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The A4 rectangle

I think that most paperfolders by now know the significance of a rectangle which has its sides in the ratio of one X square root of two. (approximately 1.414). It has many fascinating features and paperfolders are increasingly exploring its attributes.

It is, of course the shape of the International-sized papers (A3, A4, A5, B3, B4 and the like), which are designated by the International Standards Organisation for all sorts of purposes. A4 has become the standard-sized business letter paper in all countries except in North America, where the stick to "letter-sized" paper (8X5 - not so very far off A4).

Reading David Mitchell's splendid new book, "Exploring Mathematical Ideas with Origami" (Which I wholeheartedly recommend to every one interested in the extraordinarily rich geometry of simple shapes of paper.) has reminded of the august origin of the name "Silver Rectangle" for a rectangle of the proportions 1 X sq rt 2.

Chasing through John Cunliffe's BOS booklet "The Silver Rectangle" (Booklet no. 21, 1983) led me to issue 75 of "British Origami", the magazine of the BOS, for April, 1979. I think that what I found is worthy of being dusted down and reprinted. It comes in the form of a short note by John Cunliffe, better known these days as the International President of the Envelope and Letter Fold Association (ELFA). Here is what he wrote:



"An item in the Sunday Telegraph about the need for a suitable word to describe the study of flags also mentioned that the British Origami Society requires a word for the 1 X sq rt 2 rectangle. Among several letters received was one from Miss Muriel Smith of Maidenhead who suggested root-o'-two or rutatu to make it look more Japanese. Mr. Flack of Norfolk thought Dina would be appropriate since it was the Germans who had introduced it to their D.I.N. paper sizes, and Mr. Hinchliffe, head of maths at Windsor School, thought tortangle or tortagon applicable, this being derived from the Greek letter T (tau) and used in a similar ratio. Dr. Juby of Birmingham felt that as a quarter was involved it should somehow be incorporated i.e. quartangle or quartogram, while Mr. Appleyard of Oxshot, though not putting up a name, interestingly illustrated the result of the rectangle's long side being the square root of a number other than two.

"Among British Origami Society members who put forward names were John Smith, rootangle, and Roberto Morassi, true or uni rectangle. But the writer's own preference is for that of the Oxford Dictionaries organisation. On an analogy with the Golden Rectangle (1 X 1.618), they suggest that a 1 X 1.414 rectangle might be called a Silver Rectangle, and it is proposed to use this for the time being."


The name (Silver Rectangle) was the one that stuck and John Cunliffe used it as the title of his booklet eight years later in 1983. Since then, it has become the generally accepted name in the British Origami Society and, indeed, in the wider Origami world and also, it would appear, in the further world outside.

On a point of correction, I understand that the Germans were not the first to suggest the use of the Silver Ratio for paper sizes; it had previously been previously suggested by the French during the Revolution, but like their new calendar system, it was one of their ideas which didn't succeed at the time. Incidentally, DIN stands for "Deutsch Industrie Norm" which is the name of the German standards organisation.

Being a "flag wallah" I am also interested that the Sunday Telegraph had also investigated a name for the study of Flags, although no suggestions are put forward. The name that eventually stuck was "vexillology" would you believe it? That is if you can get your tongue round it!

I was not aware that Mr. Appleyard of Oxshot had suggested rectangles with the long side being the square root of a number other than two. A few years ago, when I was investigating the geometrical division of rectangles which involved equilateral triangles, I discovered that paper of the size 1 X sq rt 3 was particularly significant. I proposed that name this ratio should be the Bronze Rectangle. Alas, I now find from a recent posting to Origami-L that I was not the only one to put forward this very appropriate name and for exactly the same purpose, namely the 1 X sq rt 3 rectangle. And, in fact, David Mitchell in his recent book "Exploring tMathematical Ideas with Origami" discusses not only the bronzE Rectangle, but also the bronze Triangle under those names. Let's hope that the name "Bronze Rectangle" also sticks.

Curiously, in Issue 75 of British Origami and purely by coincidence, there is also my own article with the title "Golden and Divine". It is mainly on the Golden Section (also known as the Divine proportion) and the Golden Rectangle, but it also mentions the International Paper Sizes and hints at rectangles of the proportions: ! X sq rt 4 (which is, of course, two - a bit dull, really, but not as dull as all that) and 1 X sq rt 5 and how all these rectangles can be linked in a simple diagram (which, however, was not my own.)

I always have been fascinated by geometry and even the simplest geometry is extraordinarily rich. Don't forget to go out and buy David Mitchell's latest book. You'll enjoy it.

In reply to this posting about the Silver Rectangle, The Origamimaster asks:

"Well then, how many different coloured titled rectangles are there?"

I suspect that The Origamimaster is pulling my leg, but I'll take his question seriously.

At the moment, I know of only three special rectangles on which the accolade of a metallic name has been bestowed.

The Golden Rectangle is a very old designation, dating at least from Ancient Greece. The Golden Rectangle has its sides in the proportion of the Golden Ratio which is irrational, but approximates to 1.6180 and which apparently occurs in nature. The Golden Ratio is closely related to the Fibonacci Series of numbers, which certainly occurs in nature.

The Silver Rectangle is an analogous play on the Golden Rectangle and dates only from the 1970s, although the particular rectangle it refers to was known centuries earlier.

The Bronze Rectangle was proposed very recently (possibly not more than five years ago). Again the particular rectangle has been known for many years (I don't know how many), but it has only recently been found to have any application in paperfolding.

Obviously, this trio of rectangles echoes the now ubiquitous Gold, Silver and Bronze medals of sporting contests.

I don't believe there are any more metallic colours to bestow. Platinum is beloved by credit card companies, but in colour is hardy distinguishable from silver. But there is no reason why rectangles of other colours should not be adopted if anyone feels a passionate need for them. There are enough special rectangles to support a whole rainbow, if not a kaleidoscope of colours. But do they mean much in origami? Considerable work has been done in origami with the Silver Rectangle, but so far, very little with the Bronze Rectangle. It is surprising that, ancient and ubiquitous though the Golden Rectangle and its Ratio may be, it has so far (but not entirely) been ignored by paperfolders. Would The Origamimaster like to do some research into it?

David Lister
Grimsby, England.

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