Irrational numbers and pyramids

by Gay Robins and C.C.D.Shute

Discussions In Egyptology 18, 1990, pp 43-53 Reproduced with permission

Interpretations of the external form of pyramids are of two sorts, according to whether the slope of the faces or the height of the apex is considered to be the more important determinant. Both approaches have recently been expounded in Discussions in Egyptology. What we believe to be the more orthodox view was developed in our paper on the 14 to 11 proportion. [1] This holds that the size and shape of a pyramid was predetermined by the lengths of the sides of its square base, and by the slope of its triangular faces usually measured in sekeds which were units of lateral displacement in palms for a vertical drop of seven palms or one royal cubit. The slope would be controlled throughout construction and would decide the eventual height. A seked has been ascribed to a number of Egyptian pyramids by Lauer, although sometimes he gives other ratios when he thinks that the seked did not apply. [2] The Great Pyramid is believed to have a seked of 5 1/2, which converts into an inclination of 51° 51’ 35″. This pyramid has been surveyed more often and more carefully than any other, and estimates for the angle of inclination agree closely with the seked-derived figure.

It is interesting that Perring, a century and a half ago, before the seked was known to modern scholars, thought that the Great Pyramid was constructed to give a perpendicular height to base ratio of 5:8 and an inclined height (apothem) to base ratio of 4:5. [3] The first ratio gives a gradient of 5:4 or 1:25, and the second is equivalent to a gradient of 1:249. According to modern ideas, the gradient is 7:5 1/2 = 1.273. Perring’s inference can be said to show, on the one hand his ingenuity, and on the other the inadequacy of a purely numerological approach.

The other interpretation of pyramidal form lays particular emphasis upon the height. It can be considered to be more theoretical, since the height unlike the slope, cannot be checked directly at all stages of building. The pre-eminent importance of the height has been championed by numerologists anxious to find encoded in the Great Pyramid the irrational numbers pi and phi. The supposed significance of pi in relation to the 14 to 11 proportions of the Maidum Pyramid, resembling those of the Great Pyramid, has now been urged in DE by Legon. [4] Some numerologists have argued that the Great Pyramid encapsulates a squaring of the circle, since the 14 to 11 proportions mean that the area of a circle whose radius is equal to the Pyramid’s height is nearly the same as that of a rectangle whose sides are equal to the circle’s diameter and the width of the pyramid’s base. [5] Against this, the Rhind Mathematical Papyrus makes it clear that the Egyptians obtained an approximation for the area of a circle by equating it, not to a rectangle, but to a square whose side is equal to eight-ninths of the circle’s diameter. [6]

The involvement of pi can also be adduced from the observation that in a pyramid with 14 to 11 proportions the perimeter of the base is nearly equal to the circumference of a circle whose radius is the pyramid’s height. If half the width of each side of the base is 5 1/2 units and the height is 7 units, the perimeter is 44 units and the circumference is 14 pi units, giving 22/7 as a good approximation for the ratio (pi) of the circumference of a circle to its diameter.

Following Legon, [7] one must agree that the Egyptians could have drawn a circle with a diameter of a royal cubit equal to 7 palms, measured the circumference, and found it to be 22 palms or 3 1/7 royal cubits, so obtaining empirically a value for pi. Nevertheless, we do not believe that it was the purpose of the 14 to 11 proportion to celebrate such a discovery, since we consider the explanation that it derives from the relative heights and widths of risers and treads in the steps of the Maidum Pyramid core and the Pyramid of Djoser to be simpler and more plausible. We concur with Lauer, who wrote as follows:

Nous avons ainsi la preuve évidente que cette pente [14/11] a resulté directement de la proportion même du profil des grandes pyramides à degrés de la IIIe dynastie; et rien n’autorise, par consequent, à supposer que l’architecte de Khéops ait pu, plus que celui de la pyramide de Meidum, avoir conscience de l’existence de rapports phi ou pi recelés dans ces proportions de la pyramide. [8]

Legon in the course of his article has tried to play down the whole concept of the seked, saying that:

it is not known … whether this unit was used as a practical measure during the IVth dynasty. [9]

