Geometry 5: Easy application of numerical ratios

above: Le Manio Quadrilateral

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.

The last lesson showed how right triangles are at home within circles, having a diameter equal to their longest side whereupon their right angle sits upon the circumference. The two shorter sides sit upon either end of the diameter (Fig. 1a). Another approach (Fig. 1b) is to make the next longest side a radius, so creating a smaller circle in which some of the longest side is outside the circle. This arrangement forces the third side to be tangent to the radius of the new circle because of the right angle between the shorter sides. The scale of the circle is obviously larger in the second case.

Figure 1 (a) Right triangle within a circle, (b) Making a tangent from a radius.

Figure 1 (a) Right triangle within a circle, (b) Making a tangent from a radius.

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The Tetraktys as plan of planetary harmony and the four Elements

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.

In a previous post 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.

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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Megalithic application of numeric time differences

Natural time periods between celestial phenomena hold powerful insights into the numerical structure of time, insights which enabled the megalith builders to access an explanation of the world unlike our own. When looking at two similarly-long time-periods, the megalithic focussed on the difference between them, these causing the two periods to slide in and out of phase, generating a longer period in which the two celestial bodies exhibit a complete ensemble of variation, in their relationship to each other. This slippage of phase between celestial periods holds a pattern purely based upon number, hidden from the casual observer who does not study them in this way. Such numerical patterns are only fully revealed through counting time and analysing the difference between periods numerically.

For example, the solar year is longer than the lunar year by 10 and 7/8 days (10.875 days) and three solar years are longer than three lunar years by three times 10.875 days, that is by 32 and 5/8th days (32.625 days), which is 32/29 of a single lunar month of 29.53 days.

The earliest and only explicit evidence for such a three year count has been found at Le Manio’s Quadrilateral near Carnac (circa 4,000 BCE in Brittany, France) used the inches we still use to count days, a “day-inch” unit then widespread throughout later megalithic monuments and still our inch, 1/12 of the foot [Heath & Heath. 2011]. The solar-lunar difference found there over three years was 32.625 day-inches, is probably the origin of the unit we call the megalithic yard and the megalith builders appear to have adopted this differential length, between a day-inch count over three lunar and solar years, in building many later monuments.

Figure 1 (in plan above) The monumentalising of a three-year day inch count at Le Manio as a right triangle based upon its southern kerb (in profile below), automatically generating the megalithic yard.
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Lunar Counting from Crucuno Dolmen to its Rectangle

Figure 1 The entrance of Crucuno’s cromlech, which opens to the south-east
[Summer Solstice, 2007]

It is not immediately obvious the Crucuno dolmen (figure 1) faces the Crucuno rectangle about 1100 feet to the east. The role of dolmen appears to be to mark the beginning of a count. At Carnac’s Alignments there are large cromlechs initiating and terminating the stone rows which, more explicitly, appear like counts. The only (surviving) intermediate stone lies 216 feet from the dolmen, within a garden and hard-up to another building, as with the dolmen (see figure 2). This length is interesting since it is twice the longest inner dimension of the Crucuno rectangle, implying that lessons learned in interpreting the rectangle might usefully apply when interpreting the distance at which this outlier was placed from the dolmen. Most obviously, the rectangle is 4 x 27 feet wide and so the outlier is 8 x 27 feet from the dolmen.

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Planetary Resonances with the Moon

Readers of my article [post2post id=”327″] will be familiar with the finding that in 32 lunar months there are almost exactly 945 days, leading to the incredibly accurate proximation (one part in 45000!) for the lunar month of 945/32 = 29.53125 days.

In the previous article on Seascale I noticed that 36 lunar months (three solar years) divided by 32 lunar months is the Pythagorean tone of 9/8. This led me to important thoughts regarding the tuning matrix of the Moon within the periods of the three outer planets, since the synod of Jupiter divided by the lunar year of 12 lunar months is the same tone, the tone that on “holy mountains” of Ernest G. McClain’s ancient tuning theory. Such tones are only found between two tonal numbers separated by two perfect fifths of 3/2, since 3/2 x 3/2 = 2.25 which, normalised to the octave of 1 to 2, is 1.125 or 9/8.

Figure 1 If the matrix unit is one tenth of the lunar month, then three lunar years becomes 360 units which, taken to be high do or D” = the harmonic limiting number, presents the matrix above, in the style proposed as indicative of Ancient Tuning Theory by Ernest McClain (see his The Myth of Invariance).  This Harmonic Matrix for 360 = 36 months shows that the 32 lunar month period starts row 2 as 320.
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