Form implied by the Kaaba’s Walls

The Kaaba appears to express a geometrical progression of adjacent odd numbers starting with one and three. This differs from the super-particular ratios found within the right triangles of astronomical time periods formed by the Megalithic, in which the ratio pairs separated by only one rather than by two, between only odd numbers. However, the multiple-square rectangles used by the megalithic to approximation celestial ratios, made use of the three-square rectangle. In one of the smallest of these rectangles, it simultaneously approximates two pairs of ratios: The eclipse year (346.62 days) to the solar year (365.2422 days) and the solar year to the thirteen lunar month year (384 days).

Figure 1 The three-square approximations in a triple series of astronomical periods. Note that the diagonals relative to the base are the result of having three squares in a rectangle then one high and three along – and two different. Two such rectangles geometrically sum (their angles) to give that of the (First Pythagorean) 3-4-5 triangle, 36.8 degrees.
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The Golden Mean compared to PI

In reviewing some ancient notes of mine, I came across an interesting comparison between the Golden Mean (Phi) and PI. They are more interesting in reverse:

A phi square (area: 2.618, side: 1.618) has grown in area relative to a unit square by the amount (area: 0.618) plus the rectangle (area:1 ). This reveals the role of phi’s reciprocal square (area: 0.384) in being the reciprocal of the reciprocal so that in product they return the unity (area: 1).

On the right, the phi squared square showing how the reciprocal of phi and its square uniquely sum to unity (area: 1), a property that is scale invariant between structures who share the same units and grow according to the Golden Mean.
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Astronomical Time within Clava Cairns

In North East Scotland, near Inverness, lies Balnuaran of Clava, a group of three cairns with a unique and distinctive style, called Clava cairns; of which evidence of 80 examples have been found in that region. They are round, having an inner and outer kerb of upright stones between which are an infill of stones. They may or may not have a passageway from the outer to the inner kerb, into the round chamber within. At Balnuaran, two have passages on a shared alignment to the midwinter solstice. In contrast, the central ring cairn has no passage and it is staggered west of that shared axis.

This off-axis ring cairn could have been located to be illuminated by the midsummer sunrise from the NE Cairn, complementing the midwinter sunset to the south of the two passageways of the other cairns. Yet the primary and obvious focus for the Balnuaran complex is the midwinter sunset down the aligned passages. In fact, the ring cairn is more credibly aligned to the lunar minimum standstill of the moon to the south – an alignment which dominates the complex since, in that direction the horizon is nearly flat whilst the topography of the site otherwise suffers from raised horizons.

Cairns at Balnuaran of Clava. plan by A. Thom and pictures by Ian B. Wright
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Use of foot ratios in Megalithic Astronomy

The ratios of ancient metrology emerged from the Megalithic innovations of count&compare: counting time as length and comparing lengths as the longest sides of right triangles. To compare two lengths in this way, one can take a longer rope length and lay it out (say East-West), starting at the beginning of the shorter rope length, using a stake in the ground to fix those ends together.

The longer rope end is then moved to form an angle to the shorter, on the ground, whilst keeping the longer rope straight. The Right triangle will be formed when the longer rope’s end points exactly to the North of the shorter rope end. But to do that one needs to be able to form a right angle at the shorter rope’s end. The classic proposal (from Robin Heath) is to form the simplest Pythagorean triangle with sides {3 4 5} at the rope’s end. One tool for this could then have been the romantic knotted belt of a Druid, whose 13 equally spaced knots could define 12 equal intervals. Holding the 5th knot, 8th knot and the starting and ending knots together automatically generates that triangle sides{3 4 5}.

Forming a square with the AMY is helped by the diagonals being rational at 140/99 of the AMY.
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Old Yard’s Mastery of the Square Root of 2

The old yard was almost identical to the yard of three feet, but just one hundredth part smaller at 2.87 feet. This gives its foot value as 99/100 feet, a value belonging to a module very close to the English/Greek which defines one relative to the rational ratios of the Historical modules.

So why was this foot and its yard important, in the Scottish megalithic and in later, historical monuments?