We disagree: there is evidence from actual monuments besides the Great Pyramid that it was so used. For instance, although the inclination of the lower part of the Bent Pyramid was given by Perring a century and a half ago as 54° 14’ 46″, [10] it is now generally agreed that the value should be 54° 27’ 44″, to conform with a seked of 5. [11] The discrepancy is not great considering the practical difficulties that arise in getting accurate and consistent results in the field. Furthermore, the risers of the steps of the pyramid of Djoser and the Maidum Pyramid core had almost certainly an original seked of 2. Perring’s estimate for the inclination of the risers in Djoser’s pyramid is about half a degree flatter than this, [12] but the condition of the monument, than as now, was not such as to favour accurate measurement, as can be seen from Perring’s own drawing. [13] In the case of the Maidum Pyramid core, Legon quotes measurements by Petrie giving different slopes for different risers. [14] Since the outer shell of the Maidum Pyramid collapsed in antiquity, it would not be surprising if the inner core had also undergone some shifting.

In view of the structural damage it is possible that some other findings by Perring relating to Djoser’s pyramid should be treated with caution. According to him, as Legon points out, [15] the steps diminish in height as they ascend. Since according to Perring, [16] the widths of all the treads except the lowest is constant, this would mean that the slope between the edges of successive steps becomes flatter at higher levels. But if this were so, the generally held view that the steps if infilled would produce a true pyramid with flat faces, implied by the regular series of gradation shown in countless diagrams and in Lauer’s model of the Step Pyramid, would be incorrect.

As a corollary to what has been said, it follows that the horizontals placed by Perring on his cross-sectional drawing showing the north and south profiles of Djoser’s pyramid may not indicate the true levels of the steps. [17] As to whether the treads were originally horizontal, it is probably now impossible to say. The angle of isolated casing stories is not a sure guide, since all depends on how they were set into the masonry courses, which are not themselves horizontal but slope in a downwards and inwards direction.

When considering the status of the seked, it is necessary to take account of the Third Pyramid at Giza, that of Menkaure (Mycerinus). The inclination ascribed by authorities following Lauer to this pyramid is 51° 20′ 25″, [18] which corresponds not to a true seked where the units of lateral displacement relate to a drop of 7 units, but to a lateral displacement of 4 units for a drop of 5. One has to ask, is this measurement correct? The slope of Menkaure’s pyramid is difficult to measure on account of stone loss from the upper region, and in consequence many different estimates have been given for its slope; Lauer in selecting a gradient of 5:4 was perhaps over influenced by the theories of his compatriot, the 19th century French architect and restorer Viollet-le-Duc. [19] Our own observations, based on our photographic method [20] reinforced by goniometric measurements of the casing stones round the entrance to the pyramid where they are smoothest, suggest a seked of 5 1/2, the same as that of the Great Pyramid.

There are some points relating to the seked in Legon’s paper that we do not understand. First, in spite of having just said that it is not known whether the seked was used as a practical measure in the 4th dynasty, he remarks that estimates for the external casing angle of the Maidum pyramid (which he attributes to the 4th dynasty) made by Petrie are within a few minutes of arc on either side of the angle demanded by a seked of 5 1/2. [21] What better evidence could one hope for, short of the architect’s blueprint on a papyrus, that this seked was in fact used, and how else could the slope have been obtained? Secondly, in spite of having dealt with many measurements himself all greater than a cubit, he makes the baffling assertion that the seked was not suited to the accurate control of a casing angle because a cubit rod was not long enough. [22] Of course the Egyptians could have had measuring devices corresponding to a set square or protractor even if they do not survive today. It is well known that for longer measurements they stretched cords or ropes. The “stretching of the rope” ceremony is one of the reasons for rejecting Mendelssohn’s suggestion that the apparent involvement of pi in the dimensions of the Great Pyramid resulted from the base having been measured not linearly but by rolling a drum. [23]

We shall now give consideration to some irrational numbers other than the transcendental number pi. In an earlier paper aiming to prove that the pyramid site at Giza had a predetermined ground plan, Legon employed measurements involving the square roots of 2 and 3. [24] As a justification for their introduction he wrote:

The use of these dimensions implies that the builders could calculate the numerical values of square roots, a fact which is already well known.

