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Occurence of fundamental mathematical constants

  1. Aug 29, 2007 #1
    0, 1, phi (the Golden ratio), e, pi, delta (Feigenbaum's constant) and comparatively many other fundamental, dimensionless mathematical constants occur on the interval 0 to 5. With a potential infinity of numbers to choose from, why does such a confluence exist?
     
  2. jcsd
  3. Aug 30, 2007 #2
    For the physical constant.. perhaps they lie on [0,5] due to the strength of the interactions (weak, strong) in many cases using 'Natural units' you can set them equal to value '1'
     
  4. Aug 30, 2007 #3
    But physical constants are in no way limited to [0,5] like mathematical constants appear to be. Just consider Planck's constant and Avogadro's number in SI units.
     
  5. Aug 30, 2007 #4

    CRGreathouse

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    Maybe small ones are easy to discover. Maybe our idea of 'small' is tied to the size of these constants (no, really -- 1 is the measure by which we count, so if it were 'larger' so would be our concept of 'small').

    Maybe it's just easier to find and work with small constants -- perhaps the order of the Monster group is just as fundamental, but less has been done with it since it's so large.
     
  6. Aug 30, 2007 #5
    Maybe it's because the infinite sums that start off involving small numbers like 1, 2, 3, .. or 1!, 2!, 3!, etc. yield the most interesting and general constants:

    e=1+1+1/2+1/6+1/24+1/120+... approx = 2 + 1/2 = 2.5
    pi= 4 - 4/3 + 4/5 - 4/7 +... approx = 4 - 4/3 = 2.66666...
    phi = (1 + 1/phi) = (1 + 1/(1+1/phi)) = ... approx = 1 + 1 / (1 + 1) = 1.5
    euler mascheroni gamma = (1 - ln 2/1) + (1/2 - ln 3/2) + (1/3 - ln 4/3) + ...
     
    Last edited: Aug 30, 2007
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