Month: July 2022

Counting Perimeters

above: a slide from my lecture at Megalithomania in 2015

We know that some paleolithic marks counted in days the moon’s illuminations, which over two cycles equal 59 day-marks. This paved the way for the megalithic monuments that studied the stars by pointing to the sky on the horizon; at the sun and moon rising to the east and setting in the west. It was natural then to them to see the 12 lunar months (6 x 59 = 354 day-marks) within the seasonal year (about 1/3 of a month longer than 12) between successive high summers or high winters.

Lunar eclipses only occur between full moons and so they fitted perfectly the counting of the repetitions of the lunar eclipses as following a fixed pattern, around six months apart (actually 5.869 months, ideally 173.3 day-marks apart). The accuracy of successive eclipse seasons to the lunar month can then improve over longer counts so that, after 47 lunar months, one can expect an eclipse to have occurred about one and a half days earlier. This appears to be the reason for the distance between the megalithic monuments of Crucuno, its dolmen and and its rectangle, which enabled simultaneous counting of days as Iberian feet and months as 27 foot units, at the very end of the Stone Age.

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The Best Eclipse Cycle

The anniversary of the Octon (4 eclipse years in 47 lunar months) did not provide similar eclipses and so, by counting more than four, the other motions of the Moon could also form part of that anniversary. This is especially true of the anomalistic month, which changes the changes the apparent size of the Moon within its phase cycle, recreate the same type of lunar eclipse after nineteen eclipse years. This 18 year and 11 day period is now taken as the prime periodicity for understanding eclipse cycles, called the Saros period – known to the Babylonian . The earliest discovered historical record of what is known as the saros is by Chaldean (neo-Babylonian) astronomers in the last several centuries BC.

The number of full moons between lunar eclipses must be an integer number, and in 19 eclipse years there are a more accurate 223 lunar months than with the 47 of the Octon. This adds up to 6585.3 days but the counting of full moon’s is obviously ideal as yielding near-integer numbers of months.

We noted in a past post that the anomalistic month (or AM), regulating the moon’s size at full moon, has a geometrical relationship with eclipse year (or EY) in that: 4 AM x pi (of 3.1448) equals the 346.62 days of the eclipse year as the circumference. Therefore, in 19 EY the diameter of a circle of circumference 19 x 346.62 days must be 4 x 19 AM so that , 76 AM x pi equals 223 lunar months, while the number of AM in 223 lunar months must be 239; both 223 and 239 being prime numbers.

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Vectors in Prehistory 2

In early education of applied mathematics, there was a simple introduction to vector addition: It was observed that a distance and direction travelled followed by another (different) distance and direction, shown as a diagram as if on a map, as directly connected, revealed a different distance “as the crow would fly” and the direction from the start.

The question could then be posed as “How far would the plane (or ship) be, from the start, at the end”. This practical addition applies to any continuous medium, yet the reason why took centuries to fully understand using algebraic math, but the presence of vectors within megalithic counted structures did not require knowledge of why vectors within geometries like the right triangle, were able to apply vectors to their astronomical counts.

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Vectors in Prehistory 1

In previous posts, it has been shown how a linear count of time can form a square and circle of equal perimeter to a count. In this way three views of a time count, relative to a solar year count, showed the differences between counts that are (long-term average) differential angular motion between sun and the moon’s cycle of illumination. Set within a year circle, this was probably first achieved with reference to the difference between the lunar year of 12 months (29.53 days) and the solar year of 12 average solar months (30.43 days). Note that in prehistory, counts were over long periods so that their astronomy reflected averages rather than moment-to-moment motions known through modern calculations.

The solar year was a standard baseline for time counting (the ecliptic naturally viewed as 365.25 days-in-angle, due to solar daily motion, later standardized as our convenient 360 degrees). Solar and other years became reflected in the perimeters of many ancient square and circular buildings, and long periods were called super years, even the Great Year of Plato, of the precession of the equinoxes, traditionally 25920 years long! The Draconic year, in which the Moon’s nodes travel the ecliptic, backwards, is another case.

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How Geometries transformed Time Counts into Circles

Above: example of the geometry that can generate one or more circles,
equal to a linear time count, in the counting units explained below.

It is clear, one so-called “sacred” geometry was in fact a completely pragmatic method in which the fourfold nature of astronomical day and month counts allowed the circularization of counts, once made, and also the transmission of radius ropes able to make metrological metrological circles in other places, without repeating the counting process. This “Equal Perimeter” geometry (see also this tag list) could be applied to any linear time count, through dividing it by pi = 22/7, using the geometry itself. This would lead to a square and a circle, each having a perimeter equal to the linear day count, in whatever units.

And in two previous posts (this one and that one) it was known that orbital cycles tend towards fourfold-ness. We now know this is because orbits are dynamic systems where potential and kinetic energy are cycled by deform the orbit from circular into an ellipse. Once an orbit is elliptical, the distance from the gravitational centre will express potential energy and the orbital speed of say, the Moon, will express the kinetic energy but the total amount of each energy combined will remain constant, unless disturbed from outside.

In the megalithic, the primary example of a fourfold geometry governs the duration of the lunar year and solar year, as found at Le Manio Quadrilateral survey (2010) and predicted (1998) by Robin Heath in his Lunation Triangle with base equal to 12 lunar months and the third side one quarter of that. Three divides into 12 to give 4 equal unit-squares and the triangle can then be seen as doubled within a four-square rectangle, as two contraflow triangles where the hypotenuse now a diagonal of the rectangle.

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The Octon of 4 Eclipse Years

Having seen, in the last post, that three eclipse years fitted into the three-year count at Le Manio, another eclipse fact has come to light, recorded within the nearby site of Crucuno, between its dolmen and rectangle. The coding of time at Crucuno was an evolution of a new metrology based upon the English foot in which, the right triangle of longest integer side lengths was replaced by fractions of a foot using the same two numbers as the sides would have had. This allowed the measurement of a time period to be simultaneously seen in both days and months. That this was possible can be seen at Le Manio, where it could be noticed that 32 lunar months equaled exactly 945 day-inches.

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