The Design of the Great Pyramid

In 1988-9, I published two articles on the planning of the Great Pyramid - the first in Discussions in Egyptology [i] and the second in Göttinger Miszellen.[ii] Both of these articles were written in response to theories which had recently been published in those journals, but which were found to be lacking owing to the failure of the authors to consult the primary survey data, the use of inaccurate measurements, and a willingness to vary the length of the Egyptian royal cubit in order to obtain the desired results. Although my research has advanced significantly since 1989, my two early articles seem worth reproducing because many of the basic design factors were accurately established in them for the first time.[iii] The following text combines the content of the original papers.

In a recent article describing the geometry of the Great Pyramid of Giza, Jorge Trench [1] has presented a model which, it is claimed, could have been used by the ancient Egyptian builders to determine the configuration of the passage-system. Although a high degree of accuracy is asserted, the validity of Trench's data may in some respects be disputed; while the complexity of his geometry also renders his model unlikely to have been envisaged by Khufu's architects of the Fourth Dynasty. The present paper puts forward an alternative development which is considerably more accurate, and also shows that many of the dimensions in the Great Pyramid were set to meaningful whole numbers of royal cubits.

To establish his geometrical model, Trench assumes that the sloping passages were all intended to have the same angle of slope, when the measured angles vary over a range of about half a degree, or from 26° 2' to 26° 34'. Within this range, many definitions of the slope are possible, and it is evident that each passage must be considered as a separate entity if any degree of accuracy is to be demonstrated. Trench also assumes that the angle of the Descending Passage as given by J. and M. Edgar,[2] and referred to by Maragioglio and Rinaldi,[3] resulted from an actual measurement of the passage, when in fact this angle of 26° 18' is a hypothetical value which was derived from the 'pyramid-inch' model, and was based on the mean of the different passage-angles obtained by Piazzi Smyth.[4]

For the dimensions of the various passages, Trench refers to the measures of Maragioglio and Rinaldi,[5] which are mostly metric conversions of the careful measurements made by W.M.F. Petrie in inches.[6] Trench gives these measurements only after conversion into cubits, using a length of cubit that has evidently been allowed to vary from 0.5235 to 0.524 metres. Very few of his theoretical values correspond to whole numbers of cubits, and the lengths given for the Ascending Passage and Grand Gallery diverge from the measured dimensions by about 1.3 and 1.6 cubits respectively. As shown below, however, these lengths actually correspond to significant whole numbers of cubits to within 0.05 cubit.

In another recent article in which the dimensions of Khufu's Great Pyramid are described in royal cubits, Werner Hönig [7] has unfortunately made use of some measurements that are metric conversions of dimensions which Dr I.E.S. Edwards stated to the nearest foot.[8] As a result, errors amounting to as much as one cubit have been introduced, and the whole numbers of cubits which have been ascribed to the Descending Passage, for example, are incorrect. When the exact dimensions are computed, however, the logic of the architect's design is brought to light; and the major factors will now be considered step by step.

The Dimensional Planning
In the following description of the Great Pyramid's design, reference will be made to the measurements in inches which were obtained by Flinders Petrie in his precise survey of 1880-2. Except for the meticulous survey of the base which was carried out by J.H. Cole in 1925,[9] Petrie's findings have not been superceded, and the accuracy of his work is proven by the fact that the later survey yielded a mean side-length which is only 1.5 cm longer than Petrie's result. For the length of the royal cubit, Hönig employed Petrie's determination of 20.620 ±0.005 inches, or 0.52375 metres,[10] and the same value will be used throughout the present paper.

With regard to the external form of the Great Pyramid, it is generally accepted that the sides of the base were intended to measure 440 cubits, although only the south side has exactly this length according to the survey by Cole and the cubit of 20.62 inches. The slope of the casing being 14 rise on 11 base, the height of the pyramid will be 280 cubits, with a theoretical casing-angle of 51° 50' 34". This is close to Petrie's finding of 51° 52' ± 2'. [11] For the mean side-length we have:

Mean Side of Base of Great Pyramid, Petrie, 9068.8 ins = 439.81 cubits

Following Petrie,[12] Hönig ascribes the divergence from 440 cubits to the intention of the builders to express the 'π proportion' of the pyramid in relation to the height of 280 cubits, with greater accuracy than the approximation to π of 22/7. The theoretical side-length is then 140π or 439.82... cubits.

