Geometry 3: Making a circle from a counted length

The number of days in four years is a whole number of 1461 days if one approximates the solar year to 365¼ days. This number is found across the Le Manio Quadrilateral (point N to J) using a small counting unit, the “day-inch”, exactly the same length as the present day inch. It is an important reuse of a four-year count to be able to draw a circle of 1461 days so that this period of four years can become a ouroboros snake that eats its own tale because then, counting can be continuous beyond 1461 days. This number also permits the solar year to be counted in quarter days; modelling the sun’s motion within the Zodiac by shifting a sun marker four inches every day.

Figure 1 How a square of side length 11 will equal the perimeter of a circle of diameter 14

Our goal then is to draw a circle that is 1461 day-inches in perimeter. From Diagram 1 we know that a rope of 1461 inches could be divided into 4 equal parts to form a square and from that, an in-circle to that square has a diameter equal to a solar year of 365¼ days. Also, with reference to Figure 1, we know that the out-circle will have a diameter of 14 units long relative to the in-circle diameter being 11 units long, and this out-circle will have the perimeter of 1461 inches that we seek.

Figure 3 A general method, using the equal perimeters model, applied to a 4 solar year day count of 1461 day-inches, found as a linear count at the Manio Quadrilateral. A square, formed from this linear count, can be transformed into an outer circle of equal perimeter using the simple geometry of π as 22/7.

For this, the solar year rope (the in-circle diameter) needs to be divided into 11 parts. Start by choosing a number that, when multiplied by 11, is less that 365 (and a 1/4). For instance, 33. A new rope will be formed, 11 x 33 = 363 inches, marked every 33 inches to provide 11 divisions. Through experience, we discover we need 2 identical ropes so as to make practical use of the properties of symmetry through attaching ropes to both ends of the solar diameter rope.

Place one rope at the West side of the in-circle diameter and swing it up until it touches the in-circle. Place the other rope at the East side of the in-circle diameter and swing it down until it touches the edge of the in-circle. Now connect the 33 inch marks between the 2 ropes. This will divide the 365 1/4 diameter into 11 segments.

Seven of those segments are the new radius to create the 1461 inch outer-circle.

Figure 3 Division of the in-circle into eleven equal parts so as to select 7 units as a radius rope to then form the circle of diameter 14 units and perimeter 1461 inches.

This novel application of the equal perimeters model, rescued from Victorian textbooks by John Michell and applied by him most memorably perhaps to Stonehenge and the Great Pyramid (in Dimensions of Paradise) is a general method for taking a counted length and reliably forming a radius rope able to transform that counted length into a circle of the same perimeter as the square, easily formed by four sides ¼ of the desired length.

The site survey at the start, drawn by Robin Heath, appeared in our survey of Le Manio.

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.

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Units within the Great Pyramid of Giza

There is a great way to express pi of 22/7 using two concentric circles of diameter 11 and 14 (in any units). Normally, a diameter of 7 gives rise to a circumference of 22, when pi is being approximated as 22/7 (3.142587) rather than being the irrational number 3.141592654 … for then, the 14 diameter should have a circumference of 44, which is also the perimeter of the square which encloses a circle of diameter 11.

The square of side 11 and
the circle of diameter 14
will both have the same perimeter.

Figure 1 The Equal Perimeter model of two circles, the smaller of which has an out-square of equal perimeter to the greater circle
Continue reading “Units within the Great Pyramid of Giza”