The Golden Ratio In Ptolemaic Temple Design

Published in Göttinger Miszellen 194 (2003), pp. 59-63.

The golden ratio in Ptolemaic temple design Gyula Priskin Apart from some dubious claims about the Great Pyramid (built ca. 2500 BC), little proof has so far emerged that would indicate the ancient Egyptians knew the division in extreme and mean ratio (golden ratio). The examination of the dates in the constructional history of the Ptolemaic temples at Edfu and Dendera (237–20 BC) reveals a 200-year-long temporal scheme in which the individual dates – signifying the different stages of the building work – were positioned by the golden ratio. This is clear evidence from antiquity for the practical application of the golden ratio in a religious setting, suggesting a possible Egyptian origin for the concept.

The division in extreme and mean ratio – the division of a line in two segments so that the shorter segment is to the longer one as the longer segment is to the whole line – was first explicitly stated in the history of mathematics in Hellenistic Egypt, in Euclid’s Elements (ca. 300 BC) (Herz-Fischler 1998, 1). This concept of proportionality, popularly known as the golden ratio or golden section and expressed with the irrational number  = 1.618033… (or 1/ =  = 0.618033…) has since been associated with a wide array of phenomena outside mathematics, ranging from aesthetics to the growth patterns of plant life (Dunlap 1999, 1-2). As far as ancient Egypt is concerned, it is pre-eminently referred to in connection with the Great Pyramid (ca. 2500 BC) (Herz-Fischler 2000, 80-111; Dunlap 1999, 3). Although there is now some textual evidence suggesting otherwise (Priskin 2001, 110-14), reigning scholarly opinion no doubt still attributes it to a mere coincidence that the golden ratio shows up in the pyramid’s proportions (Robins & Shute 1985, 110-11). Further attempts to make a case for the use of the golden ratio in Egyptian architectural design (Badawy 1965) have also been received with much – and I hasten to add, well justified – scepticism (Kemp & Rose 1991, 111-26). However, the dedicatory inscriptions describing the progression of building work through time on the twin temples at Edfu and Dendera (237-20 BC) (De Wit 1961; Amer & Morardet 1983) provide us with a solid set of data clearly indicating the use of the golden ratio in the temples’ temporal layout. Originally, in the inscriptions the dates assigned to a constructional event are recorded according to a simple formula: the scribe first counts the regnal year of the current ruler, then he specifies in which one of the twelve months of the 365-day-long Egyptian civil year the event took place, identifying it either by a name or by a number from one to four and the name of the season (Akhet, Peret or Shemu), and finally he gives the day of the month by a number from one to thirty. The five epagomenal days, if needed, are named as one of the days above the year. Altogether, on the basis of the inscriptions twelve dates can be identified and determined to the day with almost absolute certainty. To make the sequence of these dates intelligible for the modern reader, Table 1 puts them – with the help of established chronologies (Skeat 1954 & 1993) – in two modern chronological frames: (1) the Julian calendar that is used by historians to report events prior to AD 15 October 1582 Gregorian, and (2) the Julian day, a continuous count of days starting on 1 January 4713 BC (= JD 0, at 12h 00m 00s universal time), which is a convenient aid to reckoning the time between any two or more dates.

Published in Göttinger Miszellen 194 (2003), pp. 59-63.

Table 1 Dates of construction at Edfu and Dendera. Dates A–I are not in doubt and are widely accepted (Cauville & Devauchelle 1984, 32-47; Depuydt 1997,123-5). Date J is also absolutely certain (Amer & Morardet 1983, 256; Devauchelle 1985, 173). Date K can be read II Peret 19 or III Peret 19 with equal certainty in the original hieroglyphic text (Devauchelle 1985, 173), and thus equals either 15 February or 16 March 29 BC. Here 15 February is opted for on the force of the presented mathematical scheme. Date L is calculated by adding 34 civil (365-day-long) years to date J in accordance with the relevant building inscription (Amer & Morardet 1983, 256). That date L is anchored to date J must be due to the fact that Augustus abolished the Egyptian civil calendar between the two dates, in around 30-29 BC (Skeat 1993, 1-4) or 26 BC (Hagedorn 1994, 213). The Roman conquest of Egypt must also be responsible for the lack of dedicatory inscriptions in the pronaos at Dendera, with the exception of an ambiguous line in Greek (Winter 1989, 76).

