On Mesoamerican Mathematics
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1 On Mesoamerican Mathematics Pablo Noriega B.V. IIIA-CSIC / LANIA. Barcelona / Xalapa.. Spain / Mexico
2 On Mesoamerican Mathematics (A quick and not too scientific overview for Actuaries) Pablo Noriega B.V. IIIA-CSIC / LANIA. Barcelona / Xalapa.. Spain / Mexico
3 Contents 1. Domain of Discourse 2. Mathematical Applications 3. Final Remarks
4 Contents 1. Domain of Discourse Basic Assumptions Implications Mesoamerica 2. Mathematical Applications 3. Final Remarks
5 Basic Assumptions (1. Theoretical Postulates) 1. Ontological Platonism Mathematical entities are culture-independent 2. Epistemological Relativism At a given time, in a given culture, only some mathematical concepts and practices are known
6 Basic Assumptions (1a. Theoretical Postulates Implications) 1. Ontological Platonism Mathematical entities are culture-independent Valid to reify 2. Epistemological Relativism At a given time, in a given culture, only some mathematical concepts and practices are known Look for mathematical content into cultural developments: Tribute (Economics, Demography) Calendrics (Astronomy) Architecture (Geometry)
7 Basic Assumptions (2. Methodological Restrictions) 3. Extensional Treatment Not concerned with intensional aspects (mystical meaning, religious implications, political uses, ) 4. Not-Ethnographic Monuments Ceramics and other archeological objects Original Documents Contemporary Accounts 5. Restrict to Mesoamerica (Maya & Aztec) Area Time-frame Cf. Ethnomathematics
8 Mesoamerica (P. Birkoff 1943) Aztec (-100,1521) Tenochtitlan Tula Tajin Teotihuacan Cholula San Lorenzo Olmec (-1500, 0) Tres Zapotes La Venta Palenque Tikal Chichen Itza Maya (-200,1519) Monte Alban Zapotec Mixtec (-500, 1000) Copan
9 Contents 1. Domain of Discourse 1. Mathematical Applications: Tribute Calendrics 2. Final Remarks
10 Contents 1. Domain of Discourse 2. Mathematical Applications: Tribute: Aztec Numerical Notation Cadastral Registers Calendrics 3. Final Remarks
11 Aztec Numerical Notation Mendoza 36r (frg) Vigesimal
12 Aztec Numerical Notation Mendoza 37r (frg.) Vigesimal Additive?
13 Mendoza 42r More Compound Numbers
14 Standard Aztec Numerical Notation 1 Basic Numbers Compound Numbers Vigesimal Additive
15 Contents 1. Domain of Discourse 2. Mathematical Applications: Tribute: Aztec Numerical Notation Cadastral Registers Calendrics 3. Final Remarks
16 Aztec Notational Variants (Texcoco) P1: Census P2: Land Plots P3: Tribute Base C. SantaMaríaAsunción
17 P1: Census male female boy girl Simple Family Unit male female girl girl boy male female Compound Household Household Demographic Data
18 Additive P2: Land Plots 12 Vigesimal Ordered 1 1/2 20 Fractions 20 1/7 Perimeteral Measures
19 P3: Tribute Base
20 P3: Tribute Base More Examples
21 Additive Vigesimal Ordered Fractions Positional Zero P3: Tribute Base AREA Place Holder Third Position: 20 x n Second Position: 20 x n First Position: n
22 Contents 1. Domain of Discourse 2. Mathematical Applications: Tribute and Cadastral Registers Calendrics: Calendrical Conventions Maya Numerical Notation Maya Astronomy 3. Final Remarks
23 Calendrics Observation of Cyclical Astronomical phenomena: Equinox/solstice cycles Cenital transit Eclipses (moon /sun cycles) Ecliptic constelations ( Zodiac ) Venus and Mars synodic cycles Time = combination of cycles Two basic calendars: 260 day ritual calendar ( Tzolkin ) 20 day bearers of 13 day weeks 365 day solar calendar (Haab) 18 months of 20 days + 5 extra days Other time-keeping conventions Calendar Round Long Count Distance Numbers
24 Tzolkin (260-day calendar) 20 day symbols 13 numerals Use numbers 1-13 Move both each increment Olmec invention? Zapotec (piye) Maya ( tzolkin ) Aztec (tonalpohualli) Madrid 75-6
25 Tzolkin Day Glyphs They appear only with numbers ONE to 13
26 Tonalpohualli
27 Tzolkin (260-day calendar) Mälstrom Zenithal-Passage Hypothesis
28 Haab (365-day calendar) day months 0-19 Numerals 18 Month symbols (used sequentially) Reverse lexicograhical ordering 1 5-day month This month continues the next year The 366th day may begin only with one of four month symbols (1, 6, 11, 16) year bearers
