Points
(a)
The latitude of the apex of the structure is about 29 degrees 58 minutes 51 seconds north; this is discussed in point (m) later. That the Great Pyramid exhibits the fraction 22/7 (proportion, that is) in its design is beyond dispute (Petrie, TPTG p. 183): the quotient of half the base perimeter 1760/2 royal cubits divided by the height 280 royal cubits is 22/7; the fraction is still used in schools today. A manifestation of the proportion can be see in Figure 3. (b) In primal form (refer to slope details) the starting point for the design of the Great Pyramid is a right angle triangle with a base measuring one unit, its height 14/11 units.
In cross section the primal Great Pyramid has a full base of two units (double the triangle
base) and a height of 14/11 units; it therefore has a primal volume of Note the following for later (the number 220 below is derived from the half base measure of the Great Pyramid, 220 royal cubits):
If the royal cubit really is 20.625 inches, and it is, then 56/33 linear royal cubits is 35
inches or
(c)
The fraction 99/70 is readily found in the Great Pyramid. The floor level of the King's
Chamber is situated 82 royal cubits above base level (1691.25 inches, design intention -
Petrie, TPTG, p. 95. 1692.8±0.6 inches). There is a reason for the location. At this height
the Great Pyramid is 311 1/7 royal cubits wide: see
(d)
There are two measures that have a bearing on the Great Pyramid's layout. At a point
(e)
The volume of the Great Pyramid is 18,069,333 1/3 cubic royal cubits. The quotient of
this number divided by (f) Petrie reports (TPTG p. 86) the measures of the ends of the sarcophagus as: outer width 38.5 inches (mean); outer height 41.31 inches (mean).
Without doubt the height is intended to be two royal cubits (41.25 inches). The 38.5-inch
outer width is 1 195/225 (1.8666 ...) royal cubits, conveniently equivalent to 52.5
The area of the end is thus The quotient of 8 48/225 cubic royal cubits, the contents of the sarcophagus, divided by 2.2 is 3 165/225 (3.7333 ...): again see point (b) above. A further design intention is clear; a strategy has emerged. Note that the quotient of the measure 108 8/9 royal cubits, see point (d), divided by 3 165/225 is 29 1/6 (29.1666 ...). The latter number is 14 7/12 (14.58333 ...) multiplied by two: 14 7/12 inches is the measure of the remen - see Figure 2.
Furthermore, the quotient of 171 1/9 (royal cubits) - also see point (d) - divided by
3 165/225 is
(g) The only measure that has been the subject of conjecture in the King's Chamber is the design intention for the wall height. According to Petrie (TPTG p. 195) the length of the room is 20 royal cubits (412.24±0.12 inches), the width 10 royal cubits (206.12±0.12 inches).
General layout of the King's Chamber (sarcophagus shown)
Petrie thought the height of 230.09±0.15 inches, that is,
The royal cubit is divided into 28 digit divisions
(Petrie, IM p. 56): cf. the 224/165 × 8 48/225 = 11 169/1125 royal cubits
And 11 169/1125 royal cubits is 229 219/225 (
The volume of the chamber is 2230 10/225 (2230.0444 ...) cubic royal cubits
(11 169/1125 × 20 × 10). This is 271 17/33 times the volume of the sarcophagus contents
8 48/225 cubic royal cubits; 271 17/33 is the number The area of either the north or the south wall is 223 1/225 (223.00444 ...) square royal cubits: height 11 169/1125 multiplied by length 20. This area has been inventively used in the Giza pyramid layout: see the next point.