Abbr. Julian calendar

Julian day

Egyptian civil date

Event

EDFU A

23 Aug 237 BC

1 635 094

III Shemu 7 (Year 10 of Ptolemy III)

1st foundation of naos

B

17 Aug 212 BC

1 644 219

III Shemu 7 (Year 10 of Ptolemy IV)

2nd foundation of naos

C

3 Feb 176 BC

1 657 173

I Peret 1 (Year 5 of Ptolemy VI)

installation of gates

D

10 Sep 142 BC

1 669 810

IV Shemu 18 (Year 28 of Ptolemy VIII)

consecration

E

2 Jul 140 BC

1 670 471

II Shemu 9 (Year 30 of Ptolemy VIII)

construction of pronaos starts

F

5 Sep 124 BC

1 676 380

IV Shemu 18 (Year 46 of Ptolemy VIII)

pronaos finished

G

28 Jun 116 BC

1 679 233

II Shemu 11 (Year 54 of Ptolemy VIII)

death of king; construction of minor elements

H

7 Feb 70 BC

1 695 893

II Peret 1 (Year 11 of Ptolemy XII)

consecration

I

5 Dec 57 BC

1 700 943

IV Akhet 1 (Year 25 of Ptolemy XII)

decoration ends; installation of gates

DENDERA J

16 Jul 54 BC

1 701 896

III Shemu 14 (Year 27 of Ptolemy XII)

foundation of naos

K

15 Feb 29 BC

1 710 876

II Peret 19 (Year 1 of Augustus)

temple put in service

L

7 Jul 20 BC

1 714 306

III Shemu 14 (J + 34  365 days)

construction of naos finished

Published in Göttinger Miszellen 194 (2003), pp. 59-63.

When we examine the relations between certain of these dates, we will discover that they have been laid down in various but simply discernible ways in accordance with the golden ratio. As two of these relations involve a set of ‘mixed’ dates from Edfu and Dendera (equations 1 and 4 below), they give a new dimension to the close ties between the two temples, which culminated yearly in the ritual unification of Horus and Hathor, the divinities of Edfu and Dendera, respectively. This unity of the two cult places was well documented by the Egyptians themselves on the walls of the Edfu temple (Chassinat 1930, 124-136). So, for example, by looking at the time running from date A to date J, we will find that date F divides this period in extreme and mean ratio (accuracy to four decimal places is observed in all subsequent equations): F–A JD 1 676 380 – JD 1 635 094 41 286 = = = 1.61804 =  J–F JD 1 701 896 – JD 1 676 380 25 516

(1)

That the placement of date F is not fortuitous is further shown by its relation to date E. By calculation JD 1,676,380 – JD 1,670,471 = 5909 days pass between the two dates, but the relevant dedicatory inscription states that the length of time needed for the construction of the pronaos at Edfu was 16 years 2 months and 10 days (De Wit 1961, 293), that is, 5910 days. This period is not proportionate in terms of the golden ratio with any other period in Table 1, but does display the characteristic number of the division in extreme and mean ratio when set against the supposed length of the solar year of the Egyptians (365.25 days), first documented in the Canopus Decree, a royal edict of Ptolemy III Euergetes aimed unsuccessfully at reforming the civil calendar and issued just one year before the commencement of the construction of the Edfu temple, 238 BC (Clagett 1995, 326-31): 5 910 ÷ 10 = 1.61806 =  365.25

(2)

Date K is also positioned so that the double division in extreme and mean ratio of the period J–K equals K–L: K – J JD 1 710 876 – JD 1 701 896 8 980 = = = 2.61807 = 2 L – K JD 1 714 306 – JD 1 710 876 3 430

(3)

Yet two more dates can be accounted for with the concept of the golden ratio if we assume a date 25 years prior to date J, the foundation of the Dendera temple (J-25; JD 1,692,771), just as at Edfu there were two foundation ceremonies (dates A and B) separated by 25 civil years exactly.1 Making a one day adjustment to J-25 (= JD 1,692,772; cf. comments on date E above), we will find that date H divides the period J-25–I in extreme and mean ratio: I–H JD 1 700 943 – JD 1 695 893 5 050 = = = 1.61807 =  H – J-25 JD 1 695 893 – JD 1 692 772 3 121

