Unscheduled Post: The Silver Ratio

John Cook on The Endeavour has just mentioned the wonderful silver ratio. As this is probably my favourite number I can’t resist the chance to put up some pictures. The silver ratio: $\Psi = 1+\sqrt{2}$, is as John mentions the value of the continued fraction with just 2’s, it is also the larger solution of the equation $\Psi^2 - 2 \Psi - 1 = 0$. This goes directly into its geometric interpretations, as the diameter of an octagon and the size of a rectangle that gives a smaller version of itself when you remove two squares:

Geometric interpretations of the Silver Ratio.

In terms of tiling the golden ratio of course has the Penrose tiling, with its five fold rotational symmetry, the silver ratio plays the same role for the Ammann-Beenker tiling, with 8-fold rotational symmetry:

Version of my Ammann-scaling artwork

In fact if you find things with 8-fold rotation (Islamic art for example) the silver ratio will be lurking around. I have a personal theory that the silver ratio was as much in Christopher Wren‘s work as the golden. I have not studied it in depth, the floor under the great dome of St. Paul’s has an giant octagon. Anyone know any good studies that might mention this, it would be good to have evidence!