(h)
Layout of the Giza pyramids
In an article entitled
The north-south distance from the southern side of Khephren's Pyramid to the southern
side of Menkaure's Pyramid (
The east-west measure
Return for a moment to the area of the north or the south wall in the King's Chamber:
223 1/225 (223.00444 ...) square royal cubits. Multiply this number by one thousand and an
area of 223,004 4/9 square royal cubits is created. The quotient of this area divided by GX
353 53/99 royal cubits (500 remen) is 630.784 royal cubits or In NADIAE the reasons for the distances between all three pyramids are explicated. The east-west distance is of further interest: 353 53/99 royal cubits or 500 remen is the equivalent of 625 Roman feet, the Roman stade. It is also the equivalent of the 600-foot Greek stade. A Roman foot thus measures (like the "forearm") 11 2/3 inches and a Greek foot 12 11/72 (12.152777 ...) inches: see Figures 2 and 5. Zupko (p. 6) records the measures as [about] 11.65 inches and 12.15 inches respectively. The east-west distance may well be the first identified appearance of what was later known as the Greek-Roman stade. Two hundred and sixteen thousand (216,000 = 60 cubed) stades is equal to 1575,000,000 inches or 24,857 21/22 miles, the measure of the world mentioned in the Preamble and elsewhere. Of outstanding interest is the fact that ten stades or 72,916 2/3 inches is virtually identical with the U.S. international nautical mile of 6076.1033 feet or about 72,913 1/4 inches; the difference is less than 3.5 inches.
(i)
The Greek philosopher Plato who is known to have visited Egypt wrote a colourful
account of a mythical region called Atlantis. A physical description of the place is provided
in his unfinished work In total, the circular design is 27 stades in diameter (Lee, p. 152). But now the true value of the stade is known: 7291 2/3 inches (see point (h)). Here is one of many mathematical feats concealed by Plato. There are others in NADIAE. One that is memorable is presented shortly.
The rings of water and land around the citadel (central island)
The circumference of the circular arrangement is 84 6/7 (27 × 22/7) stades or 618,750 inches or 9.765625 miles; 9.765625 is 3.125 squared. Moreover, 9.765625 miles is equal to (multiply by 3600 - see the Preamble) 35,156.25 common cubits. The latter measure is the diagonal of a square whose sides are - divide by 99/70 - 24,857 21/22 (24,857.95454 ...) common cubits. The ancient Egyptian measure of the world is this many miles: see the Preamble and point (h). A return to the Great Pyramid is germane. (j) The height of the King's Chamber was established as 11 169/1125 royal cubits in point (g). In point (d) the measure 108 8/9 royal cubits was discussed. The quotient of 108 8/9 divided by 11 169/1125 is 9.765625 or 3.125 squared: compare with the preceding point.
(k)
Petrie records the entrance to the structure as being about
The remen is 14 7/12 inches - see Preamble,
Figure 2 and point (f). If this number is
squared and multiplied by 22/7 the product is 668 29/72 (
(l) The denominator 225 is common in various fractions already discussed: 8 48/225 and 1 195/225, for example. It is productive to divide the height of the Great Pyramid by the number: 280 royal cubits divided by 225 = 1 55/225 (1.2444 ...) royal cubits which is 25 2/3 inches (royal cubit = 20.625 inches). This new measure has been given the appellation ma'at by the present writer after the ancient Egyptian goddess Ma'at, wife of the god Thoth. Intriguing mathematical outcomes can be discovered.
The ma'at of 1 55/225 (1.2444 ...) royal cubits has an interesting relationship with another commonly-found Greek measure noted by Petrie (MW p. 4). Although he confusingly refers to it as a Greek foot, it is not the Greek foot of 12 11/72 inches described earlier: see Figure 5. This alternative Greek foot, he noted, was (about) 12.44 inches. This is nothing less than the ma'at measure of 25 2/3 inches divided by 2.0625. Furthermore, the alternative Greek foot - its correct value is 12.444 ... (12 4/9) inches - was 64/75 the remen measure. The ratio is the same as that between the common cubit of 17.6 inches and the royal cubit of 20.625 inches. The alternative Greek foot's most startling connection, though, is with the common cubit: the quotient of 17.6 inches divided by 12 4/9 inches is 99/70, the square root of two fraction discussed earlier. In NADIAE the ma'at measure is shown to be a part of what is one of the greatest design feats in history.