He does not, however distinguish between irrational roots which cannot be given an exact numerical value, and rational roots for which values could be obtained from tables of squares. Two of the latter are attested, 1 1/4 as the square root of 1 1/2 + 1/16, and 6 1/4 as the square root of 39 1/16. There is no record of the use of irrational square roots, but Gillings has suggested that approximations for these could have been obtained by trial and error, working between limits one of which was too small and the other too large. [25] This approach might have appealed to numerate scribes who were adept in manipulating unit fractions. For them it could have been a challenge to find approximations that were both close and elegant. Others less theoretically minded might have preferred a more practical approach. With such in mind, we wish to propose here another possible method, depending on measurement and some knowledge of the properties of right-angled triangles, which is particularly suitable for the irrational square roots of integers such as 2 and 3, and does not involve trial and error.

The square root of 2 is the length of a diagonal of an isosceles right-angled triangle with sides of length 1 (or the length of the diagonal of a square of side 1). The scribe measures the diagonal of such a triangle in which the unit of length is a royal cubit of seven palms, and finds the length to be approximately 1 royal cubit and 3 palms. This length is equal to a “double remen,” the Egyptian unit called the remen being equal to 5 palms. If this is split into 1 + 1/7 + 2/7 cubits, the scribe can at once write, using unit fractions, 1 + 1/4 + 1/7 + 1/28 as an approximation for the square root of 2, since 1/4 + 1/28 is the value given for 2/7 in the table for doubling unit fractions in the Rhind Mathematical papyrus. [26] The approximation is not particularly close.

A better result can be got by taking the small cubit of six palms rather than the royal cubit as the unit. The measurement for the hypotenuse of an isosceles right-angled triangle with sides of length 1 small cubit is 1 small cubit, 2 palms and 2 fingers (4 fingers make a palm), or 1 1/3 + 1/12 cubits, so that the scribe can now write the much more elegant expression 1 + 1/3 + 1/12 as a near equivalent for the square root of 2. The closeness of the approximation can be found by squaring. The square of 1 + 1/3 + 1/12 is 1 + 2/3 + 1/6 + 1/9 + 1/18 + 1/144. [27] Scribes were adept in the process known as “completing to 1”, so it would have been recognised at once that the fractional series 2/3 + 1/6 + 1/9 + 1/18 sums to 1. In other words, the expression 1 + 1/3 + 1/12 when squared differs from 2 by only 1/144. [28]

The overall east-west dimension in cubits said by Legon to occur at the Giza site is equal to the square root of 2 multiplied by 1000. [29] Using our second approximation, which for closeness combined with elegance is probably the best that can be got, the figure becomes 1416 2/3. [30] The designers of the site, if they did make use of the square root of 2, would have presumably have got it, as Legon implies, by measuring the diagonal of a square. It is less clear how he imagines they would have obtained the overall north-south dimension, which he says is exactly equal to the square root of 3 multiplied by 1000. An approximation can be got by measuring the side of a right-angled triangle whose hypotenuse is 2 small cubits and whose other side is 1 small cubit. This is equivalent to finding the height of an equilateral triangle whose side has length 2. The measurement is found to be 1 small cubit, 4 palms and 1 1/2 fingers, or 1 2/3 + 1/16 small cubits. It is thus possible to write as an approximation for the square root of 3 the expression 1 2/3 + 1/16. This approximation multiplied by 1000 yields 1729 1/6. [31]

As to whether the Egyptians did have these or similar values for the square roots of 2 and 3, and whether they were incorporated deliberately into the Giza site plan or whether they arise as a result of mathematical coincidences, one has to keep an open mind. Next, however, we would like to consider whether the Egyptians had the knowledge of right-angled triangles that would have enabled them to undertake the measuring procedures that we have suggested. Some historians of mathematics, wishing to decry the mathematical ability of the Egyptians as compared with the Babylonians and the Greeks, have doubted whether the Egyptians even knew of the 3:4:5 right-angled triangle. We do not concur with this view, since we think that the best explanation for the appearance of a 5 1/4 seked with the pyramid of Khephren in the 4th dynasty and its seemingly universal adoption in the 6th dynasty is that it gives 3:4:5 proportions to the half-base width, height, and apothem of the face. This, pace Legon, would have facilitated the cutting of casing stones.