(4)

The period between dates A and B, the only element in the temporal constructional scheme of the Edfu and Dendera temples which I have so far come to realize and already drawn attention to (Priskin 2001, 115 and in press), gives us a clue why the division in extreme and mean ratio was the organizing principle of the scheme, because 25 Egyptian civil years comprised 9125 days, which in turn included almost precisely 309 lunations. This equation – firmly evidenced only in a papyrus from AD 144, but surely known several centuries earlier (Clagett 1995, 298)

Published in Göttinger Miszellen 194 (2003), pp. 59-63.

– may be rephrased as a simple relation between the length of the Egyptian civil year (365 days) and the length of the mean synodic month (29.53059 days) involving the golden ratio: 365 ÷ 20 = 0.61800 =  29.53059 All things considered, nine out of the twelve dates in Table 1 (A, B, E, F, H, I, J, K and L) are determined in connection with the division in extreme and mean ratio. Date D is precisely 18 civil years before date F, and is thus in all likelihood also part of a premeditated scheme – hinted at by the Egyptians themselves (De Wit 1961, 285-6) – that must have been set up in its entirety before construction at Edfu started, and that in practice could have been carried out by a simple day count from the initial date A. Date G records the death of Ptolemy VIII Euergetes II, and therefore cannot have belonged to a preconceived plan. This leaves only date C unaccounted for – a hiatus that possibly results from the loss of some dedicatory inscriptions putting this date into context. Besides being a crucial point in the temporal scheme, date J is also marked by the most important astronomical event in ancient Egypt, the annual heliacal rising of the star Sirius (Clagett 1995, 307-9). This meant the star’s appearance above the horizon immediately before sunrise after a 70-day period of invisibility. We know that on 16 July 54 BC the axis of the Dendera temple was reorientated towards the direction of the heliacal rising of Sirius (Leitz 1991, 58-9). What made this astronomical event special in 54 BC was the fact that in the civil calendar it fell on III Shemu 14, the festival day when the ritual bondage between Edfu and Dendera was celebrated each year by the journey of Hathor to Edfu (Grimm 1994, 124-7). This leads us to conjecture that the Egyptians had already foretold and prepared for this coincidence about 200 years earlier, when they must have laid out the constructional dates for the Edfu and Dendera temples. In conclusion, by revealing a hitherto undisclosed aspect of a first-order temple building project of the Ptolemaic era we now have firm evidence that the ancient Egyptians grasped the concept of the golden ratio at some point in their history prior to 237 BC. Although built at a time when the Egyptian state had lost its independence and was ruled by a Hellenistic dynasty, the Ptolemaic temples were beyond doubt the products of the indigenous priesthood of Egypt and served as repositories of their knowledge (Kurth 1997, 153; Arnold 1999, 143). Indeed, no serious foreign influence can be detected in the temples’ purpose, role or design (Finnestad 1997, 188). The occurence of the golden ratio in the Egyptian civil calendar, introduced some time in the third millenium BC (Clagett 1995, 28), and the similar temporal use of this ratio in the construction schedule of the Edfu and Dendera temples – unparallelled in the ancient world – strongly suggest a native origin for this concept. Furthermore, the overall spatial configuration of the Dendera temple also derives from a wish to represent the golden ratio (Priskin, in press). It is therefore not at all impossible that the division in extreme and mean ratio found its way into Euclid’s Elements from direct or indirect Egyptian sources, through the Pythagoreans. Since, however, it is quite unlikely that straightforward evidence to this effect will come to light from the thin corpus of ancient Egyptian mathematical literature, further research should clarify whether – besides the fiercely debated example of the Great Pyramid – there were any other precursors to the use of the golden ratio in other areas of Egyptian culture. As I have demonstrated it here and elsewhere (Priskin 2001), investigations should perhaps primarily focus on written sources which appear in a religious context, yet

Published in Göttinger Miszellen 194 (2003), pp. 59-63.

contain enough data-like information (numbers, dates or measurements) for a mathematically based analysis.

