2012-05-24

Fun with the ISO 216 Standard

ISO 216 is a lovely standard. It's the one that defines the A4 paper size, and all its friends. Bengt likes papers, not just for writing on, but also for other things, like making paper airplanes. He claims that the A4 size is particularly good for making paper planes, but that would most likely be a bit of a mess to try to prove, so we'll leave this out of this problem.

One other thing they can be used for is to measure lengths. See, the meter isn't really a very convenient unit of measurement - it's historically based on the circumference of the Earth, which to be honest isn't something most people have an intuitive grasp on. The foot is better in that sense, but since people's feet aren't all the same length, that's also not quite ideal. In fact, there may not be any obvious things in nature to base a length unit on, because most things in everyday life vary in size. But a standardised paper size - well, it may not be "in nature", but papers sure are pretty much everywhere these days, and they're a convenient size too. And we could potentially set the definition of the paper size so that it fits with some scientific constant - say, a nanolightsecond; that way, the definition would be convenient for both scientists and normal people.

But there is a slight problem with the A4 as it stands: It's based on area, not length. A0 has the area 1 m2, and each subsequent one has half that area. It would probably be better to define them by saying that, for example, the short side of the first one would be one meter. It's a little known fact that the same ISO standard actually defines that as well - it's called the B series. So, for example, the B4 is a quarter of a meter wide.

Still, there's another slight problem: Every other size will have half the length, but the metric system is based on powers of ten, not two. So once you get down to B10, the width is 31.25 mm - or rather, it's 31 mm, since the standard requires that they be rounded off to the nearest millimeter, which is clearly an abomination.

If you ask Bengt, this is further proof that the decimal system is inherently flawed, and we should use base sixteen instead - in that case, the area of an A4 would be exactly one d(m2), which is admittedly a confusing unit, as it's easily confused with the much smaller (dm)2, but anyway. And the B8 would be one dm wide, so we could for example give all children a little notepad in that size, and use that as our unit of measurement, so we would all grow up with a remarkably accurate intuition for length units, and there would be no need to fight over it, which would probably lead to world peace.

Now, just to make fun of the decimal system and all that, let's try to construct a paper size which does the same thing but with each subsequent area being a tenth of the previous area, rather than half. It should have such proportions that if you fold the short side in half, and the long side in fifths, you get the same proportions again. It would be awfully tricky to fold anything like that, but that's sort of the idea.

  1. What is the width of an A0?
  2. On a side note - how long is a nanolightsecond, in, say, feet?
  3. What proportions would the "decimal" system have? Suppose the short side is 1 m, how long would the long one be?
  4. Is there a simple similar system where both the areas and lengths have integer ratios?
  5. How far does the A series go, before they get rounded off to nothing?