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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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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Models of Time within Henges and Circles

Presenting important information clearly often requires the context be shown, within a greater whole. Map makers often provide an inset, showing a larger map at a smaller scaling (as below, of South America) within a detailed map (of Southern Mexico).

This map is shown in the context of South America with a yellow rectangle which is the part blown up in scale. The subject is the Quetzal birds range which corresponds well to the Olmec then Maya heartlands leading to the god named Quetzalcoatl or Feathered Serpent. (see chapter 8 of Heath, 2018.)

Megalithic astronomy generated maps of time periods, using lines, triangles, diameters and perimeters, in which units of measure represented one day to an inch or to a foot. To quantify these periods, alignments on the horizon pointing to sun and moon events were combined with time counting between these events,where days, accumulated as feet or inches per day, form a counted length. When one period was much longer than another, the shorter could be counted in feet per day and the smaller in inches per so that both counts could share the same monumental space. In this article we find the culture leading to megalithic astronomy and stone circles, previously building circular structures called henges, made of concentric banks and ditches.

Thornborough Henge
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Palsson’s Sacred Image in Iceland

Extracted from The Structure of Metrology, its Classification and Application (2006) by John Neal and notes by Richard Heath for Bibal Group, a member of which, Petur Halldorsson, has taken this idea further with more similar patterns on the landscape, in Europe and beyond. Petur thinks Palsson’s enthusiasm for Pythagorean ideas competed with what was probably done to create this landform, as he quotes “Every pioneer has a pet theory that needs to be altered through dialogue.” Specifically, he “disputes the Pythagorean triangle in Einar’s theories. I doubt it appeared in the Icelandic C.I. [Cosmic Image] by design.” Caveat Emptor. So below is an example of what metrology might say about the design of this circular landform.

Figure 1 of Palsson’s (1993) Sacred Geometry in Pagan Iceland
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Megalithic Measurement of Jupiter’s Synodic Period

Though megalithic astronomers could look at the sky, their measurement methods were only accurate using horizon events. Horizon observations of solstice sunrise/set each year, lunar extreme moonrises or settings (over 18.6 years) allowed them to establish the geometrical ratios between these and other time periods, including the eclipse cycles. In contrast, the synod of Jupiter is measured between its loops in the sky, upon the backdrop of stars, in which Jupiter heads backwards each year as the earth passes between itself and the Sun. That is, Jupiter goes retrograde relative to general planetary direction towards the east. Since such retrograde movement occurs over 120 days, Jupiter will set 120 times whilst moving retrograde. This allowed megalithic astronomy to study the retrograde Jupiter, but only when the moon is conjunct with Jupiter in the night sky and hence will set with Jupiter at its own setting.

Figure 1 The metamorphosis of loop shape when Jupiter is in different signs of the Zodiac
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