THE 14:11 PROPORTION AT MEYDUM
First published in Discussions in Egyptology 17 (1990), 15-22.
As shown a century ago by Flinders Petrie, the 14:11 proportion of the Great Pyramid of Giza was used earlier in the IVth Dynasty in the Pyramid of Meydum. Following J-Ph. Lauer's theory for the development of this proportion in the steps of the Meydum Pyramid, Robins and Shute have recently attempted to trace its origin back further to the Step Pyramid of Djoser, with reference to the seked or measurement of slope used in the faces of the steps; and they have thereby argued against the mathematical interpretations which have been attached to the use of this profile in the Great Pyramid.
To support their theory for the origin of the 14:11 proportion, Robins and Shute refer to a plan of the Step Pyramid prepared from the measurements and drawings made by J.S. Perring in 1839. They have chosen this plan because it indicates, if rather tentatively, that the tops of the steps could have been finished horizontally, as in the Meydum Pyramid; but in this choice they have overlooked Lauer's discovery of the specially-cut casing-stones with which the tops of the steps or 'treaders' of the Step Pyramid were completed - not horizontally, but at an incline of more than 20°. At present, this slope is at least partially represented by solid masonry and not merely by debris as Robins and Shute have implied.
Let us suppose, however, that the steps of the Step Pyramid might have been finished horizontally, and that Perring's measures provide a sound basis for analysis. Robins and Shute then proceed to claim that according to Perring's drawing, 'the distance between the outer edge of each tread and the foot of the 'riser' of the step above is exactly half the height of the step, and so, as in the case of the Meidum pyramid, the average seked of the Step Pyramid defined by the edges of the steps is 5 1/2.' As Robins and Shute explain, this seked corresponds to the slope of 14 rise on 11 base, which would be indicated in the outline of the Step Pyramid provided also that the faces of the steps have a seked of 2, or a horizontal tilt inwards of 2 palms for each cubit of vertical measurement.
It is unfortunate for this hypothesis, however, that the text adjoining Perring's drawing tells us that "The degrees or stories vary in height, and gradually diminish towards the top." From the list of the actual heights of each step which is then given, it is evident that each successive step was just one cubit less in height than that preceding it; and Perring's measures in feet and inches equate with successive heights of 22, 21, 20, 19, 18 and 17 cubits. But except in the lowest step, the double layers of masonry which form these steps were kept substantially uniform, with a thickness of some 18 feet according to Perring, or about 10 cubits. It thus follows that there was no fixed relationship between the dimensions of the treaders and the adjacent risers in the Step Pyramid, and that the exact ratio of 1:2 as required by theory, arose coincidentally for only one out of the six steps. The angle obtained by filling in the steps gradually decreases with each successive step, from about 52° at the base to about 47° for the uppermost step.
This gradation in step-height draws attention to an important distinction between the construction of an authentic step pyramid, and the stepped core of a building destined to be completed as a true or geometrical pyramid. Djoser's Step Pyramid was not meant to be finished as a true pyramid; but what of the Meydum Pyramid? In spite of the many profound differences in the arrangement of the substructures and in the external buildings, Robins and Shute suggest that the Meydum Pyramid was modelled on the Step Pyramid, and they assign it to the same dynasty. Many Egyptologists, however, would dispute this dating; and in the writer's opinion, the transformation which took place in the royal monuments during the reign of Sneferu, the builder of the Meydum and large Dahshur pyramids, more than justifies the Manethonian change of dynasty. Even so, it is true that the dimensions of the steps used in the Meydum Pyramid, broadly resemble those used in the Step Pyramid.
But in the Meydum Pyramid, so far as can be judged, the steps were able to be overlaid at a constant slope with the addition of a fairly uniform filling of masonry. The builders may therefore have intended to complete the monument in one slope as a true pyramid: for in many later true pyramids, the formation of a stepped core was considered to be an essential phase in the construction. The steps would be of equal proportions, and might sometimes be paved across horizontally with stone slabs just as in the Meydum Pyramid. Neither the dressing of the step faces, nor the temporary closures of the entrance passage, exclude the possibility that the builders of the 'step-pyramid' phases of the Meydum Pyramid hoped ultimately to enclose the steps in further masonry to create a true pyramid. But if the construction of a true pyramid had been envisaged while the stepped nucleus was being built, the proportions of the steps and hence the seked of the step faces, had to be determined in accordance with the profile chosen for the final pyramid-casing.
