RESEARCH BASE
Search 3,721 documents across 34 fields — every claim tier-rated by evidence
3,633 are the core, quality-scored corpus (34 lettered sections — see How We Work); the remaining 88 are cross-corpus synthesis documents (68 InterDocs, 12 Connections, 8 Theories) also indexed here.
221 results for "square root two" — page 10 of 12
F_4_12 — Bantu Expansion: Africa's Great Migration and Iron Age Spread
The Bantu Expansion is the most consequential demographic and linguistic transformation in African history. Beginning from a homeland in the grasslands of modern Cameroon and southeastern Nigeria around 3000 BCE, Bantu-s
ZA_2_04 — Loop Quantum Gravity: Spacetime as a Fabric of Quanta
Loop quantum gravity (LQG) is a leading approach to quantum gravity that quantizes spacetime itself — predicting that area and volume come in discrete Planck-scale quanta. Unlike string theory, LQG does not require extra
ZA_1_19 — Loop Quantum Gravity
Loop quantum gravity (LQG) is one of two leading candidate theories (alongside string theory) for unifying general relativity with quantum mechanics — the central unsolved problem of theoretical physics. [KEY FINDING] LQ
ZA_1_11 — Weak Measurements: Gentle Probes and Anomalous Values in Quantum Mechanics
Weak measurements — a formalism in quantum mechanics introduced by Yakir Aharonov, David Albert, and Lev Vaidman (AAV) in 1988 — describe measurements where the interaction between the measuring device (pointer) and the
ZA_5_10 — Superfluidity: Quantum Mechanics at the Macroscopic Scale
Superfluidity — the macroscopic quantum phenomenon in which a fluid flows with zero viscosity (no resistance to flow) and exhibits extraordinary properties including frictionless flow through narrow channels, the ability
V_1_08 — Mathematical Puzzles & Recreational Mathematics
Mathematical puzzles — problems posed for amusement, education, or intellectual challenge — have served as engines of mathematical discovery for over 4,000 years. The Rhind Mathematical Papyrus (c. 1650 BCE, Egypt) conta
V_1_12 — Chinese Mathematics History
Chinese mathematics developed independently over at least 3,000 years, producing remarkable achievements often centuries before their European counterparts. The Jiuzhang Suanshu (Nine Chapters on the Mathematical Art, co
V_4_27 — Bayesian Inference: Probabilistic Reasoning from Bayes to Machine Learning
Bayesian inference — the mathematical framework for updating beliefs in light of evidence — has become the dominant paradigm in statistics, machine learning, cognitive science, and philosophy of science. Named after Reve
V_3_12 — Statistics and Hypothesis Testing
Statistics — the science of collecting, analyzing, and interpreting data under uncertainty — underpins virtually every empirical science, from medicine and psychology to physics and economics. Modern statistical hypothes
V_2_02 — Topology & Knot Theory: Celtic Knots to DNA
Topology — the study of properties preserved under continuous deformation (stretching, bending, but not tearing or gluing) — originated with Euler's solution to the Königsberg bridge problem (1736) and evolved into one o
V_2_15 — Galois Theory and Field Extensions
Galois theory, developed by Évariste Galois (1811-1832) in the last years of his tragically short life, is one of the great triumphs of abstract algebra — a theory connecting field extensions to group theory that definit
U_1_15 — Jazz: Improvisation, African Roots, and Cultural Revolution
Jazz — America's most original and influential art form — emerged in the early 20th century from the convergence of African rhythmic and improvisational traditions, African American blues and work songs, European harmony
W_4_17 — Mississippian Culture and Mound-Builder Networks
The Mississippian culture (c. 800–1600 CE) was the most complex pre-Columbian society in North America east of the Mississippi River, characterized by flat-topped platform mounds, intensive maize agriculture, hierarchica
ZH_2_13 — Tropical vs. Sidereal Zodiac: Two Systems and Cultural Divergence
The zodiac — the band of twelve named segments along the ecliptic — exists in two fundamentally different systems that have diverged over two millennia due to the precession of the equinoxes. The tropical zodiac (used in
ZF_5_16 — Ocean Observation Networks: Global Monitoring of the Marine Environment
Ocean observation networks constitute the global infrastructure for monitoring the physical, chemical, and biological state of the world's oceans in near-real-time. The centerpiece of modern ocean observation is the Argo
Z_4_17 — Non-coding RNA Networks: Regulation Beyond the Genome
Non-coding RNAs (ncRNAs) — RNA molecules that are not translated into protein but perform functional roles in the cell — have emerged since the late 1990s as a vast and previously unsuspected layer of biological regulati
D_1_18 — Taş Tepeler: Pre-Pottery Neolithic Ritual Network of Southeastern Turkey
Taş Tepeler ("Stone Hills") is a Turkish government-sponsored archaeological research program and site network encompassing at least 12 Pre-Pottery Neolithic (PPN) sites in the Şanlıurfa Province of southeastern Turkey,
D_3_18 — Great Zimbabwe Trade Network Expansion
Great Zimbabwe — the largest stone structure in sub-Saharan Africa south of the Sahara — was the capital of a Shona-speaking state that controlled the gold-for-cloth trade between the Zimbabwe Plateau and Indian Ocean po
N_2_13 — Islamic Esoteric Orders: Ismaili, Sufi, and Heterodox Networks
The Islamic world developed elaborate esoteric (bāṭinī) traditions organized through hierarchical spiritual orders, initiatory lineages, and secretive organizational structures that closely parallel Western secret societ
N_1_07 — Ancient Egyptian Priesthoods and Temple Networks
The Egyptian priesthood constituted one of the most powerful, long-lasting, and institutionally complex religious establishments in human history, operating continuously for over 3,000 years (c. 3100 BCE – 4th century CE
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