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 geodeticUnits of measures and monumental measurements relating to the numerical definition of the shape of the Earth by the late megalithic. 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 metrologyThe application of units of length to problems of measurement, design, comparison or calculation. 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.

441/440 is an important ratio due the Earth’s spin deforming it from spherical, reducing the Earth’s polar radius and expanding its equatorial radius, to make the mean radius of the Earth 441/440 of the polar radius. The pyramidion would have had something to do with this ratio, possible adding 1/440 to reach the true apex of its geometry. However, as figure 1 below shows, the differences between two given degree lengths, in three cases, also equals 441/440. Thus, the differences between degree lengths (rather than the ratio of two different earth radii) could also have been the reason why the actual Great Pyramid is 440 units high to its ideal apex of 441 units high (the shared units being 12/11 feet long).

Neal also noted that the side lengths of the pyramid’s base were deliberately made unequal in length, so that the apex height when multiplied by all four sides generated the latitudinal degree lengths for 31, 30, 29 and 27 degrees north; for more see this previous post. In a second post I noticed the actual height of 480 feet also multiplies with the southern side of 756 feet to give the degree length for Ethiopia. Other possibilities could have been exploited by the pyramidion by *exceeding* the apex height of the pyramid, to model the Old Kingdom’s reference latitudes seen in figure 1. These latitudes must have been measured by geodetic surveys, and these survey results were then encoded in the Great Pyramid, as a geodetic monument exploiting rectangular numbers.

The differences between latitudes were also built into the metrological system of different foot ratios, as the micro-variations found within historical metrology. However, these micro-variations were already known to the megalithic of Brittany and Britain, but they were already employed for modelling the ratio between a circle’s perimeter and the radius length which can draw a circle from a central stake, using a radius rope. From this one can deduce that a search had been made, by the epoch of the Great Pyramid’s construction, for those specific ratios that conformed to the value of π as 22/7The simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods. whilst maintaining integers between the radii and perimeters of circles. That is, these prehistoric surveys appear to have *looked for *degrees of latitude which *could be related *by 441/440, 176/175Ratio crucial to maintaining integers (see geometry lesson 2) between radii and circumference of a circle, and crucial to the micro-variation of foot modules in ancient metrology. and their product 126/125!

A number of reference latitudes were therefore established, where 176/175, 441/440 and their product 126/125, linked different degree lengths relative to each other, as per figure 1 below.

The pyramid’s apex height points to 30.5 to 31.5 degrees North, the degree for the Nile Delta, and so a pyramidion any higher than that could form rectangular numbers pointing to latitudes north of the Nile Delta. A pyramidion that exceeded 481.09 feet when multiplied by the southern base length of 756 feet, could reach the three reference latitudes *north *of the Nile Delta.

Most significant perhaps is the 51.5 degree length, that is the degree equal to every degree of the mean earth circumference. The required height above the pyramidion’s platform (480 feet high) would then be 96/35 ft, a yard of feet 32/35 ft long.

What kind of form, consistent to the Old Kingdom’s iconography, could define a set of four vertically displaced levels? The Djed column of Osiris comes to mind and, as a symbol, the Djed could have been emblematic of the world and its reference latitudes. I have therefore calculated all the dimensions between all four levels and created the diagram (figure 2) in which, a Djed sits upon the Great Pyramid as its pyramidion.

Osiris was the great god for the pharaohs, for whom the Great Pyramid was consecrated, playing out the ritual concerning the Pharaoh’s death and his resurrection as Horus, the son of Osiris, a new pharaoh. Osiris may have been conceived as the god of the geocentric world, the Primordial Hill.

#### Previous posts on this

- Units within the Great Pyramid of Giza
- Ethiopia within the Great Pyramid
- Recalibrating the Pyramid of Giza
- A Pyramidion for the Great Pyramid

### bibliography

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*Inductive Metrology*. 1877. Reprint, Cambridge: Cambridge University Press, 2013.