Aside: If I was king of Sun III (previously known as Earth), my replacement for "US Legal" sized paper would be 210mm × 340mm. This maintains the width of A4, but gives it a length similar to that of US Legal. Moreover, it achieves this using the Golden Ratio. Calling it φA4 for the time being.
US Legal..: 8.5” × 14” (Before mm conversion.)
US Legal..: 216mm × 356mm —> approx. 1.647:1 ratio
A4........: 210mm × 297mm —> approx. 1.414:1 ratio
φA4.......: 210mm × 340mm —> approx. 1.619:1 ratio
See http://i.imgur.com/S0rrRnP.png for an example.Aside: Apparently the 210mm × 340mm size is used for some folded paper towels and some pastry bags already. Also, Mexico's "Government-Legal" is 216 mm × 340 mm — same height as my proposal but with a 6mm width difference. (Theirs matches US Letter/Legal width of 216mm instead of the A4 width of 210mm.)