "microcosms" of the basic concepts of man's thought and much of his knowledge.
Given a circle, is it easier to divide it into halves, or thirds? The answer to that one is pretty obvious: but given a semicircle, is it easier to get two or three equal parts?
Simply extending the fixed linear length of a radius in the opposite direction automatically yields two halves of a circle and a length, the diameter, that is twice the length of the radius that was used to draw the circle. No effort. How would an observer within the circle know that the two halves are equal?
An observer at the center might be able to get a "feel", an "intuitive" understanding by observing that two "rays" moving away from each other from the same starting point require him to turn tha same amount to get to the point that they meet. He might also observe two points moving along the circumference and somehow "measure" the distance. What "tools", other than his intuitive sense of equality, could such an observer use, limited as he is to only two dimensions? Perhaps the same one that was used to maintain a fixed linear relation between the two points, something like a rod, an "umbilical" cord or piece of string that gives an exact linear measure when it has no slack.
Using the diameter as the unit of length, he would find one, and only one, point on the circumference that is exactly one linear unit from a fixed point on that circumference. Attempts to get a "linear measure" of the circumference by letting a string bend around the curve would lead to the conclusion that the circumference is 3 units plus some other smaller unit in length. However, using the radius as a linear unit of length he would find 6 points on the circumference that are each exactly that distance from its nearest neighbors. Thus the result is to divide each semicircle into exactly 3 equal parts.
Curious.
A circle is easily divided into halves, or sixths. Exploring this line of reasoning, it could be shown that a circle is more easily divided into thirds than into quarters: two sixths equal one third. It might seem trivial to notice that 1/2 of a circle equals 1 1/2 thirds, in the same way that 1/2 of a semicircle equals 1 1/2 thirds of a semicircle. On the other hand, couldn't it be viewed as a relationship between bisections and trisections? If so, is it an accidental or a fundamental relationship?
Taking several quantum leaps forward, I might pose that question in a different way. Given two series of numbers that are independent of each other, one derived by halving, [i.e., 1/2, 1/4, 1/8, ...], the other by trisecting, [i.e., 1/3, 1/6, 1/9, 1/27, ...], there is no intersection between them. But in a closed system like that of a circle where they are not independent, is that true?
Again taking several quantum leaps forward, I will present a geometric structure that results from bisecting an angle and produces triple angles at the same time. The structure may be worth studying because it implies constructions to "un-bisect" an angle starting from one of the triple angles to yield a trisection. It at least establishes that there is a relationship between them.
Given an angle smaller than the "natural" angle of an equilateral triangle, from a point on one of its sides, draw the line to the other side that has the same length as the distance of the (starting) reference point from the vertex. Do the same thing from the other side using the same length. Draw the line through the point of intersection of those two lines and the vertex of the angle.
That is very similar to the "traditional" bisection, except that the intersections of the secondary circles with the opposite sides of the angle are used as secondary reference points rather than using their intersection with each other directly. Then these two reference points are used to find the tertiary reference point at the intersection of the two lines. I avoided the phrase "using any radius" here, because that tends to obscure the fact that a very specific length is being used as reference.
I shall conclude this part of my presentation by labeling the structure and listing the relations in a "normal" geometric way. Thus, the primary points of reference from the vertex, V, are A and B: the secondary points of reference on the sides of the angle are C and D: the tertiary point of reference, E, is the intersection of the lines Ac and BD.
Notice that triangles VAC and VBD are both isosceles, congruent, and they overlap each other in such a way that they have a base angle in common. That base angle is bisected by the line VE. For the sake of simplicity, extend VE to some point, F, so that the external angles are made apparent, and label each half of the bisected angle as having the measure, x. Then: The base angles of the two isosceles triangles is 2x.
Angles FEC and FED, external to triangles EVC and EVD respectively, are both
equal to 3x, and since they are equal they bisect angle CED, which is then 6x.
Another way of phrasing that is:
Each of the original bisected angles is 1/3 the external angles (FEC and FED)
formed by the bisector: and, the original angle that was bisected (angle CVD) is
is 1/3 angle CED, i.e., angles DVF, CVF, and CVD are trisectors of their triple
angles, DEF, CEF, and CED, respectively.
Let me emphasize that this is not a trisection. It merely shows the kind of relationship that is being sought. To get a trisection it is necessary to start from a given triple angle. That's one of the reasons the angle chosen in this example was limited to less than 60 degrees, so that the larger triple angle would be less than a straight angle.
