[
  {
    "name": "Pi",
    "symbol": "π",
    "value": "3.14159265358979323846264338327950288419716939937510",
    "description": "Ratio of a circle's circumference to its diameter; appears throughout geometry, trigonometry, and analysis."
  },
  {
    "name": "Euler's Number",
    "symbol": "e",
    "value": "2.71828182845904523536028747135266249775724709369995",
    "description": "Base of the natural logarithm; limit of (1 + 1/n)^n as n → ∞; fundamental to calculus and exponential growth."
  },
  {
    "name": "Golden Ratio",
    "symbol": "φ",
    "value": "1.61803398874989484820458683436563811772030917980576",
    "description": "Positive root of x² − x − 1 = 0; ratio such that (a+b)/a = a/b; appears in art, architecture, and nature."
  },
  {
    "name": "Square Root of 2",
    "symbol": "√2",
    "value": "1.41421356237309504880168872420969807856967187537694",
    "description": "Length of the diagonal of a unit square; first number proven to be irrational (by ancient Greek mathematicians)."
  },
  {
    "name": "Square Root of 3",
    "symbol": "√3",
    "value": "1.73205080756887729352744634150587236694280525381038",
    "description": "Height of an equilateral triangle with side 2; appears in hexagonal geometry and trigonometry."
  },
  {
    "name": "Square Root of 5",
    "symbol": "√5",
    "value": "2.23606797749978969640917366873127623544061835961152",
    "description": "Appears in the closed-form expression for the golden ratio: φ = (1 + √5) / 2."
  },
  {
    "name": "Natural Logarithm of 2",
    "symbol": "ln 2",
    "value": "0.69314718055994530941723212145817656807550013436025",
    "description": "Logarithm base e of 2; related to the half-life formula and binary entropy; sum of the alternating harmonic series."
  },
  {
    "name": "Natural Logarithm of 10",
    "symbol": "ln 10",
    "value": "2.30258509299404568401799145468436420760110148862877",
    "description": "Conversion factor between natural logarithms and common (base-10) logarithms: log₁₀(x) = ln(x) / ln(10)."
  },
  {
    "name": "Euler–Mascheroni Constant",
    "symbol": "γ",
    "value": "0.57721566490153286060651209008240243104215933593992",
    "description": "Limit of the difference between the harmonic series and the natural logarithm; appears in number theory and analysis."
  },
  {
    "name": "Apéry's Constant",
    "symbol": "ζ(3)",
    "value": "1.20205690315959428539973816151144999076498629234049",
    "description": "Value of the Riemann zeta function at 3; proven irrational by Roger Apéry in 1978; arises in quantum electrodynamics."
  },
  {
    "name": "Catalan's Constant",
    "symbol": "G",
    "value": "0.91596559417721901505460351493238411077414937428167",
    "description": "Sum of the alternating series 1 − 1/9 + 1/25 − …; appears in combinatorics and the theory of random matrices."
  },
  {
    "name": "Feigenbaum Delta",
    "symbol": "δ",
    "value": "4.66920160910299067185320382046620161725818557747576",
    "description": "Ratio at which period-doubling bifurcations converge in chaotic dynamical systems; universal across many maps."
  },
  {
    "name": "Feigenbaum Alpha",
    "symbol": "α",
    "value": "2.50290787509589282228390287321821578638127137672714",
    "description": "Scaling factor for the width of the branches in period-doubling bifurcations; companion to Feigenbaum delta."
  },
  {
    "name": "Khinchin's Constant",
    "symbol": "K₀",
    "value": "2.68545200106530644530971483548179569382038229399446",
    "description": "Geometric mean of the partial quotients in the continued-fraction expansion of almost all real numbers."
  },
  {
    "name": "Omega Constant",
    "symbol": "Ω",
    "value": "0.56714329040978387299996866221035554975381578718651",
    "description": "Unique fixed point of the natural exponential function: Ω · e^Ω = 1; solution to the equation xe^x = 1."
  },
  {
    "name": "Twin Prime Constant",
    "symbol": "C₂",
    "value": "0.66016181584686957392781211001455577843262336028473",
    "description": "Product over primes p ≥ 3 of 1 − 1/(p−1)²; conjectured to appear in the distribution of twin primes."
  },
  {
    "name": "Plastic Constant",
    "symbol": "ρ",
    "value": "1.32471795724474602596090885447809734073440405690173",
    "description": "Real root of x³ = x + 1; the unique real solution; analogous to the golden ratio but for cubic self-similarity."
  },
  {
    "name": "Silver Ratio",
    "symbol": "δₛ",
    "value": "2.41421356237309504880168872420969807856967187537694",
    "description": "Positive root of x² − 2x − 1 = 0; equals 1 + √2; first member of the metallic means family after the golden ratio."
  },
  {
    "name": "Glaisher–Kinkelin Constant",
    "symbol": "A",
    "value": "1.28242712910062263687534256886979172776768892732500",
    "description": "Appears in the asymptotic expansion of the Barnes G-function and products of consecutive factorials."
  },
  {
    "name": "Gauss's Constant",
    "symbol": "G",
    "value": "0.83462684167407318628142973279904680899399301349034",
    "description": "Reciprocal of the arithmetic-geometric mean of 1 and √2; related to elliptic integrals and lemniscate arc length."
  }
]