Mattress manipulation mathematics

An American Scientist article addresses a question I've long wondered about: Whether there is a single mattress-flipping operation that, applied repeatedly, would step the mattress through all of its valid orientations? The answer, sadly, is no; however, they do provide a solution (the one my wife and I have used for years): Label the mattress with numbers, and each time rotate it so as to increment the number at the head of the bed. Along the way, the article provides a basic introduction to the branch of mathematics known as group theory. [via LifeHacker.]