  The relationship between period doubling bifurcations and Feigenbaum's
constants has been studied for nearly 40 years and this relationship has helped
uncover many fundamental aspects of universal scaling across multiple nonlinear
dynamical systems. This paper will combine information entropy with symbolic
dynamics to demonstrate how period doubling can be defined using these tools
alone. In addition, the technique allows us to uncover some unexpected, simple
estimates for Feigenbaum's constants which relate them to log 2 and the golden
ratio, phi, as well as to each other.
