Geometry 1: π

understanding the megalithic: circular structures: part 1
What is the relationship of a circles perimeter to its width, called its “diameter”?

It would require 3 and a bit diameters to wrap around the circle – the ratio of 3 and a bit diameters to the perimeter is known as “Pi”, notated by the Greek symbol “π”. Half of the diameter, from the circle’s center to its edge, is named its radius.

A circle is made using a staked rope whose other end can only then move in a circle of radius equal to the rope’s length. That is, if you can make a wooden stake and weave a rope out of fibres, you can make circles. You can then create an encircling fence of stakes, at a constant distance from the central stake.

How long is the perimeter (a circumference) relative to the radius? The answer will always be the same ratio and one can compare the two lengths by putting a rope once around the fence, and comparing the radius and perimeter ropes side-by-side. Units of measure do not need to exist: the radius rope could be of any length and, alongside the perimeter rope, the radius rope could be laid end-to-end. There would be more of the perimeter rope left over, an excess of X (figure 1). So the value (of 2 x π) must be greater than six radius lengths.

Figure 1

This excess over six radii could be recorded as a short wooden rod (X in length). Once more, the number of rods can be divided into the perimeter rope 22 times to leave a new excess Y, which is noticeably half of X.

The new excess divides into the radius rope 7 times, with no excess, and into the perimeter rope 44 times, with no excess; leading to the most common ancient ratio for π in the megalithic world: the circumference accurately expresses 44 times the radius as 7, accurate to one part in 2500.

It was therefore possible for the megalithic astronomers to quantify π, and arrive at a “good” formulaic ratio for π of 22/7 (3.142857), which then means that any radius rope of the same seven units “could be counted on” as having 44 of the same units on its perimeter.

Taking an arbitrary size about one foot long, the radial rope can be made seven feet long and the perimeter must then be 44 feet long. And because the diameter of this circle is naturally 14 units, one can make a square of equal perimeter by taking a perimeter rope and marking its four equal sections, each then 11 feet long. One then has four corners that just exceed the circle (figure 2, left).

Figure 2

If a further rope of 11 feet is made, half that (5.5 feet) is the radius of another circle with the same center, which fits within the square (sometimes called the in-circle to the square, figure 2 right). The outer circle has a perimeter equal to that of the square and the diameter of the inner circle is 11 whilst the radius of the outer circle is 7. A wonderful example of this is to be found in the ideal ratio between the southern base of the Great Pyramid (756 feet or 440 Giza feet of 1.718 feet) and its uncapped height (481.09 feet or 440 Sumerian feet of 12/11 feet), the common unit being 68.72 feet (see figure 3). For this to be true, the Pyramid must have a relationship to our own foot length.

Figure 3

Between prehistoric experiments with circles and π and dynastic Egypt’s greatest pyramid, lies the Megalithic Period. During this period a system of fractional units of measure evolved which could solve a number of problems involving π that afterwards would be solved instead using the emerging arithmetical methods and number notations, instead of geometrical methods using lengths. My soon-to-be-released book, Sacred Geometry: Language of the Angels, reconstructs some of this story.

The 2nd lesson is about maintaining integers within circles.