Category: Sacred Geometry: Language of the Angels

Released in January 2021, this book explores sacred building as the continuation of the megalithic: its astronomical knowledge, geometrical and metrological methodology and knowledge of the size of the Earth. The megalithic was our first numerate culture which discovered the main design design parameters for our existence, organised by higher intelligences called angels.

The Megalithic Pythagoras

Pythagoras of Samos (c.600BC) very likely gleaned megalithic number science on his travels around the “Mysteries” of the ancient world. His father, operating from the island of Samos, became a rich merchant, trading by sea and naming his child Pythagoras; after the god of Delphi who had “killed” the Python snake beneath Delphi’s oracular chasm, now a place of Apollo. The eventual disciples of Pythagoras were reclusive and secretive, threatening death on anybody who would openly speak of mysteries, such as the square root of two, to the uninitiated. It can be seen from the previous post that many such “mysteries” were natural discoveries made by the megalithic astronomers, when learning how to manipulate number without arithmetic, through a metrological geometry unfamiliar to the romantic sacred geometry of “straight edge and compass”.

As previously stated, the vertex angles of right triangles whose longer sides are integer in length, are angular invariants belonging to the invariant ratio of their sides. To create a {11 14} angle one can use any multiple of 11 and the same multiple of 14 to obtain the invariant angle whereupon, the hypotenuse and base will shrink or grow together in that ratio: any length on the “14” line is 14/11 of any length below it on the “11” base line and visa versa.

If one enlarges the base line to being 99 then the diagonal of the square side length 99 will be 140, which is 99 times the square root of two. In choosing, as I did, to enlarge 91 (the quarter year) to 9 x 11 = 99, I encountered the cubit of the Samian (“of Samos”) foot of 33/35 feet, as follows. When Heraclitus, also of Samos, visited the Great Pyramid he gave its southerly side length as 800 “of our feet” and 756 English feet (the measured length) needs to be divided by 189 and multiplied by 200 to obtain such a measurement, giving a Samian foot of 189/200 (=0.945 feet) which is 441/440 of the Samian root foot of 33/35 feet. 33/35 x 3/2 = 99/70 (1.4143) feet but its inverse of 35/33 x 4/3 = 140/99 feet.

There is then no doubt about Samos as being a center in the Greek Mysteries since, the form of the Greek temple seems first to evolve there. For example, 10,000 feet of 0.945 feet equal 945 feet, the number of days in 32 lunar months. The Heraion of Samos (pictured above) has been shown to have had pillars around a platform (a peristyle), and an elongated rectangular room (a cella), involving megalithic yards and a 4-square geometry cunningly linking lunar and solar years, to alignments to the Moon’s minimum using the {5 12 13} second Pythagorean Triangle. (diagram at top is from figure 5.9 of Sacred Geometry: Language of the Angels).

The reason for the Samian (lit. “of Samos”) foot being 33/35 feet appears to be that as a cubit of 99/70 feet, or √2 =1.4142, it is the twin of 140/99 as 1.41. In the geometrical world such foot ratios were exact, relative to the English foot; which is the root of the Greek module and of all other rational modules, such as the Royal of 8/7 feet. Such cubits could measure across the diagonal the same number as the side length in English feet. Such measures became essential for building of rectangular temple structures in Greece and further east, but when the metrological geometry, of square and circle in equal perimeter, was the focus, 140 in the diagonal can use 99 in the base (or side-length of the square).

If we remember that the 99 length must be rooted from the shared center of the square and equal circle then, the side length of the square must be twice that, or 198. This means that the perimeter of the square must be 4 times that, equal to 792, at which point readers of John Michell’s books on models of the world will recall that the diameter of the mean earth can be presented, within an equal perimeter design, if each unit is multiplied by 720 units of 10 miles, my own summary being in my recent Sacred Geometry book , chapter 3 on measuring the Earth. This model Michell called The Cosmological Prototype, where the mean earth diameter is (quite accurately) 7920 miles.

If the square of 198 feet is rolled out into a single line, it “becomes” the mean diameter of the Earth in units of 10 miles. For this sort of reason, my 2020 book was called Language of the Angels, since this model looks like a first approximation of the mean earth size which a later Ancient Metrology would improve upon as to accuracy, by a couple of miles! That is, that the earth’s dimensions follow a design based upon metrological geometry and the properties of numbers.

John Michell finalized his Cosmological Model in an Appendix to The Sacred Center, and in his text on “sacred Geometry, Ancient Science, and the Heavenly Order on Earth” called The Dimensions of Paradise, both published by Inner Traditions.

Seven, Eleven and Equal Perimeters

above: image of applications involving sacred geometry based upon pi as 22/7 and a circle of equal perimeter to a square, from a previous post.

The geometrical and other relationships between different numbers are easily found to be useful through simple experiments. The earliest approximations to pi (22/7) was key in the megalithic and later ancient cultures, for making circles of a known diameter and circumference, the foremost using the numbers 7 and 11 doubled twice. A staked rope of length seven will create a circumference of 44, to a high degree of accuracy.

