BALTICA10

More documents

Recommendations

Info

Basatanya revisited: two periods of a Copper Age cemetery in the Carpathian Basin KATALIN BARLAI References BARLAI, K., 1980. On the Orientation of Graves in Prehistoric cemeteries. Archaeoastronomy, III(4), 29-32. BARLAI, K., 2002. Microhistory at Basatanya I Copper Age Cemetery. In: M.KOIVA, I.PUSTYLNIK, L.VESIK, eds. Cosmic Catastrophies (SEAC 2002 Proceedings). Tartu, 5-8. BARLAI, K., BOGNÁR-KUTZIÁN, I., ZSOLDOS, E., 1992. Rays of Prehistoric Sun. In: S. IWANISZEWSKI, ed. Readings in Archaeoastronomy. Warsaw University Cooperative, 11-19. BOGNÁR-KUTZIÁN, I., 1963. The Copper Age Cemetery of Tiszapolgár-Basatanya. Archaeologia Hungarica, series Nova XLII. Budapest: Akademiai Kiado, Publishing house of the Hungarian Academy of Sciences. Received: 16 October 2007; Revised: 20 November 2008 NAUJAS ŽVILGSNIS Į BASATANIJĄ: DU VARIO AMŽIAUS KAPINYNO PANONIJOS LYGUMOJE LAIKOTARPIAI Katalin Barlai Santrauka Prasidedant senajam vario amžiui, maždaug prieš 6000 metų, kapinynai buvo kuriami atskirai nuo gyvenviečių. Šiuose kapinynuose kapai buvo orientuojami, atsižvelgiant į vietovės horizonte esančius taškus, kuriuose Saulė teka ir leidžiasi. Basatanijos kapinynas yra Panonijos lygumoje, Tisos upės pakrantėje. 156 kapai priklauso dviem laikotarpiams, kuriuos sieja trumpa, apie 15 metų, pereinamoji fazė. Iš viso kapinynas galėjo būti naudojamas apie 200 metų, arba 8–9 gyventojų kartų. Pirmuoju laikotarpiu, senajame vario amžiuje, mirusieji buvo laidojami galvomis į vakarus. Šių mirusiųjų (o jų daugumą sudarė vyrai, kurie bendruomenėje veikiausiai užėmė priveligijuotą padėtį) kapai buvo išdėstyti rytų–vakarų ašimi su nedideliais nukrypimais. Rastos įkapės yra būdingos medžiotojams ar net kariams. (Tiesa, ankstesnis pirmojo laikotarpio tyrimas (Barlai, 2002), pagrįstas antropologinių tipų analize, ateityje turėtų būti kritiškai peržiūrėtas.) Analogiškas tyrimas buvo atliktas ir analizuojant antrąjį kapinyno naudojimo laikotarpį. Laidosena pakito. Maždaug ketvirtis mirusiųjų buvo palaidota ne galvomis į vakarus, o priešingai – į rytus. Jų įkapės byloja apie neturtingą taikių žemdirbių pobūdžio bendruomenę. 210 Vertė Vykintas Vaitkevičius
THE PYTHAGOREAN GEOMETRY OF THE ATREUS TOMB AT MYCENAE MARCELLO RANIERI 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 “Treasury of Atreus” 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 (Pythagorean) 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. ARCHAEOLOGIA BALTICA 10 Introduction The nine Tholos Tombs of Mycenae constitute a homogeneous group in terms of their spatial disposition (all are located on the western side near the Mycenae citadel), function (all are funerary) and architectonic typology (all have corbelled vault domes accessed from a Dromos). In addition, they were all built within the relatively short time span of two centuries. In the History of Architecture, the Treasury of Atreus represents the archetype of large corbelled vault domes and was the biggest in antiquity until the construction of Rome’s Pantheon (1 st century AD, 1½ millennia later). To achieve the squaring (Ranieri 1997, 2002, 2005, 2006, 2007; Malgora 2000; Patanè 2006), three integers (a Squaring Triad) can be used, defining the lengths of sides and diagonals in terms of length-units. A reduced repertory for the present analysis is shown in Table 1. (For a wider repertory see Ranieri 1997). Analyses and Results The analyses were based on the association of the orthogonal forms as drawn on the plans with the proportions of the Squaring Triads (all plans are from Pelon 1976). Three parameters were defined in order to classify the main orthogonal geometrical features: - VA (Vault): the ratio of height to radius. VA= h/r. - CH (Chamber): the ratio (Diameter+Stomion)/Diameter. CH=(F+s)/F - DR (Dromos): the ratio of Dromos length to Diameter. DR = d/F The results for the nine Tholoi are briefly summarized in table 2. As shown in Fig.2 (with the exception of the Aegisthus Tomb for which CH = 3/W), parameter CH classifies the Tholoi into two groups: a “√2-Group” (Cyclopean, Epano Phournos, Panagia, Clytemnestra) and a “D-Group” (Kato Phournos, Lion, Genii, Atreus). In each group, there is modularity despite the diversity of sizes. Table 2 also suggests that the vault heights were all established following the proportion 15/8 of the P-triad M=8-15-17. For the Treasury of Atreus, as can be seen in Fig.3, all the most significant geometrical proportions appear to have been governed by Pythagorean Triads, namely D (CH), W (DR), M (VA), and GA. This is of special interest, because these triads are the first four terms of a series of P-triads that can be generated using the numerical algorithm reported by Diophantus (3 rd century AD) in Book II, Problem 8 of his Arithmetica 1 . This algorithm was known to the Mesopotamians 2 in Mycenaean times long before Diophantus. Units of length can be derived from the analyses. For the Treasury of Atreus, a “cubit-like” length-unit of 0.496 m ± 0.004 resulted. This value corresponds (well within the ± 0.004) to the values of known Mesopota- 1 Given M and N mutually prime with M >N of different parities, if A = M 2 – N 2 , B = 2 M N, C = M 2 + N 2 then A 2 +B 2 = C 2 . 2 Mesopotamians were highly acquainted with P-triads, as demonstrated by the clay tablet N° 322 in the Plimpton collection at Columbia University (Neugebauer 1957). V V. REFLECTIONS OF ASTRO- NOMICAL AND COSMOLOGICAL KNOWLEDGE IN MONUMENTS, LANDSCAPES AND ARCHITECTURE 211
Page 1:
ARCHAEOLOGIA BALTICA10 Astronomy an
Page 4 and 5:
UDK 902/904 Ar 46 Volume 10 Editori
Page 6 and 7:
Vytautas Tumėnas. The visual and t
Page 8 and 9:
PREFACE Jonas VaiškūnaS Preface T
Page 10 and 11:
Jonas VaiškūnaS Preface cists, ma
Page 12 and 13:
LIST OF CONTRIBUTORS List of Contri
Page 14 and 15:
List of Contributors Dimitar Kolev
Page 16 and 17:
List of Contributors 16 Dainius Raz
Page 18 and 19:
Second Parts can always be better!
Page 20 and 21:
Conference Summary: A Historian’s
Page 22 and 23:
THE MILKY WAY AND ITS STRUCTURING F
Page 24 and 25:
THE MILKY WAY AND ITS STRUCTURING F
Page 26 and 27:
CECILIA PAULA GÓMEZ Venus And The
Page 28 and 29:
CECILIA PAULA GÓMEZ Venus And The
Page 30 and 31:
Astronyms in Belarussian Folk Belie
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Traditional Astronomical Knowledge
Page 38 and 39:
Traditional Astronomical Knowledge
Page 40 and 41:
A Solar Calendar from Latvian Daina
Page 42 and 43:
A Solar Calendar from Latvian Daina
Page 44 and 45:
SAULĖS KALENDORIUS PAGAL LATVIŲ D
Page 46 and 47:
Mesopotamian Influences on Persian
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
THE BIRTHDAY OF ASAKKU ARKADIUSZ SO
Page 54 and 55:
ARKADIUSZ SOŁTYSIAK The Birthday o
Page 56 and 57:
ARKADIUSZ SOŁTYSIAK The Birthday o
Page 58 and 59:
NOEMI MIRANDA, JUAN A. BELMONTE, MI
Page 60 and 61:
NOEMI MIRANDA, JUAN A. BELMONTE, MI
Page 62 and 63:
Tracing the Celestial Deer - an Anc
Page 64 and 65:
Tracing the Celestial Deer - an Anc
Page 66 and 67:
Calendrical Deer, Time-Reckoning an
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
The Mythological Moon Horse as Refl
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
The Visual and the Mythical- Poetic
Page 80 and 81:
VYTAUTAS TUMĖNAS The Visual and th
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Some Peripheral Forms of the Medite
Page 88 and 89:
JONAS VAIŠKŪNAS Some Peripheral F
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Were the 185 A.D. and 369 A.D. “G
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Indigenous Astronomical Traditions