The Internal Design
Inside the Great Pyramid, a primary division takes place in the central axis, vertically below the apex of the pyramid (see fig.1). Petrie found that this central plane is exactly marked by the face of the Great Step, at the head of the Grand Gallery, and by the apex of the Queen's Chamber. As a result, the distances southwards from the northern base-line of the pyramid are slightly less than the round number of 220 cubits. The following dimensions in inches are as stated by Petrie [8]:

A second major division is marked by the level of the King's Chamber; and we can confirm Petrie's finding that this chamber was placed at the height in the Great Pyramid at which the area of the horizontal section is equal to one-half the area of the base.[14] In addition, however, this level is found to correspond to an almost exact whole number of cubits. Geometrically, the upper height of the Pyramid above the floor of the King's Chamber is the side of a square, the diagonal of which is given by the total height of 280 cubits:

That this placing of the King's Chamber was significant to the builders is shown by the fact that at least four other dimensions in the Great Pyramid were evidently determined by it. First, the south wall of the King's Chamber is found to be within 0.07 cubit of 3 × 82 equals 246 cubits southwards from the north base of the pyramid (see table below). Representing this relationship by three modular squares of 82 cubits as shown in fig. 1, the centre of the southern square marks the level of the Queen's Chamber, and falls vertically above the lower end of the Descending Passage, which is just 2.5 × 82 equals 205 cubits southwards from the north base. The actual dimensions show an average error of only 0.06 cubit:

The Geometry of the Passage-System
We can now consider the points of division which take place within the semi-base of the Great Pyramid, at the north wall of the Grand Gallery where a junction is formed with the Ascending and Queen's Chamber Passages, and at the lower end of the Ascending Passage at the junction with the Descending Passage. Both of these divisions involve the ratio of 14:25, and can be derived from the 11:14 profile of the pyramid as 14:(11 + 14). The choice of this ratio may be associated with its use in calculating the position of the entrance to the pyramid, where the Descending Passage rising with a gradient of 1 rise on 2 base,[20] intersects the pyramid's outer casing rising with a gradient of 14 rise on 11 base. In the simple example of a passage beginning at the level of the base (see fig. 2), the horizontal distance between the starting points of passage and pyramid-casing will be divided into parts of 14, 14 and 11, the whole amounting to 39 parts. It will be seen in this figure that the 14:25 division corresponds to the mid-point of the passage-line.

Now taking the division at the north wall of the Gallery, this is such that the semi-base of 220 cubits is divided in the ratio of 14:25 as follows:

220 × 25/(14+25) = 141.0256...

Petrie's measurement in inches[21] shows that the north wall of the Gallery was in fact placed at almost exactly the whole number of 141 cubits horizontally southwards from the north base of the Great Pyramid:   

N. Base of Pyramid to N. wall of Gallery, 2907.3 inches = 140.99 cubits.

Precisely the same division occurs at the lower end of the Ascending Passage, where the floor-line and roof, respectively, intersect the floor and roof of the Descending Passage. Here, however, the parts are reversed from north to south, and the dimension which is divided in the ratio of 14:25 is the distance, equal to the round-figure semi-base of the Second Pyramid [22], of 205 cubits horizontally southwards from the north base to the foot of the Descending Passage:

205 × 14/(14+25) = 73.589...

This division is represented by Petrie's placing of the junction between the Descending and Ascending Passages,[23] to within 0.02 cubit:

N. Base of Pyramid to N. end Ascending Passage, 1517.8 inches = 73.608 cubits

Note that whilst decimal fractions have been used here for the sake of convenience, the architect would probably have used unit fractions in the customary Egyptian manner, and have converted the result to cubits, palms and fingers. The above dimension would then have been 73 cubits 4 palms 1 finger.

We can now turn our attention to the sloping and vertical lengths of the passages which result from the defined horizontal lengths and the chosen angles of slope. The architect was able to obtain suitable whole numbers of cubits in both sloping and vertical lengths, by selecting angles of slope slightly adjusted from the basic profile of 1 rise on 2 base. It may be supposed that these adjustments were estimated graphically by carefully plotting the dimensions on a suitable scale.

It is interesting to find that the sloping length of the Grand Gallery floor and ramps, from the junction with the Ascending Passage up to the Great Step in the centre of the pyramid, is just 88 cubits or one-fifth of the pyramid's base of 440 cubits - a fact which has hitherto gone unnoticed:

Sloping length of Gallery floor, 1815.5 inches = 88.05 cubits

It now follows that when calculated in due proportion to the division of the semi-base in the ratio of 14:25, the length of the sloping floor-line from the north wall of the Grand Gallery down to the vertical plane of the pyramid's north base, will be just 50 π cubits:

Slope from Gallery N. wall to N. Base = 88 × 25/14 = 157 1/7 = 22/7 × 50 cubits

The constant π is thus expressed in terms of a module of 50 cubits, by the extension of the Gallery floor-line over the complete semi-base of the Great Pyramid. The portion of this sloping length which is actually represented by the Ascending Passage is also significant, since as noted by Petrie, it is just 75 cubits:

Sloping length of Ascending Passage, 1546.8 ins = 75.01 cubits

Now subtracting this length from the sloping distance of 157 1/7 cubits, we obtain a remainder of 82 1/7 cubits, which is practically the module of 82 cubits as defined by the level of the King's Chamber.