29 Haab Month Symbols Numbers 0 ( seating glyph ) to 19 precede each number glyph
30 Other Time-keeping Conventions Calendar Round (one date, two calendars) Combination of Tzolkin & Haab Long Count Number of days elapsed since origin Distance Numbers Number of days elapsed since a given date (in the same text) Lunar Series Day of lunation, number of lunation (1-6) and deity in a sixlunations period, length of lunation. All this for a given date
31 Calendar Round Dates (in a 52 year cycle) Combination of Haab and Tzolkin cycles Haab = 365 days Tzolkin = 260 days Calendar Round = lcm(365,260) = days CR = 52 Haab CR = 73 Tzolkin
32 Long Count Dates Distance (in days) between a date of the calendar round and a fixed ( origin ) date Maya origin : 4Ahau, 8Cumhu (Aug 13, 3113 BC) Earliests long count dates known: Chiapa de Corzo (36 BC) Tres Zapotes (32 BC) Tikal (292 AD) full positional, without zero Uaxactun (357 AD) badly preserved, uses two zeroes
33 Calendrical Notation Tzolkin Date (Earliest known example of a the use of a number glyph) Day glyph ( flint ) Number glyph ( 1 ) Zapotec San José del Mogote, Oaxaca. Estela 1 (300 BC)
34 Calendrical Notation (Early use of number / day-sign combinations) year 4 wind (haab fragment) 1 flint (tzolkin) Zapotec Monte Albán, Oaxaca South Platform Stelae 12, 13 (AD 300)
35 Calendrical Notation (Earliest complete long count calendric date) Olmec Tres Zapotes, Veracruz Estela C (fragment) (32 BC)
36 Calendrical Notation (Earliest full long count date) ISIG (1) Uo (haab) Bar & Dot Notation Bar = 5 units Dot= 1 unit Vigesimal Positional Value Olmec Tres Zapotes, Veracruz Estela C (fragment) (32 BC) 6 Edznab (Tzolkin)
37 Calendrical Notation (Early long count inscription) ISIG 15 Pop Manik Epi-Olmec La Mojarra, Veracruz (AD 100)
38 Calendrical Notation (Early distance number inscription) Epi-Olmec La Mojarra, Veracruz (AD 100)
39 Long Count (5125 years) Date=number of days from origin Maya Quiriguá, Stela C AD 800
40 Classic Period Inscription
41 Calendrical Arithmetic P1: Given two dates calculate the distance between them P2: Given a date (e.g.,in CR), obtain the equivalent (e.g., in LC) P3: Given a date, find other dates with interesting relevant features (astronomical events, cycles, nearest date such that )
42 Contents 1. Domain of Discourse 2. Mathematical Applications: Tribute Rolls: Cadastral Registers Calendrics: Calendrical Conventions Maya Numerical Notation Maya Astronomy 3. Final Remarks
43 Two Facts about Maya Positional Notation 1. All known examples of native positional Mayan numbers are calendrical or astronomical examples. 2. All known Mayan numbers in positional notation use at most five positions Brickner & Brickner, p193
44 Numeric Icons in Maya Inscriptions 5 units ( bar ) 1 unit ( dot ) Zero units, full count, place holder Maya Quiriguá, Stela C AD 800
45 Notational Conventions in Maya Inscriptions Additive (bar & dot) 13= Baktuns 5 4 Multiplicative (time units) Value Calendric unit 0 Tuns 0 Kins Positional? Canonical order - Zero (a Maya invention) Maya Quiriguá, Stela C AD 800
46 Conjecture: Basic Mayan Positional Notation Basic Symbols = 5 units = 1 unit Whole Numbers Number Position Value n 4 5th. Position n n i є [0,19] = 0 units 4th. Position n 3 n th. Position n 2 n nd. Position n 1 n st. Position n 0 n
47 Conjecture: Basic Mayan Positional Notation Basic Symbols = 5 units = 1 unit Whole Numbers Number Position Value n 4 5th. Position n i є [0,19] n n 4 є [0,19] = 0 units 4th. Position n 3 n n 3 є [0,19] 3th. Position n 2 n n 2 є [0,19] 2nd. Position n 1 n n 1 є [0,17] 1st. Position n 0 n n 0 є [0,19]
48 Basic Mayan Calendrical Positional Notation Basic Symbols = 5 units = 1 unit Whole Numbers Number Position Value n 4 5th. Position n = 0 units 4th. Position n 3 n th. Position n 2 n nd. Position n 1 n st. Position n 0 n
49 Numeric Icons in Maya Codices Bar & dot symbols Symbols for zero Symbol for twenty
50 Deciphering Maya Numerical Notation: Negative (Ring) Numbers... used for counting dates previous to the origin Fösterman 62