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
A new attempt to date the Xia, Shan
Page 108 and 109:
GÖRAN HENRIKSSON A new attempt to
Page 110 and 111:
Eleventh-Century Supernovae: Anothe
Page 112 and 113:
Eleventh-Century Supernovae: Anothe
Page 114 and 115:
The Date of Purim and Calendars in
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
L’abbé Jean Lebeuf, (1687- 1760)
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Orientations at the Minoan Peak San
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
GREGORY E. MUNSON, BRYAN C. BATES,
Page 134 and 135:
Reading, Writing, and Recording the
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Reading, Writing, and Recording the
Page 142 and 143:
David W. Pankenier, Ciyuan Y. Liu,
Page 144 and 145:
The Xiangfen, Taosi site: a Chinese
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
On the Origin of the Roman Idea of
Page 152 and 153:
GIULIO MAGLI On the Origin of the R
Page 154 and 155:
On the Origin of the Roman Idea of
Page 156 and 157:
The Nuraghic Well of Santa Cristina
Page 158 and 159:
ARNOLD LEBEUF The Nuraghic Well of
Page 160 and 161: The Nuraghic Well of Santa Cristina
Page 162 and 163: The Nuraghic Well of Santa Cristina
Page 164 and 165: ANDREI PROKHOROV Images on the Ston
Page 166 and 167: Images on the Stones at Scebiaraky
Page 168 and 169: Images on the Stones at Scebiaraky
Page 170 and 171: Thracian Dolmens and their Orientat
Page 172 and 173: Antonio César González García, V
Page 174 and 175: Thracian Dolmens and their Orientat
Page 176 and 177: JOHN MCKIM MALVILLE, MICHAEL ZAWASK
Page 178 and 179: JOHN MCKIM MALVILLE, MICHAEL ZAWASK
Page 180 and 181: Cosmological Motifs of Peruvian Hua
Page 182 and 183: Cosmological Motifs of Peruvian Hua
Page 184 and 185: MARCELLO RANIERI A Geometrical Anal
Page 194 and 195: A Geometrical Analysis of Mesoameri
Page 196 and 197: LUCILLA LABIANCA, IDA SCIORTINO, SI
Page 198 and 199: An Archaeoastronomical Study of the
Page 200 and 201: Some Cosmological Aspects оf Catho
Page 204 and 205: RIMVYDAS LAUŽIKAS Some Cosmologica
Page 208 and 209: Fig. 1. Distribution of the graves
Page 212 and 213: The Pythagorean Geometry of the Atr
Page 214 and 215: The Pythagorean Geometry of the Atr
Page 216 and 217: Measurement Systems for Mathematics
Page 218 and 219: Measurement Systems for Mathematics
Page 220 and 221: Integrating Archaeoastronomy with L
Page 222 and 223: LIONEL SIMS Integrating Archaeoastr
Page 224 and 225: Integrating Archaeoastronomy with L
Page 226 and 227: From Umm Al Qab to Biban Al Muluk:
Page 228 and 229: Juan Antonio Belmonte, Antonio Cés
Page 230 and 231: Juan Antonio Belmonte, Antonio Cés
Page 232 and 233: From Umm Al Qab to Biban Al Muluk:
Page 234 and 235: The Orientations of Central Alentej
Page 242 and 243: Cardinal and Lunisolar Alignments a
Page 244 and 245: Cardinal and Lunisolar Alignments a
Page 246 and 247: Searching for the Astronomy of Abor
Page 254 and 255: Alternative Archaeoastronomies - An
Page 256 and 257: Alternative Archaeoastronomies - An
Page 258 and 259: THE CASE STUDY OF THE BOORONG The C
Page 260 and 261:
The Case Study of the Boorong JOHN
Page 262 and 263:
The Case Study of the Boorong JOHN
Page 264 and 265:
Nationalism and Politics in the Rec
Page 266 and 267:
Nationalism and Politics in the Rec
Page 268 and 269:
Cosmic Catastrophes and Cultural Di
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
In Memoriam Izold’s interests wer
Page 276 and 277:
Klaipėdos universiteto leidykla AR
Page 278 and 279:
Plates 2 6 1 3 Plate II. Antonio C
Page 280 and 281:
Plates 12 11 10 9 8 7 Plate IV. IV
Page 282 and 283:
Plates 1 11 Plate VI. VI Antonio C
Page 284:
Plates 2 3 Plate VIII. VIII Lucilla
show all

BALTICA10

Create successful ePaper yourself

Delete template?

Save as template?