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Really interesting problem in recreational mathematics - try to reverse a unit line segment via translations/rotations - but in doing so sweep out a minimal area.
PI/4 is the obvious minimum - but it turns out you can do so in an arbitrarily small area.
What's odd to me, is that any combinations of moves that are either a) rotations about the center and b) translations, CAN NOT achieve an area less than PI/4; so it's counter-intuitive that you can do better than that.
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They actually can sweep out less than PI/4 if they sweep over the same area twice and therein lies the key. Details at http://www.darrellplank.com/math/Besicovitch/Besicovitch.aspx.
Cool. The pictures really help - thanks for creating this page.
Fascinating problem and solution! Very counter-intuitive.