The Plan of the Giza Pyramids

My original discovery of the Giza Site Plan of Three Pyramids was first published by the Archaeology Society of Staten Island in 1979. [i] Since then, I have described the most significant elements of the site plan in a series of articles in the journals Discussions in Egyptology, [ii] and Göttinger Miszellen. [iii] The following text is based on my article in Discussions in Egyptology, Vol. 10, but includes new material and new illustrations.

Now that a detailed topographical study of the Giza Plateau is in progress, [1] it is interesting to consider the results of the excavations and survey carried out by Flinders Petrie in 1880-2, when the exact dimensions and relative positions of the pyramids of Khufu, Khaefre and Menkaure, were established by triangulation. [2]

With reference to Petrie's survey-data, the present paper reviews the evidence first put forward by the writer in 1979, [3] showing that the sizes and relative positions of the three pyramids were determined by a single unifying ground plan.

The existence of a dimensional scheme underlying the placing of the three pyramids is suggested in the first instance by the very regular arrangement of these pyramids on the Giza plateau. As a result, the sides of the bases and the distances that separate them define consecutive axial distances from north to south and from east to west. The three pyramids were accurately aligned with respect to the four cardinal points, and were displaced from one another in a configuration which satisfies the requirements of a coherent dimensional design. Certain technical difficulties relating to the site chosen for each pyramid in turn also suggest that there must have been some significant constraint, in addition to factors such as ease of construction or the selection of the most favourable architectural setting, which determined where each of the three pyramids was positioned.

The Survey Data

Using some of the finest surveying equipment available in his day, Petrie fixed the positions of the main stations in his triangulation to within 3 millimetres. [4] The accuracy of his work is proven by the fact that his result for the mean side of the Great Pyramid differs from the value obtained in the later meticulous survey carried out by J.H. Cole for the Egyptian Government in 1925, [5] by only 1.5 centimetres, even though nearly all of the outer casing of the Great Pyramid is now missing. The side-lengths in Cole's survey were established with the help of traces of the casing-edge still remaining on the pavement in some places where the casing itself has been destroyed.

The dimensions of the three pyramid-bases as determined by Petrie in inches are given in Table I, together with the average variations in the lengths of the sides, and the orientations of the three pyramids with respect to true north. The distances separating the centres of the pyramids, as computed by Petrie along axes constructed parallel to the mean azimuth of the Second and Great Pyramids, of -4' 52", [6] are given in Table II.

To obtain the axial components of spacing between the bases of the three pyramids, the distances between the pyramid-centres can be combined with the sides of the bases, to give the dimensions as listed in Table III. There will be small differences at the corners due to the slight variations in the azimuths of the sides with respect to the axes of the plan; but since the Second and Great Pyramids have the same orientation within two minutes of arc - a remarkably small divergence - it can be shown that these differences average only about 5 cm. The Third Pyramid, however, differs in azimuth from the Second and Great Pyramids by about 1/3 degree in a clockwise direction, so that elements of about 25 cm are generated at the corners relative to the mean components of spacing. The exact positions have been verified through an analysis of Petrie's original coordinates of survey.

When the various dimensions are expressed in terms of the Royal Egyptian Cubit, with the value of 20.620 inches or 0.52375 metres as determined by Petrie from his measurements inside the Great Pyramid [7] and as stated by Edwards, [8] it is found that almost all of the mean components of spacing correspond to whole numbers of cubits, or in some cases half numbers of cubits, to within 0.1 cubit. With reference to these potential design values, as listed in Table III, the largest difference is only 0.23 cubit or 12 centimetres.

The Great Pyramid

To investigate the possibility of an intended positional relationship between the bases of the three pyramids, it seems reasonable to assume that any dimensional scheme would have been laid out starting from the base of the Great Pyramid, which was the first of the Giza pyramids to be constructed. The dimensions would thus have been measured out from the north-eastern portion of the plateau, southwards and westwards towards the Second and Third Pyramids.

It is generally accepted that the sides of the Great Pyramid measure 440 cubits, although only the longest or south side has exactly this length according to the survey by Cole. [9]
In terms of the Giza royal cubit of 0.52375 metres, the actual mean side of 230.364 metres corresponds to 439.8 cubits, with an average variation in the sides of only 6 cm or 0.1 cubit. Petrie suggested that an adjustment may have been effected in order that the perimeter of the base should express the so-called 'pi-proportion' in relation to the height of 280 cubits, with greater accuracy than the value for pi of 22/7. [10] In this case, the theoretically exact mean side-length would be 439.822... cubits. It seems that the builders achieved this result while retaining the round number of 440 cubits in the south side.

