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67. A Metonic Giza Rectangle

Updated: Apr 20

If the dimensions of the Giza monuments and site are read in cubits, inches or metres, many lunar connections arise. For example, the width of the Giza site, taken as a rectangle drawn between the outer corners of the Great Pyramid and the third pyramid, is given as 29 227.199 inches by Flinders Petrie, can be interpreted as 80 years in inches, or 48 lunar months in cubits of 20.613777 or 20.619296 inches (the first is the ratio between 48 lunar months and 80 lunar years, the scond is closer to the Flinders Petrie dimensions). (See more lunar connections here,) But lunar numbers can also arise out of proportions at Giza, which do not rely on any unit of measure.

The Metonic cycle is one within which the cycles of the sun and the moon reconcile, as 235 lunar months are almost exactly 19 solar years. Is the large rectangle that links the centres of the Great Pyramid and third pyramid Metonic in its proportions?

Flinders Petrie gave the length of the distance between the centres of the Great Pyramid and the third pyramid as 36857.7 inches, which is 936.18558 m. If this line is taken as the diagonal of a rectangle, the length of this rectangle, north-south, is 29102 inches, or 739.1908 m, and the width, west-east, is 22616 inches, or 574.4464 m.

If we take the width and multiply it by 29.53059, the number of days in a lunation, and 38, we get the length. And conversely, if we multiply the length by 365.242199, the number of days in a year, and divide by 470, we get the width again.

22616 x 38 / 29.53059 = 29102

29102 x 365.242 / 470 = 22616

The number 47, combined with the number 38, is present in this Giza rectangle, which stretches out between the centres of the Great and third pyramid at Giza. A lunation of 29.53059 days, multiplied by 47 x 10, and divided by 19 and 2, gives the solar year, approximately, as 365.24677. This means the number of lunar months on average per year can be simplified as 470/38, which is 12.36842 and 365.242199 / 29.53059 = 12.368266. This solar and lunar ratio is slightly different to the one found at the Rollright Stones by Jim Wakefield, yet it also uses the number 47. At the Rollrights, the radius of a circle was 47 Saxon feet, and the circumference was approximately 10 lunar months in days expressed as Saxon feet also. The other important lunar aspect of a circle with a radius of 47 units, which Jim Wakefield found, was that the area of such a circle will have almost exactly the same number of square units as there are days in a Metonic cycle.

• Metonic cycle: 29.53059 x 235 = 6939.68865 days

• Circumference of a circle with a radius of 47 units is 295.30971

• area of circle with radius of 47 units: 47² x pi = 6939.77817 units squared

At Giza we have a rectangle, and the ratio between the width and length is 38 / 29.53059, to within about half an inch: which is roughly equal to 365.242199 / 470. The width of 22616 x 38 / 29.53059 = 29 102.29697. The length as given by Flinders Petrie is 29102. This 29102 multiplied by by 365.242199 / 470 = 22 615.47379, which is 0.5262 inches less than Flinders Petrie's figure.

We can think of the the greater Giza rectangle, which links the outer corners of the Great Pyramid and third pyramid, as lunar also, in that the width represents 48 lunar months in cubits, and the length 5 draconic years, in cubits also. The dimensions of this Greater Giza Rectangle given by Flinders Petrie are 35 713.2 and 29 227.199 inches. Also, its sides are in 9 x 11 proportion, which creates a 99 square grid, and 99 lunations are almost exactly 8 years, echoing the 80 years present in the width of this Greater Giza Rectangle, if read in inches. Another way of thinking of the ratio between width and length is to multiply the length by 29.53059 x 48 / (1000 x √3), which is even more accurate. Another link to the moon is between the length of this Greater Giza rectangle and the Great Pyramid mean base side, which is 9068.8 inches according to Flinders Petrie. This base side multiplied by the number of days in a lunation, 29.53059, and then by 4/30, gives the Greater Giza Rectangle length. It's worth noting in passing that a lunation in days multiplied by 4/3 gives something close to a metre, expressed in inches: 39.37412. Expressing the lunation as 29.53125 days then gives an ancient metre of 39.375 inches.

