In my previous blogs on rectangles I showed that the golden rectangle is not so distinctive. One feature of the golden rectangle is that it can be used to generate a spiral that approximates the golden spiral. The series of rectangles I described can each be used to generate spirals similar to a golden spiral, (or a Fibonacci spiral), but being made from quarter-ellipses, rather than quarter-circles. One of the low order rectangles might be used to approximate a hyperbolic spiral, though the higher the order, the less it is apt to look like a spiral. Each quarter turn becomes so elongated that it looks less like a spiral and more like receding quarter-ellipses.
It might be helpful to diagram some spirals from the series of new rectangles that are similar to the golden rectangle.