We can now derive the theoretical slope of the Grand Gallery and Ascending Passage, by applying the complete sloping dimension of 88 × (1 + 25/14) equals 88 × 39/14 cubits, to the round-figure semi-base of 88 × 5/2 equals 220 cubits. The resulting profile reduces to 39 slope on 35 base, which is very close to the mean of Petrie's data:

Angle of Slope, for 39 slope on 35 base = 26° 10' 37"

Mean Angle of Ascending Passage and Gallery = 26° 9' 35"

Using this definition of slope, two additional occurrences of the module of 82 cubits can be evaluated. Working in inches, Petrie noted that the total horizontal length of the Gallery, between the lower and upper end-walls, is equal to the level of the Great Step above the base of the pyramid. This being the King's Chamber Level of 82 cubits, we can obtain the theoretical sloping length for the side-walls of the Gallery as follows:

82 × 39/35 = 91.371... cubits

Length of Side Walls of Gallery, Petrie: 1883.6 inches = 91.35 cubits

As noted above, the beginning of the Ascending Passage is at the sloping distance of about 82 cubits from the north base. The theoretical horizontal dimension is found to be identical to that previously defined for this position:

82 × 35/39 = 73.589... cubits

As we have seen, Petrie's measurement was 1517.8 inches or 73.61 cubits.

As illustrated in figure 4 below, the passage-geometry can be developed schematically together with the actual dimensions, starting from a pyramid with the 14:11 proportion and a height of just 200 cubits. By constructing squares inside the pyramid-section, the base is divided into parts of 11, 14, 14, and 11, and the origin of the ratio of 14:25 is self-evident. The Gallery floor-length of 88 cubits is defined by the sides of the squares, which divide the height of 200 cubits in the ratio 11:14 as follows:

200 × 14/(11 + 14) = 112,    200 × 11/(11 + 14) = 88

Having established this basic geometry, we can consider the exact angles of slope. Petrie's measurements show that the final angles of the Ascending Passage and Grand Gallery were slightly adjusted with respect to their mean value, such that suitable whole numbers of cubits were obtained in the vertical lengths of both passages. From Petrie's survey, we find:

The vertical length of the Gallery floor is therefore 39 cubits, with a plausible allusion to the (14 + 25) or 39 parts into which the semi-base of the Pyramid is divided. The resulting profile of 88 slope in 39 rise gives a theoretical angle of 26° 18' 25". The vertical length of the Ascending Passage is clearly 33 cubits, or 3/8 of the Gallery floor-length; and since the sloping length of this passage is 75 cubits, the passage-profile is 25 slope in 11 rise with a theoretical angle of 26° 6' 14". Comparing these angles with the measures of Petrie and also Smyth,[26] we have:

Hence the observed angles of the Ascending Passage and Gallery correspond very closely to 75 cubits slope in 33 cubits rise, and 88 cubits slope in 39 cubits rise, respectively.

The Descending Passage
Reference to the series of offsets taken by Petrie to the floor of the Descending Passage from a mean axis of angle 26° 31' 23",[27] shows that the slope of this passage is divided into two parts at the junction with the Ascending Passage. The slope of the lower part is almost exactly one rise on two base:[28]

It appears that the lesser angle of the upper part was an adjustment designed to bring the vertical length to a whole number of cubits. Petrie's measurement of 495.3 inches is equal to 24.02 cubits, which is also just 1/3 of the vertical rise through the Ascending Passage and Gallery of (33 + 39) equals 72 cubits. Petrie placed the junction with the Ascending Passage at the level of 172.9 inches over the base, and this is 8.39 or nearly 8.4 equals 7/10 x 12 cubits. Multiples of 12 cubits are thus evident in these vertical dimensions, and the level of the entrance to the Great Pyramid is defined as 12×(2 + 7/10) or 32.40 cubits. Petrie's result was 668.2 inches or 32.405 cubits. The foot of the Descending Passage was placed by Petrie 1181 inches or 57.27 cubits below the base, and this is nearly 7/10 × 82 or 57.4 cubits.

