Points
 Base: 440 royal cubits (half base 220 royal cubits - see "Slope" below) Base perimeter: 1760 royal cubits Height: 280 royal cubits Slope: 280:220 or more simply 14:11 Volume: 18,069,333 1/3 cubic royal cubits (base area by height divided by 3)

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 56/33 (1.6969 ...) cubic units. This fraction can also be expressed as 35/20.625. One can immediately see the Great Pyramid's nature is receptive to the number 20.625, the number of inches in a royal cubit.

Note the following for later (the number 220 below is derived from the half base measure of the Great Pyramid, 220 royal cubits):

 Step 1. 56/33 × 220 = 373 1/3 (373.333 ... as it will appear in a calculator); Step 2. 373 1/3 × 220 = 82,133 1/3 (82,133.333 ... ); Step 3. 82,133 1/3 × 220 = 18,069,333 1/3 (cubic royal cubits, the volume of the Great Pyramid).

If the royal cubit really is 20.625 inches, and it is, then 56/33 linear royal cubits is 35 inches or 2.4 remen. Time numbers like 24 have an important role in the overall design system - more on this later.

(c) The fraction 99/70 and the square root of two

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 AB in the illustration below. The product of the half base measure of 220 royal cubits (CD below) multiplied by 99/70 is 311 1/7 royal cubits. A further striking example of 99/70 in the design is presented in point (l) later.

(d) The invisible square

There are two measures that have a bearing on the Great Pyramid's layout. At a point 108 8/9 royal cubits straight down from the apex (call it location P) the pyramid is 171 1/9 royal cubits wide - location P is also the latter distance above base level. Consequently, a square could be drawn inside the cross section: see the following illustration. The two measures now assume significance, exemplifying in succeeding material the remarks of Vitruvius on symmetry and proportion in the opening paragraphs of this paper.

(e) The sarcophagus in the King's Chamber

The volume of the Great Pyramid is 18,069,333 1/3 cubic royal cubits. The quotient of this number divided by 2,200,000 (cf. 22/7 and the 220-royal cubit half base measure) is 8 48/225 (8.21333 ...) cubic royal cubits or 72,061 11/64 cubic inches. Petrie records the volume of the contents (inner space) of the sarcophagus in the King's Chamber as being 72,000 plus or minus 60 cubic inches (Petrie, TPTG p. 195). The design intention is clear. Refer to 82,133 1/3 in point (b) above. The sarcophagus is a rectangular box-shaped object.

(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 common cubit digits or 2.1875 common cubits. Compare the latter number with KM 2.1875 inches in Figure 3. As well, 38.5 inches is equal to 2.2 Roman cubits or 52.8 remen digits: a mile contains 5280 feet.

The area of the end is thus 3 165/225 (3.7333 ...) square royal cubits.

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 45 10/12 (45.8333 ...) which is 14 7/12 multiplied by 22/7. A clever variant of this outcome was given architectural form - see point (k) later.

(g) The King's Chamber

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, 229.94 to 230.24 inches, may have represented half the diagonal measure of the floor or 11.18 ... royal cubits (230.53 inches if the royal cubit were assigned the traditional value of 20.62 inches, 230.59 inches if taken at the correct value of 20.625 inches). However, his hypothesis does not fit observed data. Here is what is going on.

The royal cubit is divided into 28 digit divisions (Petrie, IM p. 56): cf. the 280 royal cubit height of the Great Pyramid. Each digit is thus 165/224 inch. Invert this fraction and multiply by the number of cubic royal cubits in the contents of the sarcophagus:

224/165 × 8 48/225 = 11 169/1125 royal cubits

And 11 169/1125 royal cubits is 229 219/225 (229.97333 ...) inches. The fit is near perfect. Of additional interest is the fact that 8 48/225 royal cubits is 229 219/225 (229.97333 ...) royal cubit digits. Such remarkable outcomes are worthy of contemplation.

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 280 divided by 1.03125 which is one- twentieth of 20.625, the number of inches in a royal cubit.

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) The Menkaure-Khephren rectangle

Layout of the Giza pyramids

In an article entitled A Ground Plan at Giza British researcher J. A. R. Legon utilised Petrie's measurements of the rectangular area that contains the pyramids of Khufu, Khephren and Menkaure for a hypothesis on the spacing and sizing of the structures. Of especial interest in the article is the layout of the Khephren and Menkaure pyramids (GXJB in diagram), in particular, the north-south distance between the southern sides of the two pyramids (GB) and the east-west distance between their western sides (GX). The measures quoted here are from the Legon article; Legon obtained them from Petrie's book The Pyramids and Temples of Gizeh.

The north-south distance from the southern side of Khephren's Pyramid to the southern side of Menkaure's Pyramid (GB) is 13,009.7 inches. The east-west distance, the west side of Menkaure's Pyramid to the west side of Khephren's Pyramid (GX) is 7289.5 inches.

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 13,009.92 inches. This is clearly the north-south distance GB of 13,009.7 inches mentioned above. The accuracy is stunning: the layout "error" a mere 0.22 inch.

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 matter of Atlantis

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 Critias. The key interest here is the arrangement and dimensions of the concentric rings of land (two of them) and water (three of them) that surround the central feature, an island five stades in diameter which contains, amongst other things, a royal palace and a Temple of Poseidon.