The triad 3,4,5 is the simplest example of a so-called Pythagorean triple, three numbers such that the sum of the squares of two of them is equal to the square of the third. Did the Egyptians know of such numbers, or were they intellectually behind other early civilisations such as that of Mesopotamia and, later, China? [32] It has been inferred from the cuneiform tablet Plimpton no.322 that the Babylonians obtained Pythagorean triples from reciprocals. [33] The Egyptians were accustomed to calculating with reciprocals, some of them involving more than one fraction, as in problems 9, 67 and 71 of the Rhind Mathematical Papyrus. They obtained then by dividing one of the numbers into 1 to get the other. The simplest of all reciprocal pairs are 2 and 1/2. It is possible to obtain the 3,4,5 triple from these reciprocals as follows.

1 = 2 x 1/2 = (1 + 1) x (1/4 + 1/4)

= [(1 + 1/4) + (1 – 1/4)] x [(1 + 1/4) – (1 – 1/4)]

= (1 + 1/4)2 – (1 – 1/4)2

Multiplying through by 4² gives

42 = 52 – 32

or 5² = 4² + 3²

The same result can be got from the reciprocals 3 and 1/3, leading to

32 = 52 – 42

Of course, the 3,4,5 result could have been got very easily from inspection of a table of squares, and splitting 25 into 16 and 9. We do not claim that the Egyptians necessarily did use the method of calculation outlined above, although they could have, but we give it because it can readily yield any Pythagorean triple, and because of the slur, which we think to be unjustified, that the Egyptians were mathematically inept, and hampered by being harnessed to unit fractions. [34] In fact, some of the calculations in the mathematical papyri are quite complex. Moreover, it would be wrong to suppose that what survives in such student manuals represents the sum total of ancient Egyptian mathematical knowledge.

We have said nothing so far about phi, the other irrational number which numerologists love to think is, together with pi, incorporated in the design of the Great Pyramid. The number phi is derived from the “golden section.” [35] In a pyramid with 14 to 11 proportions in which half the width of the base is 1 unit, the height is approximately equal to the square root of phi. There is no reason to suppose that the Egyptians were aware of this. There are various ways of constructing a line of exact length phi, but no evidence that the Egyptians ever did so. A line with its length exactly equal to the square root of phi can be obtained from the side of a right-angled triangle whose hypotenuse is phi and whose other side has unit length. An approximate numerical value for the square root of phi can then be obtained by measuring, according to the small cubit method described in this paper. The approximate length of the line turns out to be 1 small cubit, 1 palm, and 2 1/2 fingers, or 1 1/4 + 1/48 small cubits, giving as an approximation for the square root of phi 1 + 1/4 + 1/48. This expression multiplied by 5 1/2 is equal to 7 – 1/96, showing that a right-angled triangle with one side adjacent to the right-angle equal to 1 and the other equal to the square root of phi nearly has 14 to 11 proportions, being out by a small fraction. The result explains how the square root of phi comes to relate to the height of the Great Pyramid simply as a result of the use of a seked of 5 1/2. It does not follow that the ancient Egyptians knew of this consequence, or that they had any concept of phi and its square root.

1. Gay Robins and C.C.D. Shute “The 14 to 11 proportion in Egyptian architecture” DE 16 (1990), 75-80.

2. J.-P. Lauer, Le mystère des pyramides (Paris 1974). The check list of pyramids in J. Baines and J. Malek Atlas of Ancient Egypt (Oxford 1980), 140-141 gives angles of inclination in degrees, minutes and seconds based on Lauer’s values.

3. in: H. Vyse Appendix to Operations carried on at the pyramids of Gizeh 3 (London 1842), 106.

