It has been remarked upon, that I am not very observant. To combat this, I’ve taken to saying “your hair looks lovely” at regular intervals in an attempt to avoid the molten wrath of girls who’ve recently spent a staggering amount of time and money on a haircut when I have absolutely no inkling that their hair was ever, at any point in the many years I’ve known them, any different to how it currently looks.

But, if I ever do say that to you, it’s because your hair really does look beautiful at the moment I say it.

It should come as little surprise, therefore, that it took me 8 months to notice that paper over in the USA is wrong. I’d like to be generous to my new home and say different or alternately sized but I can’t bring myself to do it. Paper in America is simply wrong. Allow me to elaborate.

A sheet of standard American letter paper has the short, squat dimensions 215.9mm × 279.4mm. An alternative to this travesty is available: they also believe in such as thing as legal paper at an inordinately lanky 216mm x 356mm. Now that my eyes have been opened to this fact I’m unable to even look at a sheet of paper here without feeling slightly nauseated; it’s a real boon for the environment.

I’ve created the below image of pink letter paper sitting atop blue A4 paper to show the problem:

But surely, you say, surely this is a mere difference, not a problem or a mistake. Read on.

Firstly, there appears to be no rhyme nor reason to the dimensions set by the Americans. The American Paper Association – who appear to be accepted as the authority in the history of these choices – in their engaging guide to paper sizes claim that the length of letter paper, 279.4mm, is a quarter of the length of a 17^th century Dutch vatman’s arms. No corollary claims are made regarding the width and/or aspect ratio of letter paper and no claims at all regarding the origins of legal paper. The best full answer to be found is that some government committees debated the matter and Ronald Reagan unilaterally settled their disputes.

ISO 216 (publication 216 by the International Organisation for Standards, an organisation that set a bewildering standard for the abbreviation of its own name) specifies the standards used by the rest of the world, namely the A- and B- standards. A sheet of A0 paper is of area 1m² and has an aspect ratio of √2.

From this aspect ratio, it follows that any sheet of A0, cut along an edge midway along its long side and parallel to its short side, will produce two sheets with the same aspect ratio and half the area of the original sheet. In this case, that paper size would then be known as A1. Cutting a sheet of A1 in the same manner would produce a sheet of A2 and so on until we reach the ubiquitous, aesthetically pleasing and sublime A4 sheet of paper — the cornerstone of modern society even in this digital age.

Considering, as you no doubt now are, the ISO 216 set of paper sizes (A0: 841mm × 1189mm; A1: 594mm × 841mm and so on) and mentally calculating the areas of each sheet of paper size, you’ve no doubt already reached the conclusion that something is not quite right here: there’s been some rounding going on. I’ve taken the time to calculate the actual aspect ratios of common sheet sizes and their variance from √2.

I don’t have a spare 66 Swiss Francs to get an original copy of ISO 216, so I can only hope that the luminaries behind the document knew full-well the crazy and dangerous implications of the varying aspect ratios they were setting. May God have mercy on their souls.

The ideal √2, though aesthetically pleasing, endures the unfortunate circumstance of being an irrational number: an unending decimal, if you prefer. It will always be suborned in the popular imagination to its prettier and better dressed irrational counterpart, π, but it’s been my personal favourite since the Babylonians first approximated it on a rock in ~1850BC. Let me tell you just one brief reason why.

If one were to remove the largest possible square from a rectangle with ratio 1:√2 (such as a sheet of ISO 216 paper) they would be left with a rectangle of proportions 1:√2-1, which, naturally, you will recognise as the silver ratio δS. A silver rectangle, as such a shape is known, corresponds exactly to the inner dimensions of the of a split regular octagon. If such an octagon is split into two isosceles trapezoids and a rectangle, then that rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS.

With such mathematical beauty at the heart of the ISO 216 paper sizes, I think you’ll agree that the Americans, much though I love them, are in this case simply wrong. And so, I long for the day that they learn to make steak & ale pies, and adopt the international standard for paper sizes — on that day, I can truly be happy here.

For further information, I recommend A.A. Dunn’s thorough 55 page guide (printed on ISO A5 paper,) Notes on the Standardisation of Paper Sizes.

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