Dividing the radius of any circle into 47 units always allows for ten lunations to be expressed in the circumference, in that same unit, and the number of days in a Metonic cycle in the area, in that unit, squared. This works because, serendipitously, pi works with the number 47 to produce the number of days in both a lunation and a Metonic cycle, so that you could almost define an approximation of pi, in Metonic terms, as 365.242199 x 19 / 47², to about 99.99746 % accuracy. The number 19 is the number of years in a Metonic cycle, and the number 47 multiplied by 5 is the number of lunations in a Metonic cycle. However, there is no single ratio that defines the length and width of a rectangle. The rectangle formed by the centres of the Great Pyramid and the third pyramid expresses a similar idea to Jim Wakefield's Rollright circle: the sun and moon cycles are expressed through geometry. Multiplying by 38 / 29.53059 is roughly equal to dividing by 365.242199 / 470, or, to simplify, 19 / 29.53059, to pick up on the 19 in the number of years in a Metonic cycle, is roughly equal to 365.242199 / 235, to pick up on 235 the number of months in a Metonic cycle. When the circle so magically expresses the Metonic cycle, why chose to use a rectangle? The Great Pyramid's base and height are linked by pi, so perhaps we can think of the more angular geometric shapes as connected to the circle. For example, the mean base side of the Great Pyramid taken as 9068.8, as per Flinders Petrie, multiplied by √2 to obtain the diagonal of the square that makes up the base, is 12 825.21995 inches, equivalent to 64 x 9 x 1.08 cubits of 20.61667 inches. If we take this diagonal as the diameter of a circle, and divide it by 2, then by 47, we obtain 136.4385 inches. The circumference of this circle will measure 295.3097 x 136.4385 inches. These 136.4385 inches are 29.53980 x √3 x 8/3 inches. To obtain 29.53059 instead of 29.53980 we would need the mean Giza base to measure 9065.97 inches instead of 9068.8. If we think of the mean side of the Great Pyramid as 5 000 π / √3 inches, and the perimeter of the Great Pyramid as 20 000 π / √3 inches, we can arrive at a number close to the number of days in a lunation with pi, the square root of 2, and 47.

10 000  π x √2 / (47 x 32) = 29.5404451. Alternatively, the mean base side of the Great Pyramid, as 5000π / √3, multiplied by √2 to obtain the diagonal of the base square, multiplied by 3 / (47 x √3 x 16) = 29.5404451, or more simply (5/4)² x 1 000 π x √2 / 235, the 235 being the number of lunations in a Metonic cycle, and (5/4)² x 1 000 / 235 being approximately equal to 47 x 10 x √2 . The Great Pyramid base diagonal can be expressed as 29.54044 x 47 x 16 / √3 inches. If we start from 10 000 inches and divide it by √3 to obtain the base, multiply by π/2 to obtain the base side, and so on, the results are less accurate than if we start with 10 001.48222 inches (see algorithm diagram below). It is possible that the inch used at Giza, if it was used at Giza, was slightly different to the modern inch. If we divide the 29.54044 inch measure from the diagonal by 1.000148222 we get a slightly smaller value for the lunation, of 29.536066, which is closer to the actual value of a lunation. Interestingly, if we use an inch that is linked to the modern inch by 29.54044 / 29.53059 instead, at the start of this algorithm, 29.54044 / 29.53059 x 10 000 / √3 = 5 775.42846 inches, for the height of the Great Pyramid, and x π/2 gives 9072.02181 inches for the mean base side, both of which are acceptable, and indeed, 9072 inches is widely accepted as the optimal value. However the rest of the Giza dimensions algorithm then produces values that are a little too big. We saw that the width of the Giza site, taken as a rectangle drawn between the outer corners of the Great Pyramid and the third pyramid, can be interpreted as 80 years in inches, or 48 lunar months in cubits of 20.613777 or 20.619296 inches, the first value being the ratio between 48 lunar months and 80 lunar years, the second value the one that matches most closely Flinders Petrie dimensions. The ratio between these two cubit values is quite close to the ratio between the two lunation values. 29.54044 / 29.53059 = 1.00033355 and 20.619296 / 20.613777 = 1.00026773.

Just as Jim Wakefield's Rollright circle allowed us to hypothesise about pi being defined in relation to the Metonic cycle, here would can do the same with √2, which can be approximated as (47 x 8)² / 10 000 = 1.41376, or (5/(4 x 47))² x 2000 = 1.1414667. And 1.41376 x 10 000 π x /(47 x 32) = 29.53097. Or with Metonic pi, the result is 29.53022.

The Metonic rectangle offers another lunar dimension to interpreting the Giza design, which does not rely on any unit, nut only on proportion. It raises questions on the use of geometry to express the Metonic cycle not just with a circle, but also with a rectangle. Looking into the Great Pyramid base as part of a circle also raises questions on the use of the day-inch at Giza.

To summarise, the rectangle formed by the centres of G1 and G3 is that it is Metonic in character, both in measurement and in proportion. The width given by Flinders Petrie is 22 616 inches, which can be interpreted as 1900 x 29.53059 x 2π/(3 x √3) x 1/3 = 22 616.3054 inches. And length x 365.242199 / 470 gives width. The length is 29102 inches. So to get from length to width: 365.242199 /470 x 29 102 = 22 615.4861. 19 is the number of years in a Metonic cycle, 29.53059 is the number of days in a lunation, 470 is twice 235, the number of lunations in a Metonic cycle, and π / (3 x √3), for me, symbolises the system of time keeping, because: civil year x draconic year / (lunar year x lunation) ≈ 20π /(√3 x 3), with civil year: 365 days, draconic year: 346.6201 days, and lunar year: 354.36708 days. Also: number of lunations per year ≈ π /√3 x 300 / 44.