The Placing of the Second Pyramid

Now turning to the relative position of the Second Pyramid, the analysis of Petrie's survey data has shown that the mean north-south spacing from the Great Pyramid is 250.2 cubits. It seems likely, therefore, that the builders intended to place the north side of the Second Pyramid on a line just 250 cubits southwards from the south side of the Great Pyramid. Taking further distances along the north-south axis, the impression of a deliberate design is strongly supported. The distance southwards from the north side of the Great Pyramid to the south side of the Second Pyramid is 1101 cubits, or only 0.1% greater than the round-figure of 1100 cubits. This is just 2½ times the side of the Great Pyramid of 440 cubits, and the south sides of the two pyramids are separated by an axial distance from north to south of close to 3/2 × 440 equals 660 cubits.

Based upon these plausible formative dimensions of 440 and 250 cubits, we can obtain a provisional design-value for the side-length of the Second Pyramid as follows:

Initial Side-length for Second Pyramid = 440 × 3/2 - 250 = 410 cubits

I have termed this the initial side, because Petrie's survey-data gives an actual mean side for the Second Pyramid of 8474.9 inches or exactly 411 cubits. The average variation in the sides is only 1.5 inches or 4 cm. Thus although the above derivation accounts for a side of about 410 cubits in preference to say, just 400 cubits, an adjustment of one cubit appears to have been made to the actual value. Reasons for this adjustment will be given shortly.

Now taking dimensions along the east-west axis, Petrie's data places the west side of the Second Pyramid 624 cubits westwards from the west side of the Great Pyramid. Again with an adjustment of one cubit, therefore, this dimension may be equated with an initial value of just 2½ × 250 or 625 cubits, making it analogous to the distance of 2½ × 440 or 1100 cubits along the north-south axis.

The side-length of the Second Pyramid and the position relative to the Great Pyramid may thus be ascribed to a simple modular scheme, based on dimensions of 440 and 250 cubits. In this design, two-and-a-half squares of 440 cubits are arranged from north to south so as to define the base of the Great Pyramid and the position of the south side of the Second Pyramid; while two-and-a-half squares of 250 cubits are arranged from east to west to give both the north-south spacing between the two pyramids of 250 cubits, and the east-west dimension of 2½ × 250 or 625 cubits. The side-length of the Second Pyramid is thereby defined as (660 - 250) or 410 cubits, while the east-west spacing from the Great Pyramid is found to be (625 - 410) equals 215 cubits:

With reference to this modular scheme, however, the builders evidently subtracted one cubit from the dimension of 625 cubits, and added one cubit to the dimension of 1100 cubits, making the side of the Second Pyramid equal to 411 cubits, and the east-west spacing from the Great Pyramid, (624 - 411) or 213 cubits (see Table III). These adjustments suggest that further factors must have influenced the final choice of dimensions; and indeed these factors will now be found to have anticipated the inclusion of the Third Pyramid in the completed ground plan.

The Overall Dimensions of the Site Plan

As shown by the survey-data in Table III, the Third Pyramid extends the scheme of the Second and Great Pyramids by 631 cubits towards the south, and 353.5 cubits towards the west. The overall dimensions of the Giza site plan are thereby defined along the two axes. Now computing these enclosing dimensions from the component parts, we find:

Overall East-West Dimension = 440 + 624 + 353.5 = 1417.5 cubits

Overall North-South Dimension = 440 + 661 + 631 = 1732 cubits

For the mathematician, these dimensions are of striking significance, for they represent the square roots of the first two prime numbers, two and three, multiplied by a factor of 1000. The theoretical values are:

1000Ö2 = 1414.21...

1000Ö3 = 1732.05...

The north-south dimension thus expresses the value of 1000Ö3 exactly, while the east-west dimension displays a discrepancy of about 3.3 cubits with respect to the true value of 1000Ö2. Nonetheless, this error amounts to only about 0.2%, and will be explained in due course. Given that a planned relationship between the Second and Great Pyramids is already clearly indicated, the combination of these fundamental square-root values in the sides of the enclosing rectangle provides compelling evidence that the Third Pyramid was included in the site plan.

The question therefore arises as to how these values of square roots were understood by the pyramid architects. Contrary to the popular assumption that the ancient Egyptians had no knowledge of such numbers, examples of square roots are found in the mathematical papyri. [11] There is no reason why the pyramid architects should not have been able to accurately evaluate the square roots of two and three by a process of trial and error.