Conclusions
Whilst this description of the Great Pyramid's passage-system does not represent a complete analysis, enough material has been put forward to show the great precision with which the dimensions of the passages were set, both in terms of the Egyptian cubit and with reference to a logical geometrical design. Some writers have stressed the use of the seked measure of slope, or relation between the vertical and horizontal lengths, as if the ancient Egyptians were unable to determine the sloping length of a passage by direct measurement. Although the use of the Pythagorean theorem may be a matter of debate, any problem involving the hypotenuse of a right-angled triangle could have been solved by the measurement of a plan drawn to a suitable scale. Whatever methods were used, the skill of the architect in selecting the sloping, horizontal, and vertical dimensions of the passage-system was clearly quite considerable, while the highly coherent quality of the design is yet another strong argument against the popular assumption that the structure was subject to a change of plan.

J.A.R. Legon

NOTES

[i]. J.A.R Legon, 'The Design of the Pyramid of Khufu', Discussions in Egyptology 12 (1988), 41-48.
[ii]. J.A.R Legon, 'The Geometry of the Great Pyramid', Göttinger Miszellen 108 (1989), 57-64.
[iii]. Soon after these articles were published, it emerged that new metric measurements of the passage-system in the Great Pyramid had become available through the work of Gilles Dormion and Jean-Patrice Goidin, Khéops: Nouvelle Enquête (Paris, 1986). See now also G. Dormion, La chambre de Chéops. Analyse architecturale (Paris, 2004). The findings of the French enthusiasts have not supplanted Petrie's measurements, and give essentially the same results.

[1]. J.A. Trench, Göttinger Miszellen 102 (1988), 85-94.
[2]. J. & M. Edgar, The Great Pyramid Passages (Glasgow, 1924), II, 12.
[3]. V. Maragioglio and C.A. Rinaldi, L'Architettura delle Pirimidi Memphite, IV (Rapallo, 1965), 10. The metric measurements in Tavs. 3-7 are largely based on Petrie's findings.
[4]. C.P. Smyth, Life and Work at the Great Pyramid (Edinburgh, 1867).
[5]. Maragioglio and Rinaldi, op. cit., 5.
[6]. W.M.F. Petrie, The Pyramids and Temples of Gizeh (London, 1883), 37-95 (first edition only for full details).
[7]. W. Hönig, Discussions in Egyptology 10 (1988), 27-32.
[8]. I.E.S. Edwards, Die ägyptischen Pyramiden (Wiesbaden, 1967), 75-79. Conversions were made from the feet of the English editions as in Edwards, The Pyramids of Egypt (London, 1985), 99-103, thus: level of entrance, 'about 55 feet', 16.76 m; length of Gallery, 'about 153 feet', 46.63 m. From Petrie's survey these dimensions are actually 16.97 m and 46.11 m respectively.
[9]. J.H. Cole, The determination of the exact size and orientation of the Great Pyramid of Giza, (Survey of Egypt, paper no.39), (Cairo, 1925).
[10]. Petrie, Pyramids and Temples, 179.
[11]. Petrie, op. cit. 42, 184; Cole, op. cit. 6. The seked was therefore 5 1/2; see G. Robins and C.C.D. Shute, Göttinger Miszellen 57 (1982), 49-54.
[12]. Petrie, op. cit. 220, proposed "some slight fractional adjustment in the 22"; but see also J.-P. Lauer, Le Mystère des Pyramides (Paris, 1974), 307-308.
[13]. Petrie, op. cit. 39, 95. For his measurements southwards from the north base, Petrie gave probable errors of ± 0.3 inch at the lower end and ± 0.8 inch at the upper end of the Ascending Passage, and ± 0.9 inch at the upper end of the Grand Gallery.
[14]. Ibid. 186-7, 220.
[15]. Ibid. 83, 95. The south wall is stated to be 537.0 ± 0.8 inches south from the pyramid's centre, which Petrie computed as 4534.1 inches from the north base.
[16]. Ibid. 59.
[17]. Ibid. 187. The pavement is missing, but was reconstructed by Petrie at half the level of the King's Chamber.
[18]. Ibid. 186.
[19]. Applying mean angle of 26° 9' 35" to horizontal distance of 1517.8 inches .
[20]. Ibid. 58, 191. The theoretical angle is 26° 33' 54", which is close to Petrie's mean result of 26° 31' 23".
[21]. Ibid. 65, 95.
[22]. See J.A.R. Legon, 'A Ground Plan at Giza', Discussions in Egyptology 10 (1988), 33-40. The side-length of the Second Pyramid is (5 x 82 + 1) or 411 cubits.
[23]. Petrie, op. cit. 65, 95
[24]. Petrie, op. cit. 65.
[25]. Calculated from the sloping length using Petrie's mean angle of 26° 16' 40".
[26]. Petrie, op. cit. 65, 71; Smyth, op. cit. II, 151, 161.
[27]. Petrie, op. cit. 58; Smyth, op. cit. II, 148, gives 26° 27'.
[28]. Petrie, op. cit. 191; see also J-Ph. Lauer, Le Mystère des Pyramides (Paris, 1974), 297.