Double squares: Venus and the Golden Mean

The humble square, with side length equal to one unit, is like the number one. It’s area is one square unit and, when we add another identical square to one side, the double square appears. Above right the Egyptian Djed column is shown within a double square. The Djed is the rotating earth which the gods and demons have a tug of war over. This is also a key story in the Indian tradition, called The Churning of the Oceans, where the churning creates both the food of the gods (soma) and every wonderful thing that emerges upon the Earth. In this, the double square symbolized the northern and southern hemispheres of the Earth. The anthropomorphic form Djed shown above has elbows indicative of the Double square.

Figure 1 The churning of the ocean (Samudra Manthan in Sanskrit)

The Djed appears to be the general principle of rotation of, and apparent motion around, the earth.

The god Isis is (as a planet) Venus and is shown (fig.2) offering up the sun disk: another Djed is below, with her Ankh symbol of Life atop the Djed, now having female arms . This sun most probably points to the practical year as 365 days which is 5/8 of the Venus synod of 584 days. (This ratio of 1.6 is the sixth note of the octave 1 to 2.)

In figure 2, two female attendants provide the duality which one might take to be her two famous manifestations of (firstly) the brightest Evening Star, as the sun goes down, and then (after that) the brightest Morning Star before the sun rises. Above there is duality again with three baboons either side of the sun, perhaps representing the six visible planets: Moon, Venus and Mercury: Jupiter, Saturn, Mars and their “tug of war”.

Figure 2 The creation of Horus-Ra from out of an ankh with female arms atop a djed. from Budge 1899, also fig. 7.8 of Richard Heath, The Harmonic Origins of the World.

The numbers 5 and 8 are Fibonacci approximations {1 2 3 5 8 13 21 34 …} to the golden mean, a transcendent number {1.618034…} which rational numbers can only approximate. Venus and the Earth have clearly settled into orbits around the sun resonant with Fibonacci ratios since the Venus orbital period (224.701 days) is 8/13 of the solar year. And it is this fact that eventuates in what we see on Earth, namely the manifestations of Venus every 8/5 of a practical year. of 365 days.

Figure 3 The double square, its in-circle and out-circle manifesting golden rectangles around itself.

In this post, I developed a result sent to me, that a square drawn within the upper hemisphere of a circle must define a golden mean rectangle either side from its height of 1 and the remaining radius of 0.618034… and so it can be seen that the divine principle of the Golden Mean emanates from the double square, either side of each square, when the double square is embraced by a circle drawn from its center. Obviously, on Earth and between orbits (of Venus and Earth), the Golden Mean (also called Phi) has to be approximated by whole number ratios but the principle is present within the geometry and its out-circle. Schwaller de Lubicz thought the dynastic Egyptians held the Golden Mean to be “the fundamental scisson” (literally “scissor cut”) in the range one to two and, its reciprocal can be seen to share the portion over 1 (figure 3).

One can see that geometry and the early numbers would have been seen as two aspects of what we call space and time, in which “things” are separate from each other in Existence but somehow conjoined within Eternity. What we call order is in fact an achievement of harmony made possible by the arranging and fitting of parts to form a coherent whole. It is this insight which gave meaning to their study of geometry and numbers from the prehistoric onwards, into the recorded history of early civilizations. The meaning for Life on Earth became encoded within ancient and prehistoric symbols, whose geometrical and numerical language of expression went to the heart of phenomena.

The Approximation of π on Earth

π is a transcendental ratio existing between a diameter/ radius and circumference of a circle. A circle is an expression of eternity in that the circumference, if travelled upon, repeats eternally. The earths shape would be circular if the planet did not spin. Only the equator is now circular and enlarged, whilst the north and south poles have a shrunken radius and, in pre-history, the shape of the earth’s Meridian between the poles was quantified using approximations of π as was seen in the post before last. In some respects, the Earth is a designed type of planet which has to have a large moon, 3/11 of the earth’s size and a Meridian of such a size that the diverse biosphere can be created within the goldilocks region of the Sun’s radiance.

It would be impossible to quantify the earth as a physical object without the use of approximations to π, a technique seen as emerging in Crucuno between its dolmen and famous {3 4 5} Rectangle where the 32 lunar months in 945 days was used, through manipulation of proximate numbers to rationalize the lunar month to 27 feet (10 Drusian steps) within which days could be counted using one Iberian foot (of 32/35 feet) as described here and in my Sacred Geometry book.

John Michell (1983) saw that different types of foot had longer and shorter versions, different by one 175th part and corresponding to the north-south width of two parallels of latitude: 51-52 degrees, which is the mean earth degree, and 10-11 degrees. The ratio 176/175 is interesting as for its primes.

  1. The harmonic primes {2 3 5} are 16/25 times 11/7.
  2. The 11/7 is half of the pi of 22/7 and the harmonic ratio is the inverse of 25/8.

From this it is clear that these two latitudes are related by the approximation to 1 of a π (22/7) and a reciprocal 1/π (8/25).

But John Neal (2000) saw that some feet also expressed 441/440 which is the ratio between the mean radius of the earth and its polar radius, visually clear in the Great Pyramid. This ratio is also the cancellation of two different πs, namely 63/20 and 7/22 since 7 x 63 = 441 and 20 x 22 = 440. From this emerged an ancient model of the earth that was embodied within the ancient metrology itself. I call this the metrological model rather than the (earlier) geometrical model based upon equal perimeters and the singular π of 22/7.

