THE PYTHAGOREAN GEOMETRY OF THE ATREUS TOMB AT

MARCELLO RANIERI BALTICA 10 BALTICA

Abstract

A geometrical analysis was performed using CAD (Computer Aided Design) tools on the plans of the nine Tholos Tombs of Mycenae and of the “” in particular. Dedicated parameters were established in order to classify the main common geometrical features of the tombs. The analyses were based on a comparison between the geometrical proportions found on the plans and those of the Squaring Triads. It appears that Mycenaean architects made use of both Perfect (Pythago- ARCHAEOLOGIA rean) and Quasi-Perfect combinations of integers. The Treasury of Atreus stands out by exhibiting all the major geometric proportions identifiable with those belonging to a series of Pythagorean Triads reported by Diophantus and known to the Mesopotamians. The unit of length for the Atreus Tholos Tomb coincides with the Lagash Gudea cubit of 0.496 m. Key words: geometry, Pythagorean Triads, cubit, Mycenae, Dyophantus.

Introduction The results for the nine Tholoi are briefly summarized in table 2. The nine Tholos Tombs of Mycenae constitute a homo- As shown in Fig.2 (with the exception of the geneous group in terms of their spatial disposition (all Tomb for which CH = 3/W), parameter CH classifies are located on the western side near the Mycenae cita- the Tholoi into two groups: del), function (all are funerary) and architectonic ty- pology (all have corbelled vault domes accessed from a “√2-Group” (Cyclopean, Epano Phournos, Panagia, a Dromos). In addition, they were all built within the ) and relatively short time span of two centuries. In the His- a “D-Group” (Kato Phournos, Lion, Genii, Atreus). tory of Architecture, the Treasury of Atreus represents the archetype of large corbelled vault domes and was In each group, there is modularity despite the diversity the biggest in antiquity until the construction of Rome’s of sizes. st Pantheon (1 century AD, 1½ millennia later). Table 2 also suggests that the vault heights were all V To achieve the squaring (Ranieri 1997, 2002, 2005, established following the proportion 15/8 of the P-triad V. REFLECTIONS 2006, 2007; Malgora 2000; Patanè 2006), three integers M=8-15-17. OF ASTRO- (a Squaring Triad) can be used, defining the lengths of NOMICAL AND For the Treasury of Atreus, as can be seen in Fig.3, COSMOLOGICAL sides and diagonals in terms of length-units. A reduced all the most significant geometrical proportions appear KNOWLEDGE IN repertory for the present analysis is shown in Table 1. MONUMENTS, to have been governed by Pythagorean Triads, namely LANDSCAPES (For a wider repertory see Ranieri 1997). D (CH), W (DR), M (VA), and GA. This is of special AND ARCHITECTURE interest, because these triads are the first four terms of Analyses and Results a series of P-triads that can be generated using the nu- merical algorithm reported by Diophantus (3rd century The analyses were based on the association of the or- AD) in Book II, Problem 8 of his Arithmetica1. This thogonal forms as drawn on the plans with the propor- algorithm was known to the Mesopotamians2 in Myc- tions of the Squaring Triads (all plans are from Pelon enaean times long before Diophantus. 1976). Units of length can be derived from the analyses. For Three parameters were defined in order to classify the the Treasury of Atreus, a “cubit-like” length-unit of main orthogonal geometrical features: 0.496 m ± 0.004 resulted. This value corresponds (well within the ± 0.004) to the values of known Mesopota- - VA (Vault): the ratio of height to radius. VA= h/r.

1 - CH (Chamber): the ratio (Diameter+Stomion)/Diam- Given M and N mutually prime with M >N of different parities, if A = M2 – N2, B = 2 M N, C = M2 + N2 then eter. CH=(F+s)/F A2 +B2 = C2. 2 - DR (Dromos): the ratio of Dromos length to Diam- Mesopotamians were highly acquainted with P-triads, as eter. DR = d/F demonstrated by the clay tablet N° 322 in the Plimpton collection at Columbia University (Neugebauer 1957). 211 Table 1. Triads related to the nine Tholoi From left: Symbol, Integers, proportion r = B/A The Pythagorean Geometry of The Pythagorean Geometry of at Mycenae Tomb Atreus the MARCELLO RANIERI

Fig. 1. Rectangular Parameters for a geometrical classification of MycenaeanTholos Tombs.