It is important to realise, however, that such numbers have geometrical significance, and occur naturally in simple geometrical constructions. Thus in a perfect square with unit side, the diagonals have a length equal to the square root of two. A rectangle constructed with a width of one unit and a length of Ö2 units, will have a diagonal of Ö3 units. The occurrence of these simple square-root values in the Giza plan, therefore, may signify nothing more than the development of the diagonals in squares and rectangles.

That the site plan was indeed based upon such simple geometry is strongly suggested by the following construction, which begins with a square of side 1000 cubits. First developing the diagonal lengths as indicated above, we obtain an enclosing rectangle with sides equal to 1000Ö2 and 1000Ö3 cubits, or with the values of 1414 and 1732 cubits (to the nearest cubit). The diagonal of the enclosing rectangle is now drawn, and is found to intersect the diagonal of the originating square such as to mark off precisely the number of 1101 cubits in the axial distance southwards from the north side of the Great Pyramid to the south side of the Second Pyramid. The major division of the north-south dimension of 1732 cubits into segments of 1101 and 631 cubits is therefore explained, and can be constructed on paper by drawing an arc as shown below:

The division of the overall north-south dimension at the south side of the Second Pyramid thus represents an immediate development of the originating geometry, and the simplicity and elegance of the design leave no doubt that the architect's intention has been correctly ascertained . When the dimensions are evaluated, the components are found to be almost exactly whole numbers of cubits, with the values of 631.03 and 1101.02 cubits.

In my initial study of the Giza Site Plan, it seemed to me that the dimension of 1101 cubits was simply derived from the modular scheme connecting the Second and Great Pyramids, and was equal to 2½ × 440 cubits adjusted by one cubit. This adjustment appeared to be necessary to obtain the overall dimension of 1732 cubits in conjunction with the dimension of 631 cubits, which relates to the scheme of the Third Pyramid.

The geometrical nature of the division only became apparent as a result of my analysis of the Bent Pyramid, which was published in 1990. Here, I demonstrated that the height of the Bent Pyramid of 200 cubits was divided at the change of slope in the ratio of Ö2 : Ö3, or into sections of 89.9 and 110.1 cubits. The analysis also showed that the mean lower slope of the Bent Pyramid represents the hypotenuse of a 1, Ö2, Ö3 right triangle, and that the upper vertical height of the Bent Pyramid is therefore equal to the lower slant height.

Now in the Giza site plan, as shown in the diagram below, the overall dimensions can be developed starting from a 1:2 rectangle measuring 1000 by 2000 cubits. The east-west dimension of 1000Ö2 cubits is given once again by the diagonal length in each component square of 1000 cubits, but the north-south dimension is constructed with reference to the diagonal of the 1:2 rectangle. This has a length of 1000Ö5 cubits, which is the diagonal of the enclosing rectangle of 1000Ö2 by 1000Ö3 cubits. Consequently, by applying the length of 1000Ö5 cubits as hypotenuse to the dimension of 1000Ö2 cubits as one adjacent side, a right triangle with second adjacent side of 1000Ö3 cubits is obtained. The diagonal of the resulting site-plan rectangle is now drawn, and is found to intersect the opposing diagonal of one of the initial squares so as to divide the length of the 1:2 rectangle in the exact ratio of Ö2:Ö3, or into parts of 899 and 1101 cubits. Thus the major north-south division of the site plan is defined as before.

The above geometrical construction would not have been laid out on the Giza plateau, but represents a working plan drawn to scale on papyrus or drawing board. If we assume that the architect was unable to compute the diagonal lengths using Pythagoras' theorem, but had instead to rely on empirical measurements, then such measurements would have been made on a suitable scale, perhaps on a levelled scribing floor. If, for example, a square of 10 cubits was set out, careful measurements would have given a diagonal of 99 palms, and hence a ratio between the side and diagonal of a square of 70 : 99, which is an excellent approximation to the square root of two. The result of scaling up this proportion to the dimensions of the site plan would have been an east-west dimension of 9900 palms, or 1414 cubits and 2 palms.

Filling in the Details

The evaluation of Petrie's survey-data has shown that the north-south overall dimension of the Giza site plan, and the division at the south side of the Second Pyramid, both satisfy the geometrical requirements with complete accuracy. The question therefore arises as to why the east-west overall dimension diverges from the value of 1000Ö2 or 1414.2 cubits by about three cubits. One good reason is immediately obvious, for the axial component of 353.5 cubits westwards from the Second Pyramid to the west side of the Third Pyramid has the exact value of 250Ö2 cubits:

250Ö2 = 353.55... cubits

This dimension is therefore just one-quarter of the theoretical overall dimension on the same axis. It is the diagonal length in the modular layout squares of 250 cubits, which define both the north-south spacing between the Second and Great Pyramids, and the dimension of about 2½ × 250 or 625 cubits on the east-west axis.

