Fourth Line

      Fourth line is one side of a square, which is the two dimensional prototype of the cube, whose three dimensions create volume, the definition of Matter. Fourth Line is the doorway to the third dimension, and where matter is created.

      Because of its equal two dimensions of height and width, a square with a side of one, is geometry’s graphic illustration of the second dimension, and as such, its diagonal 1.41421359 (the square root of 2) is the ideal number to represent the whole abstract world of the second dimension.

Similarly, because volume is the definition of the three dimensions of matter, the diagonal of a cube with a side of one, 1.732050808, (the square root of 3) is the obvious choice of a symbolic numerical representative of the height, width, and depth world of matter.

      The manner by which the measure of the circumference can be known, is at the same time the process that transforms the second dimension into the third dimension. This process is called circle-squaring, an ancient time honored practice whereby the geometrician  attempts to reconcile spirit and matter by constructing a circle and square of equal circumference and perimeter or equal area, using only a compass and straight edge.

      In this exercise, the circle’s line, which already represents Time, represents Spirit also and is female, being the mother of all straight lines. The square, with its four straight lines, represents Matter, and is male.

      The product, or child, of this metaphysical union of circle and square is a new geometric form, the pyramid. This form is developed by drawing two lines connecting the ends of the square’s diameter up to the apex of the circles upright radius making the height of the pyramid.                                            

      This figure contains both the two dimensional plan and elevation of a three dimensional pyramid. These two additional lines look like they ought to be the corner edges (vertices) but actually form the apothem, an invisible line that a raindrop would trace in its descent from the pyramids apex.

      Once a metaphysically satisfying geometric union of Spirit and Matter has been constructed with the studyable  opposites of circle and square, pi,(the ratio between the diameter and its circles circumference, First line and Second line), can be determined for that particular construction. This is done by dividing the square’s measurable perimeter, which the geometer is declaring theoretically the same as the circles un-measureable circumference, by twice the radius-height, the diameter of the circle. This determines what pi is for that particular circle-square arrangement, regardless of whether it conforms to the conventional number. Although the Great Pyramid is not the result a particular solution to this problem because it is not square, having four different sides, it never the less embodies many solutions to this problem, including as many different versions of pi as possible in its design, illustrating the point that there are many ways to look at reality.

      In this circle-squaring exercise, achieving the correct pi, 3.1415926, is NOT the goal, as it has been proved to be impossible to do with just a compass and straight edge. Instead, a different criteria for what is correct is being used. It is modeled after the genetic laws of the form trying to make the construction, the human form, where the child get exactly half of his or her 46 chromosomes from each parent. The pyramid, as child of its mother, the circle, and its father, the square, must have equal genetic material from each parent to be the legitimate offspring of its parents. In this geometric exercise, the genetic material is number, and the circle’s contribution is found in the measure of the many straight lines comprising the circle’s regular polygons.  

      Of the thirteen basic lines in a pyramid,(seventeen if the diagonals of the square base are included), only eight actually transform into edges in the three dimensional pyramid. The circle’s radius and four apothems,(the line a raindrop would trace from the apex rolling down the middle of the pyramid’s side), are lines in the two dimensional elevation drawing, but have no definition in the three dimensional pyramid just as the north -south axis line the earth rotates around is non-existent in materiality.

      Although the four apothems are not observable lines in the pyramid, they do illustrate the kind of blended equality necessary in this exercise for the whole pyramid to truly be a child of its mother and father, the circle and square. This is due to Pythagoras’s s law of right angled triangles, which dictates that the square of the circles radius (the height of the pyramid) plus the square of the square’s radius, (or an eighth of the base perimeter) equals the apothem, (hypotenuse)  squared. The circle and square are thus inseparably in the measure of the apothem, illustrating number as genes.

      These primary eight lines that are left must be considered the pyramids chromosomes. Since the base’s four edges obviously originated in the square because they ARE the square, the remaining four edges running from the corners of the base to the apex, the vertices, must have their origin in the circle for their to be true equality.

      Just as there is only one sperm that makes it into the ovum out of the millions that try, there is only one circle- square solution that delivers the same measurement for the pyramids vertices as well as for an integral line within the circles domain of polygonal lines. This important line is the height or altitude, of the equilateral triangle, one and a half times the length of the circles radius.

      The one and only circle-squaring operation that produces this chromosome number is created by circumscribing a square around the circle, dividing one side into four equal parts, and then drawing a line from the circles center up to the quarter division of the side of the square. This line,1.118033989 longer than the radius, becomes the diagonal from the center to the corner of a new square that is theoretically equal to the circle.

      The pi ratio developed by this particular configuration is 3.16227766, the square root of ten, which  geometrically is the cubic diagonal of three cubes, end to end, with a side measure of one.

      Although this number has been used by a number of mathematicians through out history, it is way too high for practical consideration. It is, however, perfect for theoretical consideration, as it is perfection itself, beautifying everything it touches. Of all numbers, it has the closest relationship to the number One, as its inverse proportion, (one divided by any number) duplicates the same infinite series of numbers as its own self. One divided by 3.16227766 is .316227766 and is the only number to do so. It also delivers a genetically based system of numbers that rectify and purify the inequity between the ratios of the polygons lines to the circle. Fourth line to First line ratio is the square root of 20 in the world according to the square root of 10 while using correct pi it is the square root of 19.73921951. The ratios of the square root of 10 world are the ones used within the Great Pyramid, it’s Fourth Line to First Line ratio being a major ingredient.

     Using this number for pi is like having the world function as one wishes it would, this being the one and only circle-squaring solution that delivers numerical equality equal to genetic equality of the very form that is examining the nature of equality, the human geometer.

      However, only one half of the human equation, male straight line, logical thinking has dominated the determination of pi and this mind set has not regarded the circle as Spirit with a need for special consideration as to its mathematical nature. Because it can’t be held under a microscope and measured as matter can, Spirit has been exclude by science as a factor in either the origin or operation of the material world. But here the circle does represent un-measureable Spirit, and beauty and equality are its nature. Its pi ratio makes sense of the chaotic universe because it makes a unified whole as no other number can.

      Fourth Line, therefor, represents the transformation of the second dimension into the third dimension via the marriage of circle and square, cementing Spirit and Matter together as equal, inseparable partners. At the same time it inaugurates a genetically based system of ratios between the straight lines of the polygons and the curved line of the circle.

      Therefore, in a circle whose circumference is One, Second Line, the diameter, measures .316227766., with 3.16227766 as its pi ratio to the circumference.

      With one ratio established, all other ratios can be determined. Fourth line measures .223606797 (square root of 5) and its ratio to its mother (First Line) is 4.47135955 (square root of 20). In the Great Pyramid this ratio times a thousand is the distance from the entrance to the Great Step, as well as a number of other places. 2236.06797 inches is the length of the Well Shaft, the only actual way of reaching the upper chambers and passages.

      Third Line measures .2738612 (square root of .075) and its very important ratio (Action to Time) is 3.651483716, which, times 100 is just two and a quarter hours short of our Earth’s yearly revolution around the Sun. The Great Pyramids invisible northwest to northeast corner socket line measures 9128.709292 inches, one quarter of  this ratio. Because it is the first number encountered when approaching the pyramid’s entrance, and has to be crossed, it has to be considered one of the most important numbers incorporated into the Great Pyramid, if not the most important.  

     These pure number ratios can be thought of as comprising an ideal perfect world that our world is clearly not, but the world that would be anyone’s first choice, if one had a choice.                                            

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