Table 2. The resulting association of the Geometrical Parameters with Squar- ing Triads

Fig. 2. The tholoi as classified using the CH “Chamber” geometrical parameter. 212 BALTICA 10 BALTICA ARCHAEOLOGIA

Fig. 3. The Pythagorean geometry of the Treasury of Atreus. mian cubits: the Assyrian cubit of 0.494 m; the Sumer- Artigianato a Marzabotto. Nuove Prospettive di Ricerca. ian cubit of 0.500 m; and the Gudea of Lagash statue Bologna: Dipartimento di Archeologia Università di Bo- cubit of 0.496 m. logna N°11, 73-87. RANIERI, M., 2006. Contenuti geometrici, numerici, metrici e astronomici del tempio nuragico a pozzo “Su Tempiesu” di Orune. Proc. 6° Convegno Annuale Società Italiana di Conclusions Archeoastronomia. Campobasso, Italy. RANIERI, M., 2007. The Stone Circles of Li Muri: geom- It appears that Mycenaean architects made use of both etry, alignments and numbers In: M.P. Zedda, J.A. Bel- Perfect (Pythagorean) and Quasi-Perfect combinations monte, eds. Lights and Shadows in Cultural Astronomy. of integers. The Treasury of Atreus stands out by hav- Isili, Sardinia: Asssociazione Archeofilia Sarda, 58-67. ing all the major geometric proportions identifiable V Received: 2 November 2007; Revised: 28 February 2008 with those of the series of Pythagorean Triads reported V. REFLECTIONS by Diophantus and known to the Mesopotamians. The OF ASTRO- length-unit for the Atreus Tholos Tomb coincides with NOMICAL AND COSMOLOGICAL the Lagash Gudea cubit of 0.496 m. ATRĖJO TOLAS MIKĖNUOSE IR KNOWLEDGE IN PITAGORIEČIŲ GEOMETRIJA MONUMENTS, LANDSCAPES AND References ARCHITECTURE Marcello Ranieri NEUGEBAUER, O., 1957. The Exact Sciences in Antiquity. Providence, Rhode Island: Brown University Press. MALGORA, S., 2000. L’uso dei numeri e la Ritualizzazione Santrauka nelle Strutture Cerimoniali nella Topografia Monumentale di Saqqara. Thesis. Bologna: Bologna University. Naudojant kompiuterinę automatizuoto projektavimo PATANÈ, A., 2006, Indagine Archeoastronomica sulla Ba- silica Sotterranea di Porta Maggiore in Roma. Thesis. sistemą CAD (Computer Aided Design) buvo atlikta Roma: “La Sapienza” University. devynių laidojimo rūsių – tolų ir atskirai kapavietės, PELON, O., 1976. Tholoi, Tumuli et Cercles Funéraires. vadinamos „Atrėjo lobynu“, geometrinė analizė. Sie- Athens: Échole Française D’athénes. kiant klasifikuoti tyrinėjamas kapavietes, buvo išskirti RANIERI, M., 1997. Triads of Integers: How Space Was bendri geometriniai tyrinėjamų laidojimo rūsių para- Squared in Ancient Times. Journal of Ancient Topography - Rivista di Topografia Antica - JAT, VII, 209-244. metrai. Analizės pagrindas buvo geometrinių laidojimo RANIERI, M., 2002. Geometry at Stonehenge. Archae- rūsių proporcijų lyginimas pagal kompiuterine analize oastronomy: The Journal of Astronomy in Culture, XVII, sudarytus planus su Pitagoro skaičių trejetais. Atrodo, 81-93. kad Mikėnų architektai statiems kampams nustatyti RANIERI, M., 2005. La Geometria del Tempio Urbano di naudojo tiek tikslius Pitagoro skaičių trejetus, tiek jų Marzabotto (Regio I–Ins.5). In: Culti, Forma Urbana e 213 artinius. „Atrėjo lobynas“ turi visas svarbiausias aiš- kiai atpažįstamas geometrines proporcijas, aprašytas Diofanto bei žinotas Mesopotamijoje ir būdingas Pi- tagoro skaičių trejetų geometrijai. „Atrėjo lobyno“ lai- dojimo rūsyje naudotas ilgio matas sutampa su Lagašo valdovo Gudėjo (Gudea) uolektimi – 0,496 m.

The Pythagorean Geometry of The Pythagorean Geometry of at Mycenae Tomb Atreus the Vertė Audronė Bliujienė MARCELLO RANIERI

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