3 ms·

I wonder what happens for powers of e. This should be different than pi, since e has continued fraction coefficients that make a nice pattern [2, 1, 2, 1, 1, 4

I wonder what happens for powers of e. This should be different than pi, since e has continued fraction coefficients that make a nice pattern [2, 1, 2, 1, 1, 4, 1, 1, ...]. e^2 does as well (https://oeis.org/A001204 https://oeis.org/A001204). (I know this fact but have never understood any of the proofs.) But e^3 (https://oeis.org/A058282 https://oeis.org/A058282) does not have a "nice pattern", and neither does e^4 (https://oeis.org/A058283 https://oeis.org/A058283)

pyrolistical 4y ago
Often a nice pattern can be found if one uses a generalized continued fraction
westurner 4y ago
/? 3blue1brown e^ipi https://m.youtube.com/results?sp=mAEA&search_query=3blue1brown+e%5Eipi https://m.youtube.com/results?sp=mAEA&search_query=3blue1bro... Given: e = limit((1 + 1/n)^n, +∞) # Euler's number i = √-1 # orthogonal; i_0^2 = -1 pi = (666/212 - 22/7)*π # circle circumference / diameter Euler's identity: e^iπ + 1 = 0 Euler's formula: e^ix = cos(x) + i*sin(x) Euler's formula: https://en.wikipedia.org/wiki/Euler's_formula https://en.wikipedia.org/wiki/Euler's_formula e (Euler's number) https://en.wikipedia.org/wiki/E_(mathematical_constant) https://en.wikipedia.org/wiki/E_(mathematical_constant)
- westurner 4y ago
  Is there something fundamental here - with e.g. radix base e - about countability and a continuum of reals, and maybe constructive interference?
  - westurner 4y ago
    When EM waves that are in phase combine, the resultant amplitude of the combined waveform is the sum of the amplitudes; constructive interference is addition. And from addition, subtraction & multiplication and exponents and logarithms. And then this concept of phase and curl ( convergence and divergence ) in non-orthogonal, probably not conditionally-independent fluid fields that combine complexly and nonlinearly. Define distance between (fluid) field moments. A coherent multibody problem hopefully with unitarity and probably nonlocality. Can emergence of complex adaptive behavior in complex nonlinear systems of fields emerge from such observable phenomena as countability (perhaps just of application-domain-convenient field-combinatorial multiples in space Z)?

pyrolistical 4y ago

Often a nice pattern can be found if one uses a generalized continued fraction

westurner 4y ago

/? 3blue1brown e^ipi https://m.youtube.com/results?sp=mAEA&search_query=3blue1brown+e%5Eipi https://m.youtube.com/results?sp=mAEA&search_query=3blue1bro... Given: e = limit((1 + 1/n)^n, +∞) # Euler's number i = √-1 # orthogonal; i_0^2 = -1 pi = (666/212 - 22/7)*π # circle circumference / diameter Euler's identity: e^iπ + 1 = 0 Euler's formula: e^ix = cos(x) + i*sin(x) Euler's formula: https://en.wikipedia.org/wiki/Euler's_formula https://en.wikipedia.org/wiki/Euler's_formula e (Euler's number) https://en.wikipedia.org/wiki/E_(mathematical_constant) https://en.wikipedia.org/wiki/E_(mathematical_constant)

westurner 4y ago
Is there something fundamental here - with e.g. radix base e - about countability and a continuum of reals, and maybe constructive interference?
- westurner 4y ago
  When EM waves that are in phase combine, the resultant amplitude of the combined waveform is the sum of the amplitudes; constructive interference is addition. And from addition, subtraction & multiplication and exponents and logarithms. And then this concept of phase and curl ( convergence and divergence ) in non-orthogonal, probably not conditionally-independent fluid fields that combine complexly and nonlinearly. Define distance between (fluid) field moments. A coherent multibody problem hopefully with unitarity and probably nonlocality. Can emergence of complex adaptive behavior in complex nonlinear systems of fields emerge from such observable phenomena as countability (perhaps just of application-domain-convenient field-combinatorial multiples in space Z)?