Apêndice: Tabelas matemáticas recalculáveis e derivações
O apêndice matemático reúne períodos planetários, codificações binárias de Fuxi, decanatos do tarô e espaços combinatórios de 78 cartas para verificação pública e replicação.
Introduction to Recomputable Tables and Mathematical Auditability
Scientific research demands transparency, auditability, and independent recomputability. This appendix brings together the foundational constants, algebraic permutation matrices, combinatorial probability figures, and algorithmic lookup tables referenced throughout the monograph. Every table includes clear derivation methods and citations, ensuring third-party researchers can replicate all calculations.
Table A-1: Five Planets: Periods and Classical Chinese Correspondences
Constants determined from NASA Planetary Fact Sheets: Mercury (Chenxing / Water) sidereal period 88 days, synodic period 116 days; Venus (Taibai / Metal) sidereal 225 days, synodic 584 days; Mars (Yinghuo / Fire) sidereal 687 days, synodic 780 days; Jupiter (Suixing / Wood) sidereal 11.86 years, synodic 399 days; Saturn (Zhenxing / Earth) sidereal 29.46 years, synodic 378 days. Sidereal periods lock chronological cycles, while synodic periods pace observational omen windows.
Table A-2: Bagua Three-Bit Binary Values (Fuxi Order)
Following the Bouvet-Leibniz binary interpretation (yang = 1, yin = 0, read top-down): Qian 111 corresponds to decimal 7, Dui 110 to 6, Li 101 to 5, Zhen 100 to 4, Xun 011 to 3, Kan 010 to 2, Gen 001 to 1, and Kun 000 to 0. Shao Yong's circular and square arrangements illustrate ascending and descending binary progressions, proving 64 hexagrams exhaust the 6-bit state space.
Table A-3 and A-3b/c: Book T Tarot Correspondences and Thirty-Six Decans
Sourced from the 1888 Hermetic Order of the Golden Dawn Book T manuscripts: 0 Fool corresponds to Aether/Air, 1 Magician to Mercury, 2 High Priestess to Moon, 3 Empress to Venus, 4 Emperor to Aries, through 21 World to Saturn. Egyptian thirty-six decans divide the zodiac into 10-degree intervals, providing a recursive 3x12 subdivision identical to astronomical timekeeping practices across Persia and Hellenistic Greece.
Table A-4a/b/c: Tarot 78-Card Combinatorial State Spaces and Permutations
Across 78 cards with independent upright/reversed orientations: 1-card draw yields 156 states; 3-card spread yields 456,456 permutations (3,651,648 with orientations); 5-card spread yields 2,533,330,800 permutations (8.10x10¹⁰ with orientations); 10-card Celtic Cross spans 4.67x10²¹ states. Popular literature erroneously applying combinations underestimates spread complexity by 3,628,800-fold (10!). Exact probabilities: 3-card contains major arcana: 63.56%; 5-card: 81.91%; 10-card: 97.17%.
Figure A-1: Liuren Cosmic Board Information Flow and Modular Lookup
The ancient Chinese Liuren cosmic board maps celestial mechanics through rotating heaven and earth plates. The derivation executes modular arithmetic: solar month plus hour establishes the heaven plate; Four Lessons identify elemental relationships; Three Transmissions extract initial, middle, and final pivots. The algorithm functions as a deterministic finite state machine with zero random sources.
Figure A-2: Sexagenary Sixty-Cycle Congruence Generation Rule
Ten heavenly stems (mod 10) and twelve earthly branches (mod 12) pair via i ↦ (i mod 10, i mod 12). Because gcd(10, 12) = 2 and lcm(10, 12) = 60, the pairs remain distinct across steps 0 through 59, closing precisely at step 60. This simple algebraic rule demonstrates the mathematical inevitability of the sixty-year calendrical cycle without arbitrary empirical intervention.