But what is pi? it actually connects two different worlds, of extensive linear measure and of intensive rotational measure. As the radius rope is made larger the circle expands from its center but it remains a whole circle, except that its circumference is made up of more “units” all according to the ratio pi = 22/7, in a good approximation.

But measuring a circumference is fiddly, it is circular! In contrast, it is very much easier to work with squares since their perimeter is four times their side length. And in many cases, one does not really need to measure the perimeter. Because of this, the megalithic looked for and discovered an easier procedure in which one could know the circumference of a circle if one could generate the square that has the same circumference now called the equal perimeter model. This was surprisingly simple to grasp and implement.

First of all, one can lay out a linear length, that divides by 4, lets say 28 which is 4 x 7. The length is made up of four lengths, each of 7 units and, a square of side length 7 will have a perimeter of 28, same as the linear length. The square is really just a rolled-up set of 4 lengths at right angles!

The diameter of a circle with 28 units on its circumference must be larger than its incircle of diameter 7 and, if pi is 22/7 then, the diameter will be exactly 14/11 of the side length. Notice that 14/11 is cancelling the seven and eleven in pi as 22/7.

The equal perimeter rope will be staked in the very center of the square. The side of 7 is then 7 x 14/11 or 98/11 units and this, times 22/7 equals 28 – the perimeter of both the circle, and square side-length 7 units. There is no need to calculate this if one draws a triangle ratio {11 14} from the center of the square. This triangle’s slope angle automatically “calculates” or reproportions the cardinal length (whatever this is) into a suitable rope (or radiant) length.

One often does not need to form the circle to know what its perimeter would be through measurement. Once one knows that every square has a twin circle of the same perimeter, this changes thinking. This is particularly significant when forming a circular model of the sun’s path in the year. If the “saturnian” year 364 days was used, it unusually divides by 28 days, and 13, and 7 days; the seven-day week. The square would have a side length of 13 weeks (91 days) and the radius rope would need to be (13 x 7) x 7/11 which, times 44/7 reconstitutes the circumference of 364 days.

My book Sacred Geometry: Language of the Angels has much to say on equal perimeter modelling, which is found throughout the ancient building traditions that followed on from the megalithic period, using the older techniques of metrological geometry alongside the development of arithmetic methods. Click on the Bookshop logo or Google, and find out more.

The Megalithic Numberspace

above: counting 37 lunar months six times to reach 222,
one month short of 223: the strong Saros eclipse period.

There is an interesting relationship between the multiple interpretations of a number as to its meaning, and the modern concept of namespace. In a namespace, one declares a space in which no two names will be identical and therefore each name is unique and this has to be so that, in computer namespaces such as web domain names, the routes to a domain can be variable but the destination needs to be a unique URL.

If sacred numbers had unique meanings then they would be like a namespace. Instead, being far more limited in variety, sacred numbers have more meanings, or interpretations, just as one might say that London has many linkages to other cities. In an ordinal number set, there are many relationships of a number to all the other numbers. This means whilst their are infinite numbers in the set of positive whole numbers, there are more than an infinity of relationships between the members of that set, such as shared number factors or squares, cubes, etc. of a number.

The mathematician Georg Cantor saw “doubly infinite” sets. Sets of relationships between members of an already infinite set, must themselves be more than infinite. He called infinite sets as aleph-zero and the sets of relationships within an infinite set (worlds of networking), he called aleph-one.

Originally, Cantor’s theory of transfinite numbers was regarded as counter-intuitive – even shocking.

Wikipedia

However, in the world of sacred numbers, although there can be large numbers, in the megalithic the numbers were quite small; partly due to the difficulty that numbers-as-lengths were physically real while later numeracy abstracted numbers into symbols and, using powers of ten, modern integers are a series of place ordered numbers (not factors) in base 10, as with 12,960,000 – possible for the ancient Babylonians but, I believe, not expected for the early megalithic.

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Knowing Time in the Megalithic

The human viewpoint is from the day being lived through and, as weeks and months pass, the larger phenomenon of the year moves the sun in the sky causing seasons. Time to us is stored as a calendar or year diary, and the human present moment conceives of a whole week, a whole month or a whole year. Initially, the stone age had a very rudimentary calendar, the early megalith builders counting the moon over two months as taking around 59 days, giving them the beginning of an astronomy based upon time events on the horizon, at the rising or setting of the moon or sun. Having counted time, only then could formerly unnoticed facts start to emerge, for example the variation of (a) sun rise and setting in the year on the horizon (b) the similar variations in moon rise and set over many years, (c) the geocentric periods of the planets between oppositions to the sun, and (d) the regularity between the periods when eclipses take place. These were the major types of time measured by megalithic astronomy.

The categories of astronomical time most visible to the megalithic were also four-fold as: 1. the day, 2. the month, 3. the year, and 4. cycles longer than the year (long counts).

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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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