51 Fosterman 24 Reading a Page of the Dresden Codex Basic Layout
52 Reading a Page of the Dresden Codex Fosterman 24
53 Fosterman 24 Reading a Page of the Dresden Codex 8A 7A 6A 5A Ahau A 3A 2A A Ahau A = = 2920 = 5 x 584
54 Fosterman 24 Reading a Page of the Dresden Codex 260V 195V 130V 65V Preface to the Venus Tables Venus Cycle corrections 60V 55V 50V 45V 40V 35V 30V 25V A = 2920 = 8 x 365 = 5 x = Venus Synodic Rev. 20V 15V 10V 5V 4A 3A 2A A
55 Fosterman 24 Reading a Page of the Dresden Codex Preface to the Venus Tables Normal Base origin 4Ahau 8 Cumhu Minus Ring Number and B Ahau 18 Kayab Distance no. on pre-zero day Terminal date, nrml.day no Ahau 18 Kayab Ring Number: Ahau 8 Cumhu 1 Ahau 18 Kayab 1 Ahau 18 Uo
56 Two Facts of Mayan Positional Notation 1. All known examples of native positional Mayan numbers are calendrical or astronomical examples. Almost True NOT True 2. All known Mayan numbers in positional notation use at most five positions Brickner & Brickner, p193
57 ...They have other very long counts, extended to infinity, counting twenty times 8000, or 160,000; then they multiply this 160,000 again by twenty, and so on until they reach an uncountable figure. They do all this counting on a flat Surface Landa Relación de las Cosas de Yucatán Sec.XXIV
58 Graz 73 Codex Madrid
59 n d ,063 Graz ,789 n/d=
60 Contents 1. Domain of Discourse 2. Mathematical Applications: Tribute Rolls Cadastral Registers Calendrics: Calendrical Conventions Maya Numerical Notation Maya Astronomy 3. Final Remarks
61 Naked-eye Astronomy Mayan Astronomical Cycles 1. Solar Cycles 2. Lunar Cycles 3. Eclipses 4. Venus 5. Mars 6. Ecliptic Constellations (Zodiac)
62 Equinox / Solstice Alignments in: Solar Cycles Uaxactún (Group E) Chichen Itzá (Caracol, Castillo) Dzibilchaltun,... O.Casares: Dzibilchaltun Shadow plays in: Chichen Itzá (Castillo) Uxmal (C. De las Monjas)
63 Solar Cycles (Alignments in Chichén Itza)
64 Solar Cycles (Alignments in Chichén Itzá)
65 Solar Cyclesw (Alignments in Chichén Itzá)
66 Solar Cycles Zenithal Transit (105-day period) Copán, Izapa Chichén Itzá (Caracol) Teotihuacán, M. Albán Xochicalco (2 correction)
67 Venus Cycles Venus Synodic Revolution = 584 days Venus as Morning star: 263 days Superior conjunction: 50 days Venus as Evening Star: 263 days Inferior Conjunction: 8 days 5 Venus Synodic Revolutions = 8 Haab Dresden pp Conmensurates (VSR, Haab, Tzolkin) i.e. lcm(584, 365, 260) = days = 584 x 5 x 13 x 8
68 Lunar Cycles days Copan = Palenque = Complementary lunar series in Long Count Dates (between Tzolkin and Haab dates): 18 months, divided into trimesters (6months each) Age of the Moon Lunar month length (29 or 30 days) Ocasional extra 30-d month
69 Eclipses Codex Dresden pp Tables for 33 years, or 405 lunar months with correction values. Fosterman 52 (fragment) Fosterman 54
70 Martian Cycles Martian Synodic Revolution = 780 days Martian Synodic Revolution = 3 Tzolkin Dresden pp (?)
71 Ecliptical Constellations 13 constellations in the Ecliptic 364 days = 13 constellations 28 days Codex Paris Lintel, Las Monjas Annex (Chichen Itza) Throne, Governor Palace (Uxmal)
72 Contents 1. Domain of Discourse 2. Mathematical Applications 3. Final Remarks An Overview Open Questions
73
74 The Maya Scribe The chilanes were charged with giving all those in the community the oracles of the demon,...casting lots in divination,...they awarded them utmost respect Landa Relación de las Cosas de Yucatán Sec. XXVII
75 Scribes in a Maya Vase
76 Scribes in a Maya Vase
77
78 Mathematical Concepts in Mesoamerica Evidence of Zero (Maya, Aztec) Negative Numbers (Maya) Fractions (Aztec) Positional notation and other conventions Modular Arithmetic (Maya) Area Calculation (Aztec)
79 Mesoamerican Mathematics Open Topics Geometric and numerical concepts in Architecture Geometric and Topological concepts in art Demographic and Geometric analysis of Cadastral Registers Economic analysis of tribute Algorithms Metrics Systematic scribal errors Hermeneutical Concerns Internal coherence Catachronism / anachronism Iconic languages
80
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