The Anti-Radian: 5.28318530751

by Charles William Johnson

The diameter of a circle may theoretically be divided into its circumference 3.141592654 times, or pi times, while, the radius, may theoretically divide into the circumference 2 (6.283185308) times. The length of the radius marked off on the circumference of the same circle is called the radian (57.29577951). There is no name for the length of the diameter marked off on the circumference of the circle as far as we know. Two radians would represent the distance of the diameter (114.591559). The history of mathematics has it, however, that the radian is significant in this sense, while the length of the diameter remains significant moreover in relation to the concept of pi (diameter: circumference).

The radian represents essentially a sector of the circle, marking off 57.29577951 degrees of a 360-degree circle. The number 360 has been historically established and conventionally accepted as representing the number of theoretical sectors/degrees within a circle. Obviously, a circle may have as many sectors/degrees as one might wish to physically divide the circle into for computational reasons. The ancients appear to have divided a circle into any number of sectors/degrees. The numbers cited above remain as constants to any relationship of the circle in terms of its diameter/radius and circumference. The number of sectors/degrees, however, shall vary in their symbolic representation, whether it be a circle of 360 degrees or 260 sectors (if we do not like to use the word "degrees" in this latter case).

Contemporary electronic calculators are engineered to function as of the historically accepted 360-degree circle. In order to translate angles from a 360-degree circle to a 378-sector/degree circle, one would have to employ another constant (1.05) as of these numbers 378/360 = 1.05). The concept of the radian is also generally offered in relation to the specific 90-degree angle of circles. In examining al most any table in any textbook on mathematics about Trigonometric Functions of Angles in Degrees, it becomes obvious that information is offered regarding angles within the 90-degree boundary. For example, on one side is listed the degrees of angles in radians for 0 to 45 degrees, and on the opposite side of the table appear the degrees of angles in radians for 90 to 45 degrees. Such a procedure can be confusing in selecting opposites which are simply complementary angles. To compare a complementary angle of 30 degrees with its complementary angle of 60 degrees has nothing to do with the concept of the radian. One has merely compared part of a radian to another radian. The comparison that is required for us to better understand the concept of the radian is to compare the radian to the remainder of the angle in the circle, that is, to the anti-radian.

Given that the radian, the minor sector, represents 57.29577951 degrees of the 360 degrees in a circle, then subtracting the radian from the circle means that the remainder of the circle, the major sector, that is, the anti-radian, represents 302.7042205 degrees.

Let us review these numbers in terms of the circle.

Now, let us subtract this figure from the total radians in a circle (2):

By thinking radian/anti-radian, the comparison of the numbers in degrees and percentages, are more relational. Whereas, by thinking in terms of a right angle, one has to relate the percentages of a radian to that of ninety degrees (.1.5708).

In the textbook, the implied comparison between a 36 degree angle and a complementary 54 degree angle is made in relation to the ninety degree angle. However, the percentages offered for each one are in relation to the radian of 57.29577951 degrees which in turn is in relation to the full circle of 360 degrees.

Obviously, then, for the sake of comparison and a full theoretical understanding of the relationship between mathematics and geometry, the tables of trigonometric functions should be better served were they offered directly in relation to the 2-pi relation (6.283185308) of the full 360-degree circle, as follows.

In this manner, with one single table of trigonometric function, one may better observe the comparisons between complementary, supplementary angles, as well as in relationship to the entire 360 degrees of the circle.

The ancients may have conceived of the significance of the anti-radian in such a comparison. For example, let us examine some of the previous angles to the anti-radian.

Now, consider a little reverse engineering of numbers. An historically significant number for the maya long count system is that of 1872000. Consider it as a fractal expression and reverse the computation as follows:

From this computation, we may observe a direct relationship with an angle that could possibly represent the 365c fractal day-count in relation to the maya long count period number (1872000).

Now consider, the 54 degree angle computation, which is suggestive of the Nineveh number/fractal 1959552.

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