[picturesal]

Not "plane"=sheet, but "plane" from the Greek "planos"=wandering. (Same root as planet=wandering star).

An aeroplane is a wanderer through the air.

[tullianus]

Etymonline disagrees; -plane here is from French planer "to soar," ultimately from Latin planus, "flat." http://www.etymonline.com/index.php?term=aeroplane