When this element of 353.5 cubits is added to the distances as already defined in our initial modular scheme along the east-west axis, however, we obtain an overall east-west dimension of (440 + 625 + 353.5) equals 1418.5 cubits, which is more than 4 cubits greater than the value of 1000Ö2 cubits. Seen in this context, the subtraction of one cubit from the component part of 2½ × 250 or 625 cubits may be understood as a judicious adjustment which improved the accuracy of the square-root value in the overall dimension, without unduly undermining the integrity of the modular scheme.

Further investigation reveals the great skill with which the final dimensions were selected in order to obtain the maximum coherence and significance for the design of the Giza plan as a whole. We have already seen that the addition of one cubit to the modular dimension of 2½ × 440 or 1100 cubits reflects some fundamental geometry, and that the side-length of the Second Pyramid was therefore increased to 411 cubits, while the east-west spacing from the Great Pyramid was reduced to (624 - 411) equals 213 cubits. Both adjustments greatly enhanced the geometrical structure of the plan.

Firstly, the axial distance westwards from the centre of the Second Pyramid to west side of the Third Pyramid, or the western boundary of the plan, is found to be:

411/2 + 353.5 = 559 cubits

250Ö5 = 559.01...

Another important square-root value is thus defined explictly in the dimensions of the site plan, with reference to the module of 250 cubits. This dimension is equal to the diagonal in a 1 : 2 rectangle measuring 250 × 500 cubits, and is also just one-quarter of the site-plan diagonal of 1000Ö5 cubits. The semi-base of the Second Pyramid can therefore be defined theoretically as 250(Ö5 - Ö2) or 205.46... cubits, thus further justifying the adjustment of one cubit which is evident in the actual side-length of 205.5 × 2 or 411 cubits.

Secondly, an exactly analogous result is found in the axial distance westwards from the centre of the Great Pyramid to the Second Pyramid. According to our initial modular plan, the east-west spacing between these two pyramids was (625 - 410) equals 215 cubits, but the above adjustments gave an actual dimension of (624 - 411) or 213 cubits. As a result, as shown by Petrie's survey-data, the distance in question is just 250Ö3 cubits:

440/2 + 213 = 433 cubits

250Ö3 = 433.01...

This is the length of the diagonal in a rectangle measuring 250 × 250Ö2 cubits, such as that contained in the Giza site plan between the north-south dimension of 250 cubits and the east-west dimension of 353.5 cubits. It is now evident that a 1 : Ö3 : 2 right triangle with hypotenuse of 500 cubits and adjacent sides of 250 and 433 cubits, can be constructed on the spacing between the Second and Great Pyramids, as shown below.

In closely connected dimensions along the east-west axis of the plan, therefore, the square roots of two, three, and five, are all accurately represented in terms of a module of 250 cubits, and correspond to one-quarter of the respective square-root values in the enclosing rectangle. Needless to say, such a coherent and precise set of relationships could never have arisen by chance, but instead prove the existence of a highly integrated and skilfully conceived design. As we have indicated above, these square-root values find an origin in the elementary geometry of squares and rectangles based on a module of 250 cubits.

We must now record one further fundamental square-root value, in the axial distance southwards from the Great Pyramid to the south side of the Second Pyramid. We have:

(250 + 411) = 661 cubits

250Ö7 = 661.437...

It will be seen, therefore, that the addition of one cubit to the assumed 'initial' side of the Second Pyramid of 410 cubits not only brought about the exact expression of 250Ö3 and 250Ö5 cubits on the east-west axis, but also resulted in a good approximation to 250Ö7 cubits on the north-south axis. Even the small error of less than 0.07% seems to have been mitigated by a slight adjustment to the mean north-south spacing between the Second and Great Pyramids, which is shown by Petrie's survey to be 5159.7 inches or 250.23 cubits. Thus the above dimension was brought to within about 0.2 cubit of the ideal value of 250Ö7 cubits.

Once again, this square-root dimension is implicit in the geometry which has already been described. It can be constructed as the diagonal in a rectangle measuring 250Ö2 by 250Ö5 cubits, or as the diagonal in a rectangle measuring 250Ö3 by 500 cubits. In conclusion, the square roots of the first four prime numbers are all defined in the Giza plan in terms of a module of 250 cubits.