The metrological model gave a set of regular reference latitudes that accurately defined the geoid of the planet’s meridian by 2,500 BC. One can ask how those developing the model came across the idea of using proximate ratios of π like 176/175 and 441/440, since the system works so well that one may say that the meridian appears to have been designed that way.

The geometric model already defined the mean radius as 3960 miles and so that gives a mean earth meridian of 22 x twelve to the power six. One 180th of this gives a degree length of 364953.6 feet and this is only found at the parallel 51-52 degrees. It is this that defines the megalithic in England, Wales, Scotland and Ireland, an obvious candidate for the metrological survey whose complementary latitude was probably 175/176 of this (362880 feet) in Ethiopia, south of the Great Pyramid. The parallel of the Great Pyramid is 441/440 longer (362704.72) than that of Ethiopia while Athens and Delphi are 440/441 of the mean earth and Stonehenge parallel, that is 364126 feet.

This system was first set by Neal in All Done With Mirrors 2000 as I was writing my first book Matrix of Creation. Are we to think Neal made it up or are we dealing with an exact science that had developed through the megalithic enterprise. And if the Egyptians had an exact science of the earth’s geiod, what are we to make of the fact that the earth appears to follow such a numerically inspired pattern of relationships still true today, in the age of global positioning satellites.

One clue lies in the mind, and how ancient number sciences focus holistically upon the balancing mean. A mean earth that did not spin never existed, since it was only the collision with another planet which created the Moon 3/11 smaller than the Earth. The mean earth radius is these days established as the cube root of the equatorial radius squared times the polar radius. This is less, by 3024/3025, than the geometric model’s mean earth radius of 3960 miles, again maintaining rationality.

It would appear that, in entering the physical and spatial, any planetary design might have been based upon precise rational approximations, about the mean size, of π. To this mystery must be added the musical harmony of the outer planets to the Moon, the Fibonacci harmony of Venus to the Earth itself and the extraordinary numerical relationships of planetary time created by the Sun, Moon and Earth documented by my heavily-diagrammed books and website. From this, more and more can be understood about our prehistory and about its monuments.

Books on Ancient Metrology

  1. Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
  2. Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.
  3. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
  4. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
  5. —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.
  6. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
  7. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

Sacred Number and the Lords of Time

Back Cover

ANCIENT MYSTERIES

“Heath has done a superb job of collating his own work on the subject of megaliths with the objective views of many other researchers in the field. I therefore do not merely recommend reading this book but can state unequivocally it is a must read.”
–John Neal, British metrologist and researcher and author of Measuring the Megaliths and The Structure of Metrology

“In Sacred Number and the Lords of Time we have an important explanation of how megalithic science was developed. This book is a long-overdue wakeup call to a modern culture that has abandoned this fully developed and astonishingly rich prehistoric model of the physical world. The truth is now out.”
–Robin Heath, coauthor of The Lost Science of Measuring the Earth and author of Sun, Moon and Earth

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A Pyramidion for the Great Pyramid

image: By 1200 BC, the end of the Bronze Age, the Egyptian map of the world (above) showed nine bows or latitudes, numbers 4 to 9 including the Nile Delta, Delphi, Southern Britain and Iceland, a map based on an ancient geodetic survey.

This post explores a pyramidion, now lost, which exceeded the apex height of the pyramid, so as to model the different reference latitudes established by geodetic surveys and encoded within their metrology and the Great Pyramid (by 2500 BC). This pyramidion would have sat on the flat top of the pyramid, 480 feet above the base of the pyramid.

In All Done With Mirrors, John Neal described how the full height of the pyramid, reaching to its natural apex, would have been just over 481 feet. Most pyramids probably had a pyramidion since a number have been found elsewhere that repeat aspects of or have a name carved on them, of a specific pyramid. Sitting on their apex, they often repeat the form of the larger pyramid, and are scale models of a specific pyramid. In the case of the Great Pyramid, exactly 441th of its natural apex is missing, and this is likely to be because a pyramidion once stood on the flat top the actual pyramid.

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Recalibrating the Pyramid of Giza

Once the actual height (480 feet) and actual southern base length (756 feet) are multiplied, the length of the 11th degree of latitude (Ethiopia) emerges, in English feet, as 362880 feet. However, in the numeracy of the 3rd millennium BC, a regular number would be used. In the last post, it was noted that John Neal’s discovery of such rectangular numbers to define degrees of latitude, multiplied the pyramid’s pointed height (481.09 feet) by the southern base length (756 feet) to achieve the length of the Nile Delta degree of latitude and, repeating Neal’s diagram relating the key latitudinal degrees of the ancient Model as figure 1, the Ethiopian degree is 440/441 of the Nile Delta degree. As shown above, the length of the 756 foot southern base is changed, when re-measured in the latitudinal feet for Ethiopia; it becomes the harmonic limit of 720 feet of 1.05 feet – normally called the root Persian foot.

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Ethiopia within the Great Pyramid

My last posting mentioned John Neal’s creative step of not averaging the Great Pyramid of Giza’s four sides, as had routinely been done in the past – as if to discover an idealized design with four equal sides. Instead, Neal found each length to have intensionally been different. When multiplied by the pyramid’s full height, the length of four different degrees of latitude were each encoded as an area. The length of the southern side is integer as 756 feet, and this referred to the longest latitude, that of the Nile Delta, below 31.5 degrees North. Here we find that the pyramid’s reduced height also indicated the latitude of Ethiopia.

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