Even accepting that the Meydum Pyramid may have passed through one or more authentic step-pyramid phases, the origin of the 14:11 profile in the proportions of the steps is by no means as certain as Lauer's convenient theory seems to suggest. The step faces are required by this theory to have a slope of 7 rise on 2 base, so that assuming the height of the steps to be just twice the breadth of the treaders, the steps could be filled in at a slope of 7 rise on (2 + 7/2) equals 5 1/2 base - this being 14 rise on 11 base or a seked of 5 1/2. The theoretical face-angle of the steps was thus:
Angle of Slope, for 7 rise on 2 base = 74° 3' 17"
Turning to the actual measurements of the step faces, however, it appears that no such definite angle was employed. On the contrary, Petrie states that "The angles of the faces are variable: the upper part of the high face is at 73° 20', the lower part 73° 54'; and the faces now built over, from the outside through to the passage, are at 74° 40' and 75°. The tendency therefore seems to be for the lower and outer parts to be steeper than the higher."
If this trend is continued down to ground-level, it is quite possible that the accretion faces started with a slope identical to the 'mastaba-angle' of 4 rise on 1 base, which Lauer ascribes to the initial mastaba in the Step Pyramid, and which was also used in the 'great mastaba' at Meydum. This gives credence to Petrie's theory that the Meydum Pyramid was itself formed around an initial mastaba, in consequence of which the first extensions of the nucleus could have been raised at the mastaba-angle. But because, as we may now suppose, this angle was too steep to be used in a stepped nucleus constructed with a riser:treader ratio of 2:1, and destined to be completed as a true pyramid with a profile of 14 rise on 11 base, it gave way to a lesser slope in the uppermost steps with a profile which was possibly defined as 10 parts rise on 3 parts base:
Angle of Slope, for 4 rise on 1 base = 75° 57' 50"
Angle of Slope, for 10 rise on 3 base = 73° 18' 3"
The steps being 20 cubits high, the horizontal subtention or tilt inwards of each step-face would have increased from 5 cubits for the lower steps to 6 cubits for the upper steps. The subtention defined by the profile of 7 rise on 2 base falls within this range, with a value of 5 cubits 5 palms; but although this slope may have been used in one or more steps, the pyramid casing-angle deriving therefrom was only one of a number that might have been selected, varying from 5 rise on 4 base or 51° 20' 25", to 4 rise on 3 base or 53° 7' 49". As illustrated in fig. 1, the construction of these angles assumes that the builders felt constrained to exactly follow the line of the steps, which was certainly not the case.
Robins and Shute interpret the 14:11 proportion as a seked of 5 1/2, and say that according to the problems given in the Rhind mathematical papyrus, the slopes of pyramids were predetermined by their seked. It is not known, however, whether this unit was used as a practical measure during the IVth Dynasty; and because of its inherently small scale, it was not suited to the accurate control of a casing-angle. A cubit-rod set vertically against the casing of the Great Pyramid, so that a horizontal offset might be measured in palms and digits, would not come half-way up the first course. The builders could not, in any case, have overlooked the fact that since the cubit contained seven palms, a seked of 5 1/2 would give a ratio of height to semi-base of 7 to 5 1/2, or 14:11; and it is evident that the builders understood the consequences of using a particular slope in terms of just such a ratio. Whilst in the Great Pyramid, therefore, a module of 20 cubits was employed in the semi-base of 20 × 11 or 220 cubits, and height of 20 × 14 or 280 cubits, so also we find that in the Meydum Pyramid, as noted by Petrie, a module of 25 cubits was employed in the base of 25 × 11 or 275 cubits, and height of 25 × 14/2 or 175 cubits.