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).

(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) The entrance to the Great Pyramid: doorway to other dimensions

Petrie records the entrance to the structure as being about 668.2 inches above the pavement (Petrie, TPTG p. 55). This previously unexplained measure is an outstanding mathematical design feature. This is how the measure was created.

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 (668.402777 ...) inches or 45 10/12 (45.8333 ...) remen, the design intention for the entrance height above ground - revisit the last paragraph in point (f). Great care has obviously been taken by ancient Egyptian builders in this instance to achieve a remarkable standard of workmanship. The measure is equivalent to 32 11/27 royal cubits (exactly 55 Greek feet). The following are some mathematical consequences:

• 32 11/27 royal cubits divided by 1 195/225 royal cubits, the width of the King's Chamber sarcophagus (see point (f)) = 17 13/36 which is 4 1/6 squared;
• 4 1/6 royal cubits is equal to 7.36607142857 ... (1650/224) "forearms" or Roman feet: a royal cubit digit measured 0.736607142857 ... (165/224) inch;
• 280 royal cubits, the height of the Great Pyramid, divided by 32 11/27 = 8.64; there are 86,400 seconds in a day: cf. remarks on 86,400 common cubit digits in a mile in the Preamble;
• 32 11/27 divided by 8 48/225 (see point (e), the volume of the sarcophagus contents) = 250,000/63,360; there are 63,360 inches in a mile;
• 7291 2/3 (the number of inches in a stade) divided by 32 11/27 = 225 (15 squared): see the following point.
• (l) The number 225

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 quotient of 1760 royal cubits (base perimeter of the Great Pyramid) divided by 1 55/225 is 1414.285714 ... (9900/7); the ancient Egyptian fraction for the square root of two was 99/70: see point (c);
• The ma'at of 25 2/3 inches is equivalent to 1.76 remen (cf. common cubit of 17.6 inches etc.) or 1.458333 ... common cubits: the remen measures 14.58333 ... (14 7/12) inches - see Figure 2;
• Doubling 25 2/3 five times produces 821.333 ... (821 1/3): cf. the volume of the sarcophagus contents 8.21333 ... (8 48/225) cubic royal cubits discussed in point (e).
• 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 and the mile

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 24 royal cubits (495 inches, design intention) off-centre. Again, the designer is pointing to something mathematically significant. Take heed of the use of the time number 24 - recall the item on 2.4 remen in point (b) and the 24-digit common cubit discussed in the Preamble.

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 3072, the number of royal cubits in a mile.

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 feet2). It is short of latitude 30 degrees North by around 0.019666 ... degree. The latter is virtually 1 1/3 miles. Now this is not disputable: a point 1 1/3 miles directly south of latitude 30 degrees North is right inside the base area of the Great Pyramid close to being under the apex.

(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 3160/63,360 miles or 364,583 1/3 feet. The difference is around 27.5 feet. Observe how the ancient Egyptian degree measure expresses the number of inches in a mile (63,360) in its make-up; the contemporary degree does not.)

Further astonishing design features can be found in Khephren's Pyramid:

• The volume of stone in Khephren's sarcophagus, which Petrie reports as being "well polished" and "dressed down to the intended height", is easily found by subtracting the inner contents volume from the overall volume. His measurements in inches (Petrie, TPTG p. 107) are shown below on the left, the design intention, which can be deduced without undue difficulty, on the right. Note the congruency and the links between numerators and denominators in the design intention.
•  Petrie min. & max. variations in measurements Mean Design intention Outer length: 103.62 - 103.73 103.68 103.7… (2800/27) Outer width: 41.95 - 42.0 41.97 41.96 … (1133/27) Outer height: 37.98 - 38.12 38.12 38.12 ... (43,200/1133) Inner length: 84.63 - 84.91 84.73 84.72 … (7,246,711/85,536) Inner width: 26.66 - 26.76 26.69 26.69 … (2376/89) Inner depth: 29.51 - 29.68 29.59 29.666 ... (89/3)

The volume of 11.264 cubic royal cubits (0.22 doubled nine times), or 98,826.75 cubic inches, is the volume of the King's Chamber sarcophagus contents of 8 48/225 cubic royal cubits multiplied by 48/35; a remen digit is 35/48 (0.7291666 ...) inch. Recall the stade measure of 7291 2/3 (7291.666 ... ) inches described in point (h) and BU in Figure 3.

• The volume of Khephren's tomb-chamber is 2800 cubic royal cubits which is 24,566,308 304/512 cubic inches (readily deduced from Petrie's measurements shown on pp. 105 - 6 of TPTG). The chamber has a gabled roof.
• General layout of Khephren's tomb-chamber

The Petrie/Vyse mean dimensions are shown below on the left, the design intention on the right:

 Petrie/Vyse inches Design intention inches as royal cubits End height: AB 244.4 244 29/684 (244.04) 11 427/513 Roof height: CD 38 37 271/342 (37.79) 1 427/513 Wall height: EF 206.4 206.25 10 Breadth: GF 195.9 195.9375 9.5 Length: FH 557.4 556.875 27

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).

Outcomes

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 De Architectura:

"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 Outcomes above, the measure of the world again, and compare 351.5625 with the Atlantis material in point (h).