Bibliografia do apêndice I: Astronomia clássica e fontes arqueológicas
Cálculos astronômicos de referência e edições de fontes arqueológicas: Bretagnon & Francou (1988) Soluções planetárias VSOP87; Espenak & Meeus (2006) Cânone da NASA de eclipses de cinco milênios (-1999 a +3000); Hunger & Steele (2019) Tratado astronômico babilônico MUL.APIN (c. 1000 a.C.); Justeson & Lowry (2025) Tabela de eclipses do Códice de Dresden maia (Science Advances 11(43), 69 estações no ciclo de 1.655 lunações); Li et al. (2026) Dataset OBIMD de inscrições em ossos oraculares (Scientific Data, 10.077 decalques); Neugebauer (1955, 1975) Textos cuneiformes e História da astronomia matemática antiga; Sivin (1969, 2009) Cosmos e computação na astronomia chinesa e reforma Shoushi; Steele (2000) Registro de eclipses; Ptolomeu (1940) Tetrabiblos (Loeb).
Bibliografia do apêndice II: Divinação chinesa, astrologia do Oriente Próximo e história binária
Modelos comparativos e fontes primárias textuais: Cullen (2011) Wu xing zhan: prognósticos dos cinco planetas de Mawangdui (SCIAMVS 12: 193-249); Ho Peng-Yoke (2003) Chinese Mathematical Astrology (RoutledgeCurzon); Needham (1959) Science and Civilisation in China, vol. 3; Pingree (1973) Origem mesopotâmica da astronomia indiana; Sun & Kistemaker (1997) O céu chinês sob os Han; Histórias dinásticas padrão (Shiji, Hanshu, Yuanshi); Bouché-Leclercq (1899) L'astrologie grecque; Cauville (1997) O zodíaco de Dendera; Catalogus Codicum Astrologorum Graecorum (CCAG); Rochberg (2004, 2010) The Heavenly Writing & In the Path of the Moon; West (1880) Textos pálavis: Bundahishn; Leibniz (1703) Explicação da aritmética binária; Ryan (1996) Sistema binário de Leibniz e o I Ching de Shao Yong; Schumann (2021) Lógica booleana em presságios mesopotâmicos.
Bibliografia do apêndice III: História do tarô, verificação estatística e protocolo de pré-registro
História dos mecanismos de amostragem e metodologia estatística: Dummett (1980) The Game of Tarot (estudo acadêmico de referência comprovando a origem cortesã no norte da Itália no século XV); Dummett & McLeod (2004) A History of the Occult Tarot; Mathers (1888) Book T - The Tarot (manuscrito da Golden Dawn mapeando 78 cartas a coordenadas astronômicas); Bates et al. (2015) Modelos mistos lineares com lme4; Cohen (1988) Análise de poder estatístico para ciências do comportamento (design pareado: d=0,4, alfa=0,05, poder=0,80); Knuth (1997) The Art of Computer Programming (embaralhamento Fisher-Yates); Rammstedt & John (2007) Inventário BFI-10; Tobacyk (2004) Escala revisada de crenças paranormais. Pré-registro de ensaio pareado e teste qui-quadrado (gl=77).
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Perguntas frequentes
Why must positional tarot spreads use permutations (P) instead of combinations (C)?
Because spread positions (such as past, present, future) possess distinct contextual definitions. Drawing the exact same cards in a different sequence produces an entirely different interpretive outcome, making order decisive.
Why does the sixty-year sexagenary cycle terminate at 60 steps rather than 120?
Because 10 and 12 share a greatest common divisor of 2. By the Chinese Remainder Theorem and elementary number theory, their least common multiple lcm(10, 12) equals 60, after which the modular sequences necessarily repeat.
How does the Liuren cosmic board connect to modern computing concepts?
The Liuren system is a geometric finite state machine. Plate rotations correspond to modular 12 addition, and extracting transmissions corresponds to strict conditional branching and matrix lookup, without supernatural randomness.
Fontes e leitura complementar
As posições astronômicas e as regras de interpretação desta página são conferidas com as obras abaixo.
- Swiss Ephemeris (Astrodienst) (a biblioteca de efemérides por trás de todos os cálculos)
- Swiss Ephemeris programmer's documentation (parâmetros de cálculo e referência da API)
- Introduction to the Tetrabiblos (Skyscript)
- Hellenistic astrology (recursos históricos e técnicos sobre a prática helenística)