BMCR 2004.09.21

Architecture and Mathematics in Ancient Egypt

, Architecture and mathematics in ancient Egypt. Cambridge: Cambridge University Press, 2004. xxii, 280 pages : illustrations ; 26 cm. ISBN 0521829542. $100.00.

Corinna Rossi’s Architecture and Mathematics in Ancient Egypt is the revised version of the author’s doctoral dissertation submitted to the University of Cambridge. The main purpose of the book is to analyze aspects of Egyptian architectural design in relation to select mathematical phenomena that are specifically Egyptian. Rossi maintains that scholars have mostly approached Egyptian architecture from modern mathematical perspectives rather than from the perspective of Egyptian mathematics itself. The other major error against which Rossi cautions time and again is the search by modern scholars for geometrical relations within plans of Egyptian monuments. The author argues that although many of these relations are possible to discover a posteriori from the plan, this does not necessarily mean that they were intended by the builders, or that they truly constituted the geometric bases of the designs. This is a problem that in fact pervades the study of ancient and medieval monuments in general. Rossi also very explicitly signals throughout the book that she does not intend to unfold ‘mysteries’ or ‘secrets’ about Egyptian monuments as many have tried to do since time immemorial, or provide formulae that will explain proportions in Egyptian architecture, arguing that whatever formulae or mathematical devices the Egyptians resorted to must have been very simple and straightforward rather than esoteric. Regardless of secrets and formulae, however, in a book with the present title, one still would have liked to see clearer elements of Egyptian mathematics, as promised, as well as a more direct relation thereof to Egyptian architecture. It is only in the final section of the book that Rossi addresses such a direct relation in dealing with a distinctively Egyptian expression of slope and triangles in the design of the pyramids, but this discussion, far from addressing the larger picture, is a case study of a highly specialized character. In any event, Rossi’s primary intention is to point out that there is something quite wrong about the way geometry and mathematics have been applied to the study of Egyptian architecture and suggest future lines of more accurate inquiry. Overall, the book is beautifully written, thoughtful, and extremely reader-friendly in its organization. Its mathematical content is nothing to be afraid of either, as it is reasonably accessible, though at times dense, with many helpful drawings, diagrams and charts.

The book comprises three principal parts, each with a designated aim. The first, “Proportions in ancient Egyptian architecture,” is where the author critically reviews the theories put forward to explain proportions in Egyptian architecture from the nineteenth century up to our time, by authors ranging from Viollet-le-Duc to the Egyptian architect and Egyptologist Alexander Badawy. This Rossi calls the ‘architectural’ approach, and it is in this part that she displays for the reader the never-never land of geometrical lines superimposed on plans and sections in search of neat proportional relations among component parts of architectural entities. One point of great interest here is a distinction drawn between “two main groups of scholars who have dealt with ancient Egyptian architecture: architectural historians and Egyptologists” (p. xv). The objective of Rossi’s Part I is hence to explore the architectural historians’ approach “corrected by the Egyptological studies,” whereas that of Part II, “Ancient Egyptian sources: construction and representation of space,” is to address the archaeological/Egyptological perspective in order to present a more “historically correct mathematical point of view.” Even though it is the author’s wish that the book act as a bridge between architects and Egyptologists, one cannot help getting the sense that she is inclined to value Egyptology over architectural history in trying to attain a healthier understanding of the matters at hand. To this end, in order to brush away the misconceptions and “faulty assumptions” generated by searching for mathematical or proportional rules in Egyptian architecture on the basis of modern plans and sections, Part II provides the reader with a number of examples of Egyptian architectural working drawings and models along with their assessment, although these by no means suffice to give one a definitive sense of the authentic Egyptian understanding and practice of architectural design and building either. The author’s main argument here is that plans drawn to scale probably never formed an integral aspect of Egyptian architectural praxis, as they do ours, since Egyptian building, like all ancient and/or traditional construction, evidently contained a significant mnemonic component that enabled contemporary builders to transfer architectural ideas to three dimensions without extensive use of architectural drawings or models understood in the modern sense of the word. What shines through most clearly in the book is perhaps this mnemonic nature of Egyptian architectural design and building praxis rather than the presence of a tangible Egyptian mathematical system which the architects drew on in their designs of monuments. The aim of Part III, entitled “The geometry of pyramids,” is nevertheless stated by the author to be an attempt to reconcile the ‘architectural’ and ‘Egyptological’ views, and this Rossi mainly carries out through combining the study of ratio and proportion with a well-documented and straightforward Egyptian manner of expressing and managing the slopes of a variety of pyramids from all ancient Egyptian history.