Note 1

This interpretation may offer an explanation for the riddle that the cartouches of Ptolemy XII Auletes can be seen in the three subterranean crypts of the Dendera temple, despite the meagre two and a half years separating 16 July 54 BC from his death in the spring of 51 BC (Devauchelle 1985, 163; Winter 1989, 77). We must agree with Devauchelle that the dedicatory inscription in the naos denotes a second building phase, which is however not the construction of the pronaos, as he believes, but the second foundation of the naos, concurring with Winter’s view that the inscription in the naos must refer to the part of the temple where it is found. This leads us to conclude that the de facto building of the Dendera temple – or at least of its underground portion (after a preliminary orientation towards the heliacal rising of Sirius) – started with the first foundation on 22 July 79 BC (JD 1,692,771), leaving enough time for Ptolemy XII to have his cartouches inscribed in the crypts. The act of this first foundation may have been recorded in a text that has not come down to us.

References Amer, H. I., & Morardet, B., 1983. Les dates de la construction du temple majeur d’Hathor à Dendara à l’époque gréco-romaine. Annales du Service des Antiquités de l’Égypte 69, 255-8. Arnold, D., 1999. Temples of the Last Pharaohs. New York and Oxford: Oxford University Press. Badawy, A., 1965. Ancient Egyptian Architectural Design: A Study of the Harmonic System. Berkeley and Los Angeles: University of California Press. Cauville, S., & Devauchelle, D., 1984. Le temple d’Edfou: étapes de la construction. Nouvelles données historiques. Revue d’Égyptologie 35, 32-47. Chassinat, É., 1930. Le temple d’Edfou vol. 5. Cairo: Institut Français d’Archéologie Orientale. Clagett, M., 1995. Ancient Egyptian Science vol. 2: Calendars, Clocks, and Astronomy. Philadelphia: American Philosophical Society. Depuydt, L., 1997. Civil Calendar and Lunar Calendar in Ancient Egypt. Leuven: Peeters. Devauchelle, D., 1985. De nouveau la construction du temple d’Hathor à Dendara. Revue d’Égyptologie 36, 172-4. De Wit, C., 1961. Inscriptions dédicatoires du temple d’Edfou. Chronique d’Égypte 36, 56-97 and 277-320. Dunlap, R. A., 1999. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific.

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Finnestad, R. B., 1997. Temples of the Ptolemaic and Roman periods: ancient traditions in new contexts, in Temples of Ancient Egypt, ed., B. E. Shafer. Ithaca (NY): Cornell University Press, 185-237. Grimm, A., 1994. Die Altägyptischen Festkalender in den Tempeln der GriechischRömischen Epoche. Wiesbaden: Harrassowitz. Hagedorn, D., 1994. Zum ägyptischen Kalender unter Augustus. Zeitschrift für Papyrologie und Epigraphik 100, 211-22. Herz-Fischler, R., 1998. A Mathematical History of the Golden Number. Mineola (NY): Dover. Herz-Fischler, R., 2000. The Shape of the Great Pyramid. Waterloo (ON): Wilfrid Laurier University Press. Kemp, B., & Rose, P., 1991. Proportionality in mind and space in ancient Egypt. Cambridge Archaeological Journal 1, 103-29. Kurth, D., 1997. The present state of research into Graeco-Roman temples, in The Temple in Ancient Egypt: New Discoveries and Recent Research, ed., S. Quirke. London: British Museum Press, 152-8. Leitz, C., 1991. Studien zur Ägyptischen Astronomie. Wiesbaden: Harrassowitz. Priskin, G., 2001. The philosopher’s stones: the Great Pyramid. Discussions in Egyptology 51, 109-20. Priskin, G., in press. On the Number Delimiting the Sacred Space of the Dendera Temple. Discussions in Egyptology. Robins, G., & Shute, C. C. D., 1985. Mathematical bases of ancient Egyptian architecture and graphic art. Historia Mathematica 12, 107-22. Skeat, T. C., 1954. The Reigns of the Ptolemies. München: Beck’sche. Skeat, T. C., 1993. The Reign of Augustus in Egypt. München: Beck’sche. Winter, E., 1989. A Reconsideration of the Newly Discovered Building Inscription on the Temple of Denderah. Göttinger Miszellen 108, 75-85. Acknowledgement. I am grateful to Michael St. John for providing me with a copy of his

date converter program which I used to cross-check chronological data. e-mail: [email protected]