## 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: , is as John mentions the value of the continued fraction with just 2’s, it is also the larger solution of the equation . 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:

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:

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!

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It’s worth noting that the ratio of sequential values in the Pell numbers, analogous with the Fibonacci sequence:

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741… (sequence A000129 in OEIS)

approaches the silver ratio. This leads naturally to a nested rectangle interpretation, with these integers as sides. See, for an example and riffs:

http://lcni.uoregon.edu/~mark/Geek_art/Simple_recursive_systems/Tesselations/Pell_spirals/Pell_spirals.html

The Pell numbers are wonderful. If you look carefully they come into the final drawing. They also come directly in the continued fraction expansion in John Cooks original post.

This leads to two questions. Firstly (and easily), what is the natural continuation of the sequence that starts Golden Ratio, Silver Ratio?

Secondly (and I would love to know the answer) why do some of the geometric interpretations (such as the relationship to regular polygons) disappear after the first few terms?

@Gelada: “This leads to two questions. Firstly (and easily), what is the natural continuation of the sequence that starts Golden Ratio, Silver Ratio?”

Golden Ratio: (1*A+B):A = A:B

Silver Ratio: (2*A+B):A = A:B

´Bronze´ Ratio: (3*A+B):A = A:B

etc….

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I learned about the silver ratio in terms of mixing efficiency: it describes the rate of blending extremely viscous fluids (such as taffy or molten glass) using efficiently organized stirring rods. A relatively accessible paper, with lots of pictures, is on the arXiv at http://arxiv.org/abs/1004.0639.

Lovely reference. Strangely I used to share an office with Matt Finn, and did not know about this!