This is a very interesting piece on the mind bending maths required to answer a simple question ‘Why is A4 paper 210mm x 297mm?”. Ben Sparks, a musician and a mathematician from the picturesque English city of Bath, answers the question and in so doing helps us understand the beauty of the A4’s dimensions. He begins gently and then leads you towards a startling insight: “I invite you to take a rectangular piece of paper that is not A4 shaped. You can always just tear a bit off your A4 paper and then neaten it up to a rectangle. With your non-A4 rectangle, try folding it in half along the shortest line of symmetry. You will observe, in a spectacular anticlimax, that you now have a piece of paper half the size, and a different shape. Possibly, you started with a ‘squarey’ rectangle and now you have a ‘long-thin-rectangle’, or vice versa.
Now do it with an A4 sheet. You probably already know what happens. You get an A5 piece of paper. It is half the size (of course it is, you just folded in half). What’s more, it is the same shape. Technically a similar shape, of course, but the sides are in the same ratio. This is something of a shock, if you ponder it, because rectangles do not normally behave like this.”
This unusual behaviour is not an accident; it is by design and it is a design which saves us a lot of money: “This is not an accident. It is possibly one of the greatest innovations of the 18th century. To take just one modern example: teachers have been using it to literally halve their photocopying budget for years. You want two copies on one page? Great – they fit exactly! Any other paper shape (say, ‘letter size’, or 8.5 by 11 inches, for all you North Americans out there) is sadly wasteful in comparison because your two half size copies leave an awkward gap on the original page.”
So who is the maths whiz behind the A4’s dimensions? “The earliest recorded discussion of the idea comes from a 1786 letter from German academic Georg Christoph Lichtenberg to author Johann Beckmann, but there is a suggestion it might have already been a problem used in a maths exam even earlier. However, it was not until the early 20th century that Germany – and then eventually most of the rest of the world – actually standardised the idea. It is now known as ISO 216, an international standard paper size.
In fact, there is only one ratio of rectangle sides that will work, i.e., one that will give a similar shape when cut in half. Consider for yourself the question: Which ratio is it?”
If you are interested in the answer then you should read Ben Sparks’ piece in full. Not only does he explain that the ratio is the square-root of two (which, interestingly, is not a ratio as all of us learnt in high school), he then goes on the show the broader implications of this answer. For example:
“Draughtsperson’s pens typically have widths that increase by factors of √2 i.e. approximately 1.4 so that the next pen in the series can have the correct width if used on a scaled up drawing on the next size of paper. It’s all nice and neat.”
The moral of the story for Finance people like us is that much of what is really useful in the world around is done by people who know maths & science much better than we do. We are just plain lucky that we get to enjoy the fruits of their mindbending labour.

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