If one forms a square with side equal to the old yard, that square can be seen as containing 9 square feet, and each of those has side length 99/100 feet. This can be multiplied by the rough approximation to 1/√ 2 of 5/7 = 0.714285, to obtain a more accurate 1/√ 2 of 99/140 = 0.70714285.

Figure 1 Forming a Square with the Old Yard. The diagonal of the foot squares is then 7/5, the simplest approximation to √ 2.
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A Lecture introducing Sacred Number and the Lords of Time

… given as the John Michell Memorial Lecture for
at Megalithomania 2015 in Glastonbury

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Thornborough Henge as Moon’s Maximum Standstill

The three henges appear to align to the three notable manifestations to the north west of the northerly moon setting at maximum standstill. The distance between northern and southern henge entrances could count 3400 days, each 5/8th of a foot (7.5 inches), enabling a “there and back again” counting of the 6800 days (18.618 solar years/ 19.618 eclipse years) between lunar maximum standstills.

Figure 1 The three henges are of similar size and design, a design most clear in what remains of the central henge.
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The Tetraktys as plan of planetary harmony and the four Elements

In "Planetary Resonances with the Moon" I explored the astronomical matrix presented in The Harmonic Origins of the World with a view to reducing the harmonic between outer planets and the lunar year to a single harmonic register of Pythagorean fifths. This became possible when the 32 lunar month period was realized to be exactly 945 days but then that this, by the nature of Ernest McClain’s harmonic mountains (figure 1) must be 5/4 of two Saturn synods.

Figure 1 The elimination of 5 as a factor in the harmonic mountain for 36 lunar years, resolved using matrix units of one tenth of a month and the limit 360 units.

Using the lowest limit of 18 lunar months, the commensurability of the lunar year (12) with Saturn (12.8) and Jupiter (13.5) was “cleared” using tenths of a month, revealing Plato’s World Soul register of 6:8::9:12 but shifted just a fifth to 9:12::13.5:18, perhaps revealing why the Olmec and later Maya employed an 18 month “supplementary” calendar after some of their long counts.

By doubling the limit from 18 to three lunar years (36) the 13.5 is cleared to the 27 lunar months of two Jupiter synods, the lunar year must be doubled (24) and the 32 lunar month period is naturally within the register of figure 1 whilst 5/2 Saturn synods (2.5) must also complete in that period of 32 lunar months.

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Number Symbolism at Table des Marchands

Table des Marchands, a dolmen at Lochmariaquer, can explain how the Megalithic came to factorise 945 days as 32 lunar months by looking at the properties of the numbers three, four and five. At that latitude, the solstice angle of the sun on the horizon shone along the 5-side of a 3-4-5 triangle to east and west, seen clearly at the Crucuno Rectangle "Lunar Counting from Crucuno Dolmen to its Rectangle".

Before numbers were individually notated (as with our 3, 4 and 5 rather than |||, |||| and |||||) and given positional notation (like our decimal seen in 945 and 27), numbers were lengths or marks and, when marks are compared to accurately measured lengths measured out in inches, feet, yards, etc. then each vertical mark would naturally have represented a single unit of length. This has not been appreciated as having been behind marks like the cuneiform for ONE; that it probably meant “one unit of length”.

Figure 1 The end and cap stone inside the dolmen Table des Marchands in which the elementary numbers in columns and rows perhaps inspired its attribution to the accounts of merchants
Locmariaquer (Morbihan, Bretagne, France) : la Table des Marchand, interieur.
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Earth and Moon within Westminster’s Coronation Pavement

Our modern globes are based upon political boundaries and geographical topography yet they have geometrical predecessors, which described the world as an image, diagram or schemata. The original idea for the form of the world was summarised within a simple two dimensional geometry, like an eastern mandala or yantra.

Such a diagram was built into the Cosmati pavement at Westminster Abbey, built by Henry III and dedicated to the Saxon King and Saint Edward the Confessor. This exotic pavement became the focus for the Coronations of subsequent English then British monarchs.

Figure 1 Photo of the Cosmati Pavement at Westminster Abbey
[Copyright: Dean and Chapter of Westminster]
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Musical Tones of the Outer Planets

My crucial entré to planetary harmony came when I noticed musical ratios in the synodic time periods of Jupiter and Saturn relative to the lunar year. This approach differs from the norms for “harmonies of the spheres” (a.k.a. Musica Universalis) which are geometrical and spatial, rather than temporally harmonic.

