Why is 1:√2 the only aspect ratio that remains the same when it is ‘folded in half’ (hence the usage in paper dimensions)

Question

1:2 becomes 1:1 or 1:4 depending on folding axis, etc

In: 10

Answer 1

Take a piece of paper with an aspect ratio of 1:√2. If you fold it in half along the long axis, then its aspect ratio becomes 1:(√2)/2. Now, to see that this is the same aspect ratio, scale both sides up by √2. Then the aspect ratio would be √2:(√2*√2)/2. √2*√2 = 2, that’s what √2 means, so the aspect ratio is √2:2/2 = √2:1. This is the same aspect ratio as before, just the other way round.