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

Just simple Math: (Maybe not for a five years old)

Original sheet: Width: W and Height: H

Folded sheet: New width: w = H and new height: h = W/2

Requested: H/W = h/w

leads to: H/W = (W/2)/H

Multiply both side by (H/W): H/W * H/W = 1/2

Then: H/W = 1/√2