It is, of course, beyond all probability that so many accurate and elegant expressions of square-roots could have arisen by chance, especially when the plan itself is framed by a rectangle measuring 1000Ö2 by 1000Ö3 cubits. These findings provide irrefutable proof that the dimensions and relative positions of the Giza pyramids were determined by a coherent plan in which simple geometry played an important part.

The Scheme of the Third Pyramid

According to Petrie's survey-data, the sides of theThird Pyramid measure 4153.6 inches or 201.44 cubits on average, with a mean variation in the sides of just 3.0 inches (Table I). This suggests a nominal dimension of 200 cubits, adjusted by 1.5 cubits. The spacing from the Second Pyramid along the east-west axis thus became (353.5 - 201.5) equals 152 cubits, or 1064 palms, and so contains the same number of palms as there are cubits in the further distance of (624 + 440) or 1064 cubits on the same axis, from the west side of the Second Pyramid to the east side of the Great Pyramid.

An all-embracing and exact solution for the dimensions of the Third Pyramid is to be found, however, in a Scheme of the Circle Squared, which is based on a square of just 500 cubits, as shown below. Since the semi-diagonals of this square measure 250Ö2 or 353.5 cubits, they give the axial distance between the west sides of the Second and Third Pyramids when aligned along the axes of the plan. The square is now placed in a circle, the circumference of which is found to be 2220 cubits for the value of pi of 3.140.

Now squaring this circle by superimposing a square with the same perimeter, or with sides of 555 cubits, we find that all of the dimensions relating to the Third Pyramid in the Giza site plan are defined. The sides of the pyramid are found to measure (555 - 353.5) equals 201.5 cubits, in close agreement with Petrie's survey results. The east-west spacing from the Second Pyramid will be (353.5 - 201.5) equals 152 cubits, while the south side of the Third Pyramid is placed (555/2 + 353.5) equals 631 cubits southwards from the south side of the Second Pyramid - again in agreement with Petrie's data. Reference to the above diagram will show how these dimensions are generated.

Finally, we must account for the slight change in the orientation of the Third Pyramid with respect to the axes of the site plan. Although only amounting to about 19 minutes of arc in a clockwise direction, there is no doubt that this rotation was deliberate and was intended to displace the corners by just 0.5 cubit with respect to the mean axial positions of the sides. As a result, significant round tens of cubits were defined in distances relative to the Second Pyramid which would otherwise have been greater or less by half a cubit.

Whilst, therefore, the mean north-south spacing between the Second and Third Pyramids is (631 - 201.5) or 429.5 cubits, the north-east corner of the Third Pyramid is in fact 430 cubits southwards from the Second Pyramid. This is just twice the initial east-west spacing between the Second and Great Pyramids of 215 cubits. At the same time, the north-west corner of the Third Pyramid is (429 + 411) or 840 cubits southwards from the north side of the Second Pyramid, and this dimension is just three times the height of the Great Pyramid of 280 cubits. These results point to the further geometrical development of the plan, as shown in my article in Göttinger Miszellen124 (1991).

NOTES

i. J.A.R. Legon, 'The Plan of the Giza Pyramids', Archaeological Reports of the Archaeology Society of Staten Island, Vol.10 No.1. New York, 1979.

ii. J.A.R. Legon, 'A Ground Plan at Giza', DE 10 (1988), 33-40; 'The Giza Ground Plan and Sphinx', DE 14 (1989), 53-60.

iii. J.A.R. Legon, 'The Design of the Pyramid of Khaefre', GM 110 (1989), 27-34; 'The Geometry of the Bent Pyramid', GM 116 (1990) 65-72, 71; 'The Giza Site Plan Revisited', GM 124 (1991), 69-78.

1. M. Lehner, 'The Development of the Giza Necropolis: The Khufu Project', MDAIK 41, 1985, 109-143.

2. W.M.F. Petrie, The Pyramids and Temples of Gizeh (London, 1883). First edition only for full details, 34-36.

3. J.A.R. Legon, 'The Plan of the Giza Pyramids', Archaeological Reports of the Archaeology Society of Staten Island, Vol.10 No.1. New York, 1979.

4. Petrie, op.cit., 24.

5. 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).

6. Petrie, op.cit.,125.

7. Ibid, 179.

8. I.E.S. Edwards, The Pyramids of Egypt, (Harmondsworth, 1947), 208.

9. Length of south side, 230.454 metres; Cole, op.cit., 6

10. Petrie, op.cit., 220.

11. R. J. Gillings, Mathematics in the time of the Pharaohs (Cambridge, Mass., 1972), 214