To consider now the measured dimensions of the Meydum
Pyramid, the results
of Petrie's survey of the base are as follow:
For the external casing-angle, Petrie's measures of 51° 49', 51° 54' and 52° 4', are within a few minutes of arc of the angle of 51° 50' 34", as required by the seked of 5 1/2, or profile of 14 rise on 11 base. The height of the Meydum Pyramid was therefore 275/2 × 14/11 or just 175 cubits, as we have seen.
The Design of the Entrance Passage
In the Meydum Pyramid, the quest for a given placing of the entrance is suggested by the fact that the slope of the entrance passage was increased from 27° 36' in the lower part to 30° 23', as the passage approached the outer casing. At the same time, since levelling lines marked in cubits above the base were found by Rowe on the step faces near the entrance, it is certain that the builders kept a check on their levels in the pyramid; and by increasing the slope of the entrance passage, they may have been attempting to bring the entrance to a particular height above the base. According to Petrie's survey, the entrance is at the level of 720.7 inches, which is 34.91 cubits for the cubit of the base; and it is therefore at practically one-fifth of the pyramid's total height, or at 175 × 1/5 equals 35 cubits above the base.
As a result of this design, as shown in fig. 2, the entrance was set 140 cubits below the apex of the pyramid, and 110 cubits northwards from the central axis. A direct reference was thus made to the 14:11 proportion, and the section of the pyramid above the level of the entrance represented one-half of the section of the Great Pyramid, with its height of 280 cubits and semi-base of 220 cubits. To this we may add that the sloping length of the entrance passage of 57.85 ms or 110.3 cubits, taken with the overall angle of slope of 28° 29', gave a vertical length of 52.6 cubits, or close to 35 × 3/2 equals 52 1/2 cubits. The foot of the entrance passage was therefore situated 17 1/2 cubits below the base - this being just one-tenth of the pyramid's height of 175 cubits.
To have achieved these relationships in a monument conceived initially as a mastaba or step pyramid would have been a matter of some difficulty, unless the builders had been able to carry out a major modification of the original design. It was established by G.A. Wainwright, however, that the base of the final true pyramid was constructed roughly 2.5 metres above the base of the internal stepped nucleus, and on gravel instead of rock. This leads us to suppose that the level of the base may have been raised so as to bring about these harmonious relationships between the position of the entrance passage and the proportions of the pyramid itself.
It now remains to consider how the 14:11 profile of the Meydum Pyramid originated - if in fact its association with the seked of some of the step faces has been misinterpreted. Theorists of the Great Pyramid have always maintained that this profile was intended to "square the circle", such that the height of the pyramid would equal the radius of a circle, the circumference of which is given by the perimeter of the base. Now although this objective cannot be proven, it does not seem unreasonable in the context of the solar cult which was emerging when the Meydum Pyramid was built. The writer would like to point out that the transference of the offering temple to the east side of the pyramid, the provision of an open-air offering-place and the absence of a ka-door or serdab, all indicate a precedence of the solar cult over that of the late king. The circle was the most familiar symbol of the solar disk; and as the square base of the pyramid was aligned with respect to the four corners of the Earth, or the 'four supports of heaven', the equation of circle and square in the form of the edifice may have symbolised a merging of solar and earthly forces.
Mathematicians may perhaps object that the approximation to pi of 22/7, which is implicit in the 14:11 proportion, is supposed not to have been known at the time of the IVth Dynasty. And yet, if the builders had attempted to measure the circumference of a circle with a diameter of one cubit, 7 palms or 28 digits, they would inevitably have found that it amounted to 22 palms, more or less; so that a deliberate squaring of the circle in the height of the Great Pyramid of 280 cubits, and semi-base of 220 cubits, is neither contentious nor particularly difficult to comprehend.
1. W.M.F. Petrie, Medum (London,
THE 14:11 PROPORTION AT MEYDUM
John A.R. Legon
The 14:11 proportion of the Great Pyramid of Giza is known to have been employed earlier in the IVth Dynasty in the Pyramid of Meydum, and is conjectured to have originated in the stepped nucleus of this monument. In the present article, however, an argument is put forward to show that this hypothesis is not well-founded; and in its place, through a relationship between the square base of the pyramid and the circle of the solar disk, support is given to the theory that the 14:11 proportion was conceived as a symbol of the solar cult.