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Fundamental Mathematical Constants and Sequences

Sep 14, 2024

Key Mathematical Constants and Sequences

Circle and Ratio Constants

  • Pi (π): Ratio of a circle's circumference to its diameter. Approximately 3.14.
  • Tau (τ): Ratio of a circle's circumference to its radius. Equal to 2π.

Important Mathematical Constants

  • Euler's Number (e): Appears in exponential contexts. Approximately 2.718.
  • Imaginary Unit (i): i² = -1, i = √-1.

Fibonacci and Related Sequences

  • Fibonacci Sequence: Starts from 0, 1; each term is the sum of the two previous.
  • Reciprocal Fibonacci Constant: Series of reciprocals converges to ~3.35.
  • Golden Ratio (φ): Limit of Fibonacci ratio. Approximately 1.618, equal to (1 + √5) / 2.
  • Super Golden Ratio: Approximately 1.46.
  • Silver Ratio: Approximately 2.41, equal to 1 + √2.
  • Plastic Ratio: Approximately 1.32.
  • Golden Angle: Related to φ, approximately 137.5° or 2.4 radians.
  • Magic Angle: Angle between cube diagonal and edges. Around 54.7°.

Tribonacci and Random Fibonacci Sequences

  • Tribonacci Sequence: Sum of three previous terms.
  • Tribonacci Constant: Ratio convergence ~1.839.
  • Random Fibonacci Sequence: Alternating addition/subtraction. Average growth related to Swanlath's constant ~1.13.
  • Embry-Trefethen Constant: Critical factor for bounded/unbounded sequence growth.

Mathematical Series and Constants

  • Harmonic Series: Alternating series starting with 1.
  • Euler-Mascheroni Constant: Approximately 0.577.
  • Meissel-Mertens Constant: Approx. 0.26 (prime reciprocals).
  • Legendre's Constant: Related to prime counting, equals 1.

Constants from Continued Fractions

  • Pythagoras Constant: √2, diagonal of unit square.
  • Locke's Constant: Precision increase in continued fractions ~0.97.

Means and Averages

  • Geometric Mean: Product of numbers rooted by their count.
  • Kinchin's Constant: Geometric mean of continued fraction coefficients ~2.685.

Prime-related Constants

  • Brun's Constant: Sum of twin prime reciprocals ~1.9.
  • Landau-Ramanujan Constant: Approx. 0.76 (numbers as sum of squares).

Mathematical Constants in Geometry

  • Universal Parabolic Constant: Ratio for parabolas, approx. 2.29.
  • Lemniscate Constant: Ratio of perimeter to diameter ~2.62.

Growth and Sequence Constants

  • Conway's Constant: Look-and-Say growth rate ~1.3.
  • Kolakoski Constant: Derived from Kolakoski sequence.

Various Other Constants

  • Mills' Constant: Approximated as 1.3, generates primes.
  • Franson-Robinson Constant: Area under reciprocal gamma function, ~2.8.
  • Polygon Circumscribing Constant: Maximum circle radius from polygon circumscribing, ~8.7.
  • Kepler-Buchamp Constant: Minimum inscribed circle radius ~0.11.
  • Dotty Number: Solution to a specific equation, ~0.73.
  • Fourier's Constant: Sequence stability, ~1.18.
  • Erdős-Borwein Constant: Sum of Mersenne reciprocal ~1.6.

Sequences and Their Constants

  • Sylvester's Sequence: Alternating sum leads to Cahen's constant ~0.64.
  • Catalan's Constant: Alternating sum of odd squares reciprocals, ~0.91.
  • Apery's Constant: Sum of cube reciprocals ~1.2.

Constants in Astronomy and Motion

  • Laplace Limit: Eccentricity limit for convergence in Kepler's equation ~0.66.

These notes cover fundamental constants and sequences, explaining their properties, relationships, and approximations essential for mathematical and scientific computation.

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