(m) Khephren's pyramid, the second largest of the Egyptian pyramids, exhibits the well- known 3.4.5 proportions in its design (Petrie, TPTG p. 202). The mean base measure of the structure is 8474.9 inches (Petrie, TPTG p. 97) or 411 royal cubits (design intention 8476.875 inches). The number 411 is based on the prime numbers 3 and 137. Three is of no interest, it is too common. The astonishing source of 137 in the design strategy is explicated in NADIAE. (It should come as no surprise that the geometric configuration described earlier has a key role in its creation.) The height of the pyramid was originally 274 royal cubits, the planned volume, accordingly, was 15,428,118 cubic royal cubits. Petrie writes of the entrance (TPTG p. 104): "The doorway of the Second Pyramid is lost, along with its casing; ... The position of the passage was fixed from a station mark near it; its axis is 490.3 (inches) E. of the middle of the N. face." (Petrie, 8471.9 inches for the north face, TPTG p. 97)
Note that the north face has a small error of workmanship of about five inches. The
entrance is thus If the base measure of 411 royal cubits is divided by 24 the quotient is 17.125 royal cubits. A cube with such dimensions contains 5022 89/512 cubic royal cubits. A stupendous design conceptualisation is imminent.
The volume of the pyramid, as stated, is 15,428,118 cubic royal cubits. The quotient of
this number divided by 5022 89/512 is There is evidence of the mile on the ground in Khephren's neighbour, the Great Pyramid; and of the ancient Egyptians' interest in 3.4.5 proportions; and of the knowledge of latitude 30 degrees North which is two-thirds of the way on a meridian from the North Pole to the equator.
Charles Piazzi Smyth observed that the Great Pyramid's apex is located around latitude
29 degrees 58 minutes 51 seconds North, give or take a few feet
(The contemporary polar circumference measure of 24,859.82 miles produces a degree of
around 69.055 ... miles or about 364,610.7 feet; the ancient Egyptian degree was 69.0498 ...
or 69 3160/ Further astonishing design features can be found in Khephren's Pyramid:
The volume of 11.264 cubic royal cubits (
General layout of Khephren's tomb-chamber
The Petrie/Vyse mean dimensions are shown below on the left, the design intention on the right:
The volume of the chamber (design intention) from the floor to the top of the wall is 2565 cubic royal cubits (22,504,493 209/512 cubic inches), the gabled roof space 235 cubic royal cubits (2,061,815 5/27 cubic inches).
The quotient of the tomb-chamber volume 2800 cubic royal cubits divided by 11.264, the
volume of stone in the sarcophagus, is 248.5795454 ... (248 102/176). This is a clear
reference to the ancient Egyptian measure of the world 24,857.95454 ... (
The renowned historical figure Eratosthenes said the circumference of the earth was
252,000 stades (Morgan, p. 27). He faked the real measure by adding 36,000 stades (cf. 360
degrees in a circle, 360 days in an ancient Egyptian year) to fool enemies or those not
initiated into the Greco/Egyptian Mysteries. Vitruvius, proven in NADIAE to be an initiate
into these Mysteries, makes a telling comment on the matter in "Some people do indeed say that Eratosthenes could not have inferred the true measure of the earth." (Morgan, p. 28) The linear measure of 11.264 royal cubits affords the opportunity of finding a series of interlocking relationships: 11.264 royal cubits is 232.32 inches or 19.36 feet; 19.36 is 4.4 squared; 4.4 feet is 52.8 inches; a mile is 5280 feet; the quotient of 1760 royal cubits (perimeter of the Great Pyramid base) divided by 11.264 is 156.25 which is 12.5 squared; 12.5 royal cubits is 257.8125 inches or 14.6484375 common cubits or 351.5625 common cubit digits; 351.5625 is 18.75 squared.
If 351.5625 common cubit digits were the measure of a diagonal in a square and the
fraction 99/70 were taken to represent the square root of two, that square's sides would
measure 248.5795454 ... (248 102/176) common cubit digits: refer to the Khephren
tomb-chamber material in |