4. J.A.R. Legon “The 14 to 11 proportion at Meydum” DE 17 (1990), 15-22.

5. P. Tompkins, Secrets of the Great Pyramid (New York 1971), ch.15.

6. See Gay Robins and C.C.D. Shute. The Rhind Mathematical Papyrus (London 1987), 44-46.

7. Legon op.cit. (n.4), 22.

8. Lauer op.cit. (n.2), 308.

9. Legon op.cit. (n.4), 19.

10. in: H. Vyse op.cit. (n.3), 67. Perring’s value for the angle of inclination of the upper part of the Bent Pyramid is 42° 59′ 26″. Our own measurements, using our photographic method, Gay Robins and C.C.D. Shute ”Determining the slope of pyramids” GM 57 (1982), 49-54, suggest that the upper part of the Bent Pyramid had a seked of 7 1/2. This corresponds to an angle of inclination of 43° 1′, to the nearest minute, which is close to Perring’s estimate.

11. Baines and Malek op.cit. (n.2).

12. in: H. Vyse op.cit. (n.3), 141.

13. ibid. pl.A, fig.1, between pp.42-43; J.S. Perring, The pyramids of Gizeh from actual survey and measurement 3 (London 1839-42), pl.10 fig. 1.

14. Legon op.cit. (n.4), 17.

15 ibid., 15-16.

16. in: H. Vyse op.cit. (n.3), 42.

17. ibid., pl. between pp.40-41; J.S. Perring op.cit.(n.13), pl.9 fig.6.

18. e.g. Baines and Malek op.cit.. (n.2).

19. E.E. Viollet-le-Duc, Discourses on architecture (trans. B. Bucknall, London 1959).

20. Gay Robins and C.C.D. Shute op.cit. (n.10).

21. Legon op.cit. (n.4), 20.

22. ibid., 19.

23. K. Mendelssohn, The riddle of the pyramids (London 1974), 64-74.

24. J.A.R. Legon “A ground plan at Giza” DE 10 (1988), 33-40.

25. R.J.Gillings Mathematics in the time of the pharaohs (New York 1972), 214-217.

26. See Robins and Shute op.cit. (n.6), 34.

27. Although in general Egyptians wrote only unit fractions, 2/3 was conceived as legitimate and had its own sign.

28. Another approximation for the square root of two involving the remen and first proposed by Petrie (see F.L. Griffith “Notes on Egyptian weights and measures” PSBA 15 (1893), 301-2, and Museum of Fine Arts, Boston Egypt’s golden age (Boston 1982), 99) states that a royal cubit is the length of the diagonal of a square whose side is one remen. This is the equivalent of equating the square root of 2 to 1 + 1/3 + 1/15, a poor approximation since the square of 1 + 1/3 + 1/15 is short of 2 by 1/25. Since the slope of the edge of a pyramid is equal to the seked of the face multiplied by the square root of 2, this approximation, equal to 7 divided by 5, accounts for the fact, noted by Lauer, that the edges of the lower part of the Bent Pyramid have a slope which is nearly 1:1. This follows from the seked of the face being 5, since 5 times the square root of 2 is approximately equal to 5 x 7/5 which is 7.

29. Legon op.cit. (n.24), 39 fig 1.

30. The figure 1416 2/3 obtained from our approximation for the square root of 2 is closer to the sum of the measurements equal to 1417 1/2 which according to Legon’s fig.1 make up the east-west dimension than the figure 1414.2 obtained by multiplying the actual square root of 2 by 1000. We do not think, however, that this necessarily proves that the Egyptians were working to a scheme involving an approximation for the square root of 2.

31. This is not particularly close to Legon’s assessment of the north-south dimension as being equal to the exact square root of 3 multiplied by 1000 = 1732 cubits, Legon op.cit. (n.24), 39, fig.1.

32. B.L. van der Waerden, Geometry and algebra in ancient civilisations (Berlin 1983), 1-5; F.J. Swetz and T.I. Kao, Was Pythagorus Chinese? Pennsylvania State University Studies no.40 (University Park and London 1977).

33. O.Neugebauer, The exact sciences in antiquity (2nd ed., New York 1957), 36-40.

34. ibid., ch.4; V.G.Child, Man makes himself (New York 1951), ch.8, pp.164,168.

35.The value of phi, 1/2(1 + the square root of 5), can be got by solving the quadratic equation (phi)² – (phi) = 1

Gay Robins
Emory University
Atlanta, GA, USA
C.C.D. Shute
Christ’s College
Cambridge, England

This article discusses whether the irrational numbers Pi and Phi were employed in pyramid design and concludes that they weren’t.