My initial findings on the Giza Site Plan of Three Pyramids were first briefly summarized in a pamphlet published by the Archaeology Society of Staten Island in 1979.[i] Subsequently, I described the most significant features of the integrated site plan in several articles in the journals Discussions in Egyptology [ii] and Göttinger Miszellen. [iii] Further research has shown that many more factors must be taken into account, without alterating the basic framework of the dimensions in royal cubits which I described in 1979. The following text was based on my article in Discussions in Egyptology Vol. 10, but now includes much 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 DataUsing some of the finest surveying equipment available in his day, Petrie asserted that he had 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 meticulous survey carried out by J.H. Cole in 1925, [5] by only 1.5 centimetres, even though nearly all of the outer casing of the pyramid is 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 had been destroyed.Further confirmation of the accuracy of Petrie's survey has resulted from an analysis by Glen Dash of the data obtained by Lehner and Goodman for the Giza Plateau Mapping Project. [6] Surprisingly, although the GPMP encompassed the base of the Great Pyramid, the dimensions and relative positions of the Second and Third Pyramids were not given. In a significant endorsement of Petrie's work, Glen Dash has converted Petrie's coordinates for the corners and centre points of the three pyramids to the GPMP grid. [7] 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 were computed by Petrie along axes constructed parallel to the mean azimuth of the Second and Great Pyramids of -4' 52", [8] and 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 - 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 can be calculated using Petrie's original survey coordinates. |
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 [9] and as stated by Edwards, [10] 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 PyramidTo 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. [11] 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 made 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 π of 22/7. [12] In this case, the theoretically exact mean side-length would be 439.822... cubits. It now seems that the base of the casing was laid out as a perfect square with the theoretical side-length of (140 × π) or 439.822... cubits, while a displacement of exactly 440 cubits from north to south and from east to west was obtained in the site plan by rotating the square base of the casing by 1' 23" with respect to the axes of the plan. This explains Petrie's observation that the pyramid-casing had been skewed by about this amount relative to the core-masonry. The Placing of the Second PyramidNow 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 PlanAs 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: 1000√2 = 1414.21...
1000√3 = 1732.05... The north-south dimension thus expresses the value of 1000√3 exactly, while the east-west dimension displays a discrepancy of about 3.3 cubits with respect to the true value of 1000√2. 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. [13] 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 occur frequently in simple geometrical constructions. In a perfect square with unit side, for example, the diagonals will have a length equal to the square root of two, and a rectangle constructed with a width of one unit and a length of √2 units, will have a diagonal of √3 units. The occurrence of these simple square-root values in the Giza plan, therefore, suggests a geometrical development involving the diagonals of squares and rectangles. That the site plan was indeed based upon such fundamental geometry is indicated 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 1000√2 and 1000√3 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 original study of the Giza Plan, it seemed to me that the dimension of 1101 cubits was simply derived from the modular scheme connecting the Second Pyramid to the Great Pyramid, and was equal to 2½ × 440 cubits adjusted by one cubit. This adjustment seemed to be necessary to obtain the overall dimension of 1732 cubits in conjunction with the dimension of 631 cubits, which relates to the position of the Third Pyramid. The geometrical nature of the division became evident 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 √2 : √3, 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, √2, √3 right triangle, and so that the upper vertical height of the Bent Pyramid is 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 1000√2 cubits is once again givenby 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 1000√5 cubits, which is the diagonal of the enclosing rectangle of 1000√2 by 1000√3 cubits. Consequently, by applying the length of 1000√5 cubits as hypotenuse to the dimension of 1000√2 cubits as one adjacent side, a right triangle with second adjacent side of 1000√3 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 √2:√3, 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 DetailsThe 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 1000√2 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 250√2 cubits:250√2 = 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
1000√2
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
250√5 = 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 1000√5 cubits. The semi-base of the Second Pyramid can therefore be defined theoretically as 250(√5 - √2) 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 250√3 cubits: 440/2 + 213 = 433 cubits
250√3 = 433.01... This is the length of the diagonal in a rectangle measuring 250 × 250√2 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 : √3 : 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
250√7 = 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 250√3 and 250√5 cubits on the east-west axis, but also resulted in a good approximation to 250√7 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 250√7 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 250√2 by 250√5 cubits, or as the diagonal in a rectangle measuring 250√3 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 1000√2 by 1000√3 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 PyramidAs we have seen, the inclusion of the Third Pyramid in the Giza site plan extends the scheme of the Second and Great Pyramids by 353.5 cubits towards the west and 631 cubits towards the south. The former dimension is just the diagonal in a square of 250 cubits, or 250 √ 2 cubits, while the latter is obtained when the larger part in the division of a length of 2000 cubits in the ratio of √ 2: √ 3 is subtracted from the overall north-south dimension of the plan of 1000 √ 3, or as (1732 - 1101) equals 631 cubits. Whilst both of these dimensions have a geometrical origin, their use together in the site plan had a further significance as defining the ratio of 1:1.7850 or practically 1: (1 + π/4). The pi-proportion of the Great Pyramid can therefore be constructed within the bounding rectangle of the Third Pyramid, by subtracting the short side of 353.5 cubits from the long side of 631 cubits to give a remainder of 277.5 cubits, which is in the ratio of π : 4 to the short side and yields the pi-pyramid slope. |
As shown in the diagram above, the dimensions of the
Great Pyramid can be
developed from the scheme of the Third Pyramid because the height and
semi-base have a sum equal to the diagonal length in the square of
353.5 cubits, or 500
cubits.
