Trisecting The Angle – The Third Great Mystery Of Geometry
In two previous essays, Squaring The Circle and Doubling The Cube, two of the three classical problems of geometry were explored. In each case, philosophical considerations appeared to associate an occult underpinning to a mathematical rationale. Trisecting the angle, the third of the great mysteries, adds to the chronicles of modern thought as it evolved from classic times.
All three were considered unsolvable, perplexing philosophers and mathematicians for centuries, largely because ancient Greek academics allowed a ruler to be used only to draw a straight line without marking it with measurements. Perhaps the real mystery is this: Why did Greek Mathematicians insist on this condition?
Geometry describes objects in space and specifies the conditions to solve a chosen problem, frequently providing diagrams for a well thought out rationale requiring the ability to draw circles around straight lines.
At the very least, any ruler makes the straight path possible, directing the line and enabling the dots to be connected, also the most efficient way of traveling from one point to the next in the shortest amount of time, as movement proceeds effecting perspective, in perpetual change. A philosophical connection between the ruler and the compass, the only other instrument allowed by the Greeks to use on this problem, was considered a symbol of precision and discernment held sacred by Freemasonry as far back as ancient Egypt.
Numerous attempts throughout history were made to trisect an angle with little success since it was only possible to estimate using the naked eye. How consistent with geometric thought it all sounds, but suppose it was also a trick question, meant to convey a simpler task than initially thought.
It is safe to assume that an angle is part of a circle, and the compass, an indispensable tool needed to construct one, is a useful device quite different than the compass used to navigate the globe. This one is designed for inscribing the circle by creating the surrounding that contains all potential angles drawn from a center point to its circumference.
Considering that 360 degrees include all angles implicit within its sphere, the diameter is a straight line that equals at least two 180 degree angles. If the line is rotated around its central axis clockwise or counter-clockwise, the circular movement evolves into the spiral as the turning continues through Time infinitely, seen from a distant “bird’s eye” perspective as “the point.”
Since the philosophers of antiquity never specified that the problem was to be solved with an angle of a particular degree, a valid solution to trisecting an angle is dividing a 360 degree circle into three 120 degree angles comprising the transcendental PI.
Further cutting of the pie into four equal sections using 2 perpendicular diameters creates a cross which becomes the square within the circle when all its endpoints on the circumference are connected with a ruler, which makes ‘squaring the circle’ and ‘doubling the cube’ possible.
Moving from any chosen origin on a straight line from point to point, one can only ask if the ancient Greeks solved another philosophical problem as exactly as they insisted for the solution of trisecting the angle. Does infinity move eternally one way, or does negative infinity exist in the opposite direction by creating an unending 180 degrees angle, an extremely long straight path of unspecified length which also precludes exact measurements.
Where did the Greeks “draw the line” when it came to precision? Or was it an arbitrary task designed to test their students and drive them nuts?