The planetary harmony I found within synodic periods became the subject of my new book The Harmonic Origins of the World (pub. 2018). These synodic ratios have been parts of my work from c. 2000, then expressed as “matrix diagrams” (Matrix of Creation, figure 2 below). In my new book, I show how ancient tuning theory seems to have presented the same information, in a different type of matrix (see figure 4).

Below I connect the outer planets using two additional (and useful) kinds of diagram, the right-angled triangle (figure 1) and the Pentad (figure 5), the latter developed in the 20th century within a discipline called Systematics. 

Figure 1 The harmonic ratios between the nearest two outer planets and the lunar year. The four square rectangle with side length of four, when equal to the lunar year gives, geometrically, the solar year as its diagonal length. The outer planetary synods are longer than the solar year as the planets have moved ahead of their last opposition to the sun. Such oppositions are marked by an outer planet appearing to travel in a loop, amongst the stars
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The Roof Axe as Circumpolar Device

This article explores the use of axe motifs within a form of carved schematic art unique to the megalithic monuments near Carnac, southern Brittany, France. First published in February 2014.

A diagram found on the underside of the capstone of a chambered dolmen called Kercado (see figure 1) appears to hold metrological and astronomical meanings. Classified as a type of AXE, local axe motifs are said to have three distinct forms (a) triangular blades, (b) hafted axes and (c) the Mane Ruthual type [Twohig, 1981[1]]. 

Figure 1 Well preserved sculpted-stone axe-head motif in Kercado dolmen

Types b and c are often found in the singular on the undersides to roof slabs and in the case of form (b), the hafted axe, I have attributed its display below the roof slab of Table des Marchands at Locmariaquer (inset right) as being used to represent the north pole between 5000 and 4000 BC, at a time when there was no star near to the pole itself. The abstract point of the north pole, the rotational axis of the earth, is shown as a loop attached to the base of the axe haft, whilst the axe head then represented a chosen circumpolar star, as this rotates counter-clockwise in the northern sky, at the fixed distance of the haft from the pole itself. Note how compatible this idea of an axe ploughing the northern skies is to our own circumpolar constellation, The Plough. Note also that the eastern horizon moves through the equatorial stars at the same angular rate as the marker star moves around the north pole.

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Sacred Latitudes of the West 1.

first published on Sunday, 19 September 2010 10:35

Adventures in Geodetic Imagination

At the heart of Sacred Number and the Origins of Civilization lay the story of the Secret Men of the North, which followed the west-to-east path of a European Michael line, in the sense of the original invasion of the Indo-Europeans of the Baltic into the Mediterranean and as the later axis this provided for the Normans to orchestrate the Crusades; ostensibly to re-take Jerusalem from the world of Islam, that was also competing over Europe, from Spain, Sicily and Levant.

Whilst working on a continuation of such a geodetic story, the concept of Sacred Latitudes emerged in which parallels of latitude might have some psycho-historical relevance, based on the original insight that, in the last century, “manifestations of Mary” have emerged, first in Garabandal in Spain and recently in Medjugorje, in Bosnia, that are on exactly the same line of latitude, 43 degrees and 12 minutes. (brought to my attention by the late John D. Kirby’s studies of “Mary places”.)

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Venus: Planet of Harmony: part 1

Venus has played a strong role in mankind’s imagination, being a bright object in the sky in the evening sky and then the morning sky, whilst also viewed as the primary female goddess of the Ancient Near East. To recent astronomers, she is covered in impenetrable clouds, whilst the invention of radar revealed a rocky sister planet to Earth but with no life as we know it. It is perennially associated with the pentagon, because its synodic periods draw out a pentagon within the zodiac in 8 solar years. The reasons it does so are intriguing to say the least, and we explore the unusual numerical characteristics of Venus seen from Earth.

(adapted from a 1994 text, using 2020 hindsight)

The Venus cycle of eight years in which a morning or evening star has five manifestations, dividing the zodiac into a pentagram (Figure 1.6 Sacred Number and the Origins of Civilization)

Part 1: A Nearly Golden Mean

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