According to Petrie's survey, the base of theThird
Pyramid has a mean
side-length of 4153.6 inches or 201.44 cubits, with an average
variation in the
sides of 3.0 inches (see Table I).
We may assume that
the round figure of 200 cubits was initially sketched out for the base;
but this
simple number conveys little meaning and it seems that the architect
wanted a
dimension which would derive from the position of the Third Pyramid
relative to
the Second and Great Pyramids, and enhance the mathematical import of
the Giza
plan.
The side-length of the square base of the Third Pyramid provides the solution to the following problem: to construct a square in the south-west corner of the rectangle of 277.5 by 353.5 cubits, such that the margin remaining on the long side of the rectangle is twice the margin remaining on the short side. As shown in the diagram above, this condition is satisified by the intersection of diagonals drawn on plans of 1:1 and 1:2 from diagonally opposite corners of the rectangle. The margins are found to be 76 and 2 x 76 or 152 cubits, leaving (353.5 - 152) = (277.5 - 76) = 201.5 cubits for the base of the Third Pyramid. It so happens that the larger margin, which represents the east-west spacing between the Second and Third Pyramids, is exactly one-seventh of the consecutive axial distance of 1064 cubits from the west side of the Second Pyramid to the east side of the Great Pyramid. At the same time, the north-south spacing between the Second and Third Pyramids is (353.5 + 76) = 429.5 cubits, or 0.5 cubit less than the round number of 430 cubits, and it is also 0.5 cubit greater than the number of 429 cubits which combines with the side-length of the Second Pyramid to give a distance of (429 + 411) or 840 cubits northwards from the Third Pyramid the the north side of the Second Pyramid. This dimension is just three times the height of the Great Pyramid. These alternative results were obtained in practice by rotating the base of the Third Pyramid by about 17 minutes of arc in a clockwise direction with respect to the azimuth of the site plan. The corners were thus moved by 0.5 cubit relative to the mean axial positions of the sides, so that the north-south spacing from the Second Pyramid varies from 429 to 430 cubits. The displacement of the base as measured along the axes of the plan, of (201.5 + 1) or 202.5 cubits, is exactly one-seventh of the overall east-west dimension of 1417.5 cubits. The size and position of the Third Pyramid may be ascribed to an all-encompassing Scheme of the Circle Squared. As shown below, this design is based on a square of 500 cubits, the semi-diagonals of which measure 250√2 or 353.5 cubits and give the axial distance between the west sides of the Second and Third Pyramids. The square is placed in a circle, the circumference of which is found to be 2220 cubits for the value of π 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 result. 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. Some notes on the further geometrical development of the plan will be found in my article 'The Giza Site Plan Revisited' in Göttinger Miszellen124 (1991). |
NOTESi. 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. Glen Dash, 'New Angles on the Great Pyramid' , Aeragram 13-2, Fall 2012, 10-19 7. Glen Dash, 'Where, Precisely, are the Three Pyramids of Giza?'. http://www.academia.edu/6056783/Where_Precisely_are_the_Three_Pyramids_of_Giza 8. Petrie, op.cit.,125. 9. Ibid, 179. 10. I.E.S. Edwards, The Pyramids of Egypt, (Harmondsworth, 1947), 208. 11. Length of south side, 230.454 metres; Cole, op.cit., 6 12. Petrie, op.cit., 220. 13. R. J. Gillings, Mathematics in the time of the Pharaohs (Cambridge, Mass., 1972), 214 |