There are a number of topics and arguments in the book that are of particular interest and at which one could take a closer look. Part I introduces two mathematical phenomena that have concerned scholars in relation to Egyptian architecture: three types of triangles and the Golden Section. The triangles are the well-known 3-4-5, the equilateral, and the isosceles triangle with the ratio 8:5 between the base and the height, and hence called the 8:5 triangle by Badawy, and ‘Egyptian’ by Viollet-le-Duc, who had erroneously identified it as the vertical section of the pyramid of Khufu. The merit of this ratio, according to Viollet-le-Duc, is its not being derived from either the half or the third or the fourth of the larger dimension, and hence its encompassing a relation which the eye could not define, resulting in “a means of obtaining the contrasts which are necessary for satisfying the first law of proportions” (p. 11). Certainly analogous in this regard is the Golden Section, not easy to define in such terms either, which Rossi deals with in some detail. She first introduces the rules and mechanics of this ratio with diagrams, and draws attention to the fact that its numerical value, 1.618033989…, is an irrational number defined by the Greek letter phi, which is also the number toward which the ratio between two consecutive numbers of the Fibonacci Series gradually converges. Rossi is rather skeptical about theories that have tried to see the Golden Section in Egyptian architectural design as the preferred ratio, even though she does acknowledge that it was one of the proportions used by Egyptian architects, along with the proportions of the triangles already mentioned. Even though the author seems to have made good use of nineteenth-century primary sources that deal with such theories in her research, their presentation in the book is over-synoptic, without clarification for the reader as to what exactly these theories entailed. Furthermore, Rossi herself seems somewhat torn between the existence and the absence of a set of clear rules in Egyptian architectural design, as she mentions the 1965 theory of Badawy as “able to explain many factors,” a theory which suggests that a number of triangles including the 8:5 and the Golden Section were used by the Egyptians in the design of their monuments among other geometric forms and ratios. Despite many points of criticism directed toward Badawy, Rossi acknowledges that “Badawy’s schemes seem to work,” and goes into a greater detail than she does for others in explaining his theory of how certain triangles seem to have been used in laying out the ground plans of certain Egyptian temples.

Within Part I, the author also addresses the question to what extent the Egyptians made conscious use of the Pythagorean theorem and the numbers phi and pi. Rossi again cautions against retrojecting modern mathematical notions into Egyptian antiquity and argues that, even though the Egyptians may have used some Pythagorean triplets, “this does not imply that the Old Kingdom architects were the creators of the Pythagorean symbolism of numbers or were aware of all the mathematical implications of Euclid’s formulation of the theorem of Pythagoras” (p. 65). Even though Rossi’s emphasis on eschewing anachronisms in dealing with the Egyptian case is well made, both the theorem of Pythagoras and the numbers phi and pi are such fundamental mathematical concepts that I wonder if it is necessary to discuss, even in the absence of contemporary sources and evidence, whether or not the Egyptians or any other pre-Hellenic ancient people in the area would have been fully familiar with such ‘universals’ and their implications. Perhaps the fundamental error is to take the theorem of Pythagoras as the ‘canonical’ or the ‘most developed’ version of this mathematical phenomenon, as if a complete understanding of the latter could not have been possible without reference to Pythagoras. Rossi actually comes back to this question in Part III, where she again states that notwithstanding the Egyptians’ knowledge of “some triplets, … it is not necessary to suggest that the Egyptians were acquainted with more or less complicated versions of the Theorem of Pythagoras as early as the Old Kingdom. … If they (the triplets) were used in ancient Egypt, it was probably in their simplest version of right-angled triangles easy to construct” (p. 218). This the author states in relation to the proportions of the pyramid of Khafre, whose half section corresponds to a 3-4-5 triangle, which she actually suggests may not have been deliberate. Even though Rossi’s overall observations on how ancient Egyptian builders must have kept design and construction matters simple, practical and mnemonic are brilliant, one wonders on the other hand how justified it is to ascribe such a degree of naiveté and fortuitousness to the monumental building activity of the Fourth Dynasty.

Coincidence in proportional configurations of Egyptian buildings is in fact a theme Rossi addresses further. She mentions, for example, that the section of the first intended version of the Bent Pyramid along the diagonal of its base can be approximated by an 8:5 triangle, which she sees as “a tantalizing coincidence” (p. 70). Pondering whether the phenomenon of matching geometrical figures drawn on plans belongs to the realm of coincidence as well, Rossi suggests a third possibility: “a general human tendency towards certain geometric patterns.” In a section entitled “Psychological experiments and involuntary trends,” Rossi briefly reviews some theories that have concentrated on “psychological experiments on the supposed preference for certain geometrical figures” or “the involuntary tendency towards patterns related to the Golden Section,” but her conclusion is again negative, namely that such tendencies could not have governed Egyptian architectural design. One of the most crucial matters the author brings up in relation to the nature of Egyptian architectural design is the presence of a kind of restricted or ‘esoteric’ knowledge among those involved in architectural praxis that would have governed many of the concepts the author tries to probe in the book. This matter is tucked into the end of Part I (p. 87), but it deserves a more extensive treatment, since it is clear from the book itself that factors other than coincidence and psychological tendencies and preferences governed Egyptian architecture, factors that are clearly very sophisticated, albeit unknown to us.

