  Anthropic arguments of Carter, Carr, and Rees give two approximate power-law
relations between the elementary charge e, the mass of the proton m_p, and the
mass of the electron m_e in Planck units: m_p ~ e^{18}, m_e ~ e^{21}. A
renormalization group argument of mine gives e^{-2} ~ -(10/pi)ln{m_p}.
Combining this with the Carter-Carr-Rees relations gives e^2 ln{e} ~ -pi/180.
Taking the exact solutions of these approximate equations gives values for e,
m_p, and m_e whose logarithms have relative errors only 0.7%, 1.3%, and 1.0%
respectively, without using as input the observed values of any parameters with
potentially continuous ranges. One can then get anthropic estimates for the
masses, sizes, luminosities, times, velocities, etc. for many other structures
in physics and astronomy, from atoms to giraffes to the universe, as simple
powers of the anthropic estimate for the elementary charge. For example, one
gets an anthropic estimate for the real part of the logarithm of the dark
energy density with a relative error of only 0.2%.
