Bianchi IX cosmologies and the golden ratio

Abstract

Solutions to the Einstein equations for Bianchi IX cosmologies are examined through the use of Ellis-MacCallum-Wainwright (expansion-normalized) variables. Using an iterative map derived from the Einstein equations one can construct an infinite number of periodic solutions. The simplest periodic solutions consist of 3-cycles. It is shown that for 3-cycles the time series of the logarithms of the expansion-normalized spatial curvature components vs normalized time (which is runs backwards towards the initial singularity), generates a set of self-similar golden rectangles. In addition the golden ratio appears in other aspects of the same time series representation.

Golden Elliptical Orbits in Newtonian Gravitation

“…Lately, the golden ratio has also appeared in the inanimate world and our work concerning Newtonian dynamics falls in this category. Two more examples concern black holes [5] and cosmological theories [2], although the interested reader may easily track down more cases in solar neutrino mixing and quantum mechanics.…”