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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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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An Angelic Geometrical Design

The above diagram contains information with can generally only be grasped by using a geometrical diagram. Its focus is the properties of a right triangle that is 4 times larger than its third and shortest side. The left hand view illustrates what we call Pythagoras’ theorum, namely that

“The squares of the shorter sides add up to the square of the longest side.”

Here this is shown as 144 + 9 = 153 because, if the third side is three lunar months long, then the 4-long base is 12 lunar months, hence the square of 12 is 144″. The longest side is then 153, the diagonal of the four squares rectangle, and the square root of 153 is 12.369 lunar months, the solar year when measured in lunar months.

Before Pythagoras, the Egyptians had a long tradition of geometrical mathematics which fed into their art in which designs can be seen to obey a grid of squares. Their view of Pythagoras’ theorum can therefore be put within a greater world of geometrical transforms using grids.

In the above, one can see this view (called Canevas by Schwaller de Lubicz, The Temple of Man) in which the larger square is seen to fit when angled into a 5-by-5 grid (see right). The extra width and height of the grid enables the smallest square to be seen in this common framework of 25 squares.

The largest square of area 153 is distinguished as an integer, rather than its square root. Thus this is not a Pythagorean triangle with all sides integral, but rather the two smaller sides being integer allows them to be placed within a grid. Somewhat rare though is the arising of an integer on the square, so that Jesus disciples in the gospel of John could comment, in being asked to throw their net on the right side, they then caught 153 fish!

If the diagram was in its least numbers, the 153 would be 9 times smaller as 17 and so the 12.369 would be √9 × √17 instead. And in sacred number science, the interaction of numbers can be seen to be determined by the prime numbers which then make larger numbers such as 153 = 9 × 17. This 17 is known to be a factor of the node cycle of 18.618 solar years, which is 6800 days long and 6800 = 400 × 17.

When two lengths of astronomical time share a larger prime such as 17, it indicates numerical compatibility between two periods, and so the solar year of √153 lunar months (in which the sun moves once around the Ecliptic) has some affinity with the 6800-day period during which its orbital nodes also move once through the Zodiac.

If the larger, yellow square has 6800 days within it, the square root is 20 × √17, whilst the square of the solar year had 153, the square root being 3 × √17.

The new imagined diagram would be 20/3 relative to the above one. Without explaining how this could be, the point is that this cannot be known by the human mind without using sacred geometry which can notate how a higher intelligence might have organised the time environment of Earth according to definite criteria. Further examples can be found in my Book, Sacred Geometry: Language of the Angels. The book is not about sacred geometry as a compendium of traditional knowledge but rather shows how it was that sacred geometry came into the human mind (and architecture) through the initial study of time periods as counted lengths, revealing angelic coincidences.

There is much else to know about the lunation triangle linking the lunar and solar years, discovered about 3 decades ago by my brother Robin Heath.

Geometry 4: Right Triangles within Circles

This series is about how the megalithic, which had no written numbers or arithmetic, could process numbers, counted as “lengths of days”, using geometries and factorization.

My thanks to Dan Palmateer of Nova Scotia
for his graphics and dialogue for this series.

This lesson is a necessary prequel to the next lesson.

It is an initially strange fact that all the possible right triangles will fit within a half circle when the hypotenuse equals the half-circles diameter. The right angle will then exactly touch the circumference. From this we can see visually that the trigonometrical relationships, normally defined relative to the ratios of a right triangle’s sides, conform to the properties of a circle.

A triangle with sides {3 4 5} demonstrates the general fact that, when a right triangle’s hypotenuse is the diameter of a circle, the right angle touches the circumference.
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