The Lister List

Golden Rectangles


I agree, it would be nice if some would give a simple dimension as for instance like 3 by 7 for an example for the silver and golden rectangle please. And why did someone come up with these sizes ratios, too?

I am reluctant to intervene: I only recently returned from holiday in Tenerife, and since then I have attended the British Origami Society spring convention. I am frantically trying to put may affairs in order before flying to Japan at he invitation of Mr. Yoshizawa on Wednesday.

At the risk of oversimplifying it Thoki Yenn has given the simplest answer to this question and I commend his postings to everyone. There may be one or two postings on the subject on the British Origami Society Web site, but for the moment I am unable to check because I have lost my link to the Web. I know I have written about the subject in the past, both in "British Origami" and elsewhere

The Golden Ratio and the Silver Ratio are not the invention of mere mortal men. They arise from the operation of mathematics. The Golden Ratio, or Golden Section (a rectangle with its sides in the proportions of the golden ratio is a golden rectangle) has exercised men's (and women's) minds since Classical times and countless books have been written on the subject. (Dover Books have several of them available at small cost). The Golden ratio has the formula: a:b = b:(a=b).

The Silver Ratio is: one to the square root of 2. It has not been so studied or discussed as the Golden Ratio, but came into prominence when this proportion was chosen for International paper sizes. I understand that the name "Silver Ratio" is a modern one and ws suggested by the compilers of the Oxford English Dictionary on the analogy of the Golden Ratio.

Both the Golden Ratio and the Silver Ratio are irrational numbers so it is hopeless to ask for them to be expressed in simple whole numbers. For practical purposes, the best you can hope to do is use and approximation, as Thoki Yenn suggests.

Just as square paper has geometrical relationships which can be exploited in paperfolding, so the Silver Rectangle has inherent geometries which can be used. For many years I have been watching the slowly-evolving techniques of folding from the Silver Rectangle, and some progress is being made. But nobody has yet written a monograph on the subject. Here is scope for someone wanting to make his or her mark on origami.

The Golden Ratio seems less promising for folding purposes than the Silver Ratio. It is more implicated and involves the square root of five. Certainly there have been less results than for the Silver Rectangle. Occasionally I have voiced my hopes that someone would work out its paperfolding geometry, but nobody has yet come forward with anything. If you deduct a square from a Golden Rectangle, the bit left over is itself a golden rectangle. One would think that something could be worked out from this.

I once identified a third kind of ratio and rectangle, which I designated the Bronze Rectangle. Unfortunately it has receded into the far back of my mind and because of the current pressures, I haven't sufficient time to dig it out. But I can do so later if anyone is really interested. It involves the square root of three.

One of the interests of origami is that it exploits the geometry of the piece of paper, whatever its shape. Square paper is not the only shape which can or should be used. Ultimately choice is a matter for personal decision. No human being lays down law for these things, but they have to be exploited by human ingenuity. I find it fascinating, because it is a hint that there is far more to be discovered about folding paper than we know already. I have previously expressed my hopes for folding from circular paper on Origami-L, but this remains an almost wholly unexplored field.

This is an important aspect of folding which deserves a better contribution than this and I apologise for its inadequacies. I thought, however that half a loaf would be better than none. Perhaps I shall be able to go into it more when I return from Japan.

Meanwhile, I'm enjoying all the contributions on the subject. They are all adding to our knowledge of it.


David Lister.

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