#Algebra - the folding of A0 paper and the square root of 2...

Question

I was listening to a podcast (it's podcast 10, and here's the transcript on his blog) by the artist Danny Gregory (ok - more like I overheard my wife listening to it and my ears perked up when Danny started talking about the mathematical development of the A, B, and C paper sizes). He said that the paper sizes (A, B, and C) as they become smaller, a ratio of square root of 2 : 1. With the paper getting folded in half (the longer side) each time, how does that ratio hold? And are there any other ratios that would work just as well as square root of 2 : 1?

This question is related to this one which looks at finding the measures of the sides of the paper. 

Answer

See below on how it is that the ratio holds. There are no other ratios that will hold.

Analysis

Ok - so how does the ratio manage to stay the same?

Let's first consider a situation where the ratio does not stay the same. Let's set up a piece of paper where the ratio is 2:1 and we fold the long side over (the 2 side) and now we have a piece of paper that has a ratio of 1:1 - not the same at all. So how does the ratio keep maintained?

We start with a ratio of:



We fold the long side in half (that's the square root side) and now we have:



We started out with the "short side" being the "1" side, so let's scale up the numbers so that the short side is 1 again and we'll see where the long side is:





We're almost done. Let's rationalize the denominator of the left hand fraction:





And taa daa! It's the same ratio.

Is there any other ratio that will hold like this? To find out, we can set up a ratio:



We know that the "short pre-fold" will be 1 as will be the long post-fold (they're the same side). The long pre-fold will be some value, x, and the short post-fold will be one half of x:



We can cross-multiply:







And so the square root of 2 : 1 ratio is the only one that will work.

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As always, questions and comments always welcome!