In Part II, Rossi examines examples of Egyptian architectural drawings and models, and, noting the lack of scale and precision in these tools, she concludes that these seem to have been not more than quick reminders of certain details within the building process. What she perhaps does not emphasize enough within this framework is the fact that architectural practice’s being primarily mnemonic and based on many simple practical devices does not preclude the presence of proportional and geometric relations in the monuments created. To demonstrate further how Egyptian designers and builders would have adapted their designs during the building process itself in accordance with various factors generated by the site, the author abruptly addresses the rock-carved royal tombs of the Valley of the Kings, pointing out that the slopes of the later ones among them may have been reduced to avoid damage to older tombs in their vicinity. One question that may be posed regarding the choice of the rock-carved tomb in discussing Egyptian architectural design and building praxis is to what extent this type of space and construction activity is representative of Egyptian architecture in general, whose mainstream manifestations are built rather than carved. Another minor conceptual discrepancy comes at the end of Part II when the author, having concluded that the planning process of Egyptian buildings cannot be fully reconstructed through the many fragmentary data available, compares what the Egyptian architects must have done to what, according to J.J. Coulton, the Greek architects did with their temples, “incomplete preliminary planning,” with many details left to be decided at a later stage while the overall dimensions and general disposition were already established (p. 175). Even though both the Egyptians and the Greeks may have followed similar methodologies in open-ended building, I believe that the highly sculptural and compact nature of a peripteral Doric temple with interconnected dimensional problems such as the angle contraction and the so-called ‘optical refinements’ would have required a tighter degree of in situ adjustments and fine-tuning, when compared to the more expansive and colossal Egyptian temple or the pyramid with a much more straightforward single form.

The greatest strength of Part III is the reference to the Egyptian concept of seked, known from Middle Kingdom sources, the horizontal displacement of a sloping line for a vertical drop of one Egyptian ‘royal cubit,’ by which the Egyptians understood and dealt with slope, rather than our angles. Rossi’s premise is that seked s, sometimes of whole numbers (in authentic Egyptian units of palms), determined the dimensions and proportions of the major as well as the minor pyramids of Egyptian antiquity. From this standpoint, there clearly was a quest for the optimum pyramidal slope, as the Bent Pyramid started out with a seked of 4 palms, corresponding to a slope of 60 degree, an equilateral triangle in section, and culminated in the ‘safer’ seked of 5 palms, a slope of about 54 degree 30′ on its steeper lower part. The slopes of the Giza pyramids fall around a seked of slightly over 5 palms, and hence about 52 degree (Khufu and Menkaure) and 53 degree (Khafre), following the optimum or the ‘safe’ slope already established, but aspiring to be slightly steeper, with the half-section of Khafre’s pyramid corresponding to the 3-4-5 triangle which the Sixth Dynasty royal pyramids also seem to have adopted. Rossi concludes that “the various seked s of pyramids work perfectly well even without the introduction of the Pythagorean triplets,” and that even if some of the triplets might have been used by the workmen, “there is no need to assume that these triplets had any symbolic meaning” (p. 221). This, I believe, is again hard to ascertain as we do not know what the Egyptian builders had in mind regarding symbols, and one cannot be so sure that the slopes of the Giza series, with a clear persistence of an angle around 52-53 degree, were not the result of any thought or concept other than a quest for the optimum form and structural safety. After all, within the confines of the book, the author’s mathematical focus on the pyramids is restricted to slopes alone, while the siting and orientation of the Giza group or the alignment of the shafts inside the pyramids with aspects of the night sky, all mathematical/astronomical feats, are not addressed. Overall, however, Part III gives an excellent, albeit dense, survey of pyramid slopes in ancient Egypt, with matters of decorum also touched upon such as the different tendencies of proportions among royal pyramids and their satellite or subsidiary components, all summarized and documented in a table as well.

In conclusion, the book does more in terms of warning the reader against the presence of a variety of nebulous theories on geometry and proportion in Egyptian architecture than of maintaining a consistent and thorough focus on the relation between architecture and mathematics in ancient Egypt. To my mind, the tendency to rule out anything ‘symbolic’ in the mathematical bases of Egyptian architecture, even with the dearth of evidence, and to place ‘symbolism’ or ‘symbology’ solely within the realm of the aforementioned eccentric or outdated theories is an aspect of the book that may be considered misleading. Notwithstanding these points, however, the book is certainly a valuable contribution to the study of Egyptian architecture, as well as a product of much serious thought and careful documentation.