Bianchi IX Cosmologies and the Golden Ratio

Special solutions to the Einstein equations in the asymptotic limit for the Bianchi IX cosmologies in the vacuum are examined using Ellis-MacCallumWainwright ('expansion-normalized') variables. Using an iterative map (the B-map) obeyed by two of the dynamical variables (the normalized shear components) in the 'asymptotic regime' close to the cosmological singularity, two period 3 solutions are constructed. These are the simplest of an infinite number of periodic solutions and represent the transition from one vacuum Bianchi I Kasner solution to another. It is shown that the full 3-cycle solutions for the remaining variables (the logarithms of the normalized curvatures) generate a set of self-similar golden rectangles in a graphical time series representation of their dynamics as the normalized